GLOBAL LORENTZIAN GEOMETRYpersonal.maths.surrey.ac.uk/st/jg0032/teaching/GLG1/... · 2013-07-10 · GLOBAL LORENTZIAN GEOMETRY JAMES D.E. GRANT Contents Part 1. Preliminary material

GLOBAL LORENTZIAN GEOMETRY

JAMES D.E. GRANT

Contents

Part 1. Preliminary material and notational conventions 2Vector bundles 2Connections 2Semi-Riemannian manifolds 3Maps between manifolds 4

Part 2. Riemannian geometry 51. Examples 52. Metric space structure on Riemannian manifold 63. First variation of arc-length 74. The exponential map and normal coordinates 95. Hopf-Rinow theorem 116. Bishop-Gromov comparison theorem 117. Second variation of arc-length 188. Cut locus 24

Part 3. Lorentzian geometry 289. Inner products of Lorentzian signature 2810. Lorentzian manifolds 2810.1. Examples 2911. Geodesics 3312. Lengths of causal curves 3513. Variation of arc-length 3713.1. First variation of arc-length 3713.2. Second variation of arc-length 3714. Causality 4214.1. Causality relations 42Appendix A. Proof of Proposition 14.8 (Non-examinable) 45References 49Index 50

Date: 31 January, 2008 (Corrected: 12 February, 2009).

2 JAMES D.E. GRANT

Part 1. Preliminary material and notational conventionsLecture 1

You should be familiar with everything in this section from previous Differential Geometrycourses. Just in case, however, I will (eventually) include an Appendix that reviews any previ-ous material that I have used. If you still have problems, standard textbooks on this subjectinclude [GHL, O’N, Cha, W].

Throughout, let M be a finite-dimensional manifold of dimension n ≥ 2. We will assume thatM is Hausdorff, connected and, except where explicitly stated otherwise, that M is smooth (i.e.C∞).

Vector bundles. Let π : V →M be a vector bundle. (Again, except where stated otherwise, wewill assume that vector bundles are smooth.) The space of sections of V will be denoted by Γ(V ),or Γ(M,V ) if we need to emphasise the manifold in question, or Cl(M,V ) (l ≤ ∞) if we wish toemphasise the amount of differentiability required. Particular cases, with special notation, are:

• TM , the tangent bundle of M . We denote the space of smooth sections of TM by X(M).An element of X(M) therefore corresponds to a vector field on M . Recall that we viewvector fields as differential operators acting on smooth functions. So, for each X ∈ X(M)and f ∈ C∞(M), we have Xf ∈ C∞(M). Given X,Y ∈ X(M), we define the Lie bracket[X,Y] ∈ X(M) by

[X,Y] f = X (Y (f))−X (Y (f)) ,

for all f ∈ C∞(M).• ∧kT ∗M is the bundle of k-forms on M . Its space of sections, the space of (differential)k-forms, is denoted Ωk(M). Given a vector bundle V → M , we will denote by Ωk(M,V )the space of k-forms with values in V (i.e. the space of sections of the bundle ∧kT ∗M⊗V ).

• T rs (M) will denote the bundle of (r, s) tensors on M (i.e. (⊗r1TM)⊗ (⊗s1T ∗M)). The setof sections of this bundle (i.e. (r, s) tensor fields on M) will be denoted by T rs (M).

Notation: Given a vector field X ∈ X(M) and a 1-form σ ∈ Ω1(M), we will denote the action ofσ on X by 〈σ,X〉 or σ(X).

Connections. A connection on a vector bundle, V , is equivalent to a covariant derivative. Thisis a map ∇ : X(M)× Γ(V ) → Γ(V ); (X, s) 7→ ∇Xs with the properties that

∇fX+gYs = f∇Xs+ g∇Ys, ∀f, g ∈ C∞(M), X,Y ∈ X(M), s ∈ Γ(V ),

∇X(as+ bt) = a∇Xs+ b∇Xt, ∀a, b ∈ R, X ∈ X(M), s, t ∈ Γ(V ),

∇X(fs) = X(f)s+ f∇Xs, ∀f ∈ C∞(M), X ∈ X(M), s ∈ Γ(V ).

Given a connection on a vector bundle, the curvature is the 2-form with values in the bundle ofendomorphisms of V , EndV , defined by(

∇X∇Y −∇Y∇X −∇[X,Y]

)s = F(X,Y)s, ∀X,Y ∈ X(M), s ∈ Γ(V ).

Hence, given vector fields X,Y ∈ X(M) and a section, s, of V , the object F(X,Y)s is also asection of V .

An affine connection is a connection on the tangent bundle TM . In this case, we define thetorsion tensor T ∈ T 1

2 (M), by

∇XY −∇YX− [X,Y] = T(X,Y) ∈ X(M), ∀X,Y ∈ X(M).

In the case of an affine connection, we have a special notation for the curvature tensor R ∈ T 13 (M):(

∇X∇Y −∇Y∇X −∇[X,Y]

)Z = R(X,Y)Z, ∀X,Y,Z ∈ X(M).

As usual, we may extend an affine connection to define a connection on each tensor bundleT rs (M).

GLOBAL LORENTZIAN GEOMETRY 3

Semi-Riemannian manifolds. A case that we will be particularly interested in is when M hasa Riemannian or semi-Riemannian metric. This is a section, g, of T 0

2 (M) that is• symmetric: g(X,Y) = g(Y,X), ∀X,Y ∈ X(M);• non-degenerate: g(X,Y) = 0 for all Y ∈ X(M) if and only if X = 0;• defines an inner product of signature (p, q) (p plusses, q minuses) on each tangent spaceTxM , for each x ∈ M . (For example, the canonical example of an inner product ofsignature (p, q) would be on Rn, with n = p + q, where, given v = (v1, . . . , vn), w =(w1, . . . , wn) ∈ Rn we define

〈v,w〉 := v1w1 + · · ·+ vpwp − vp+1wp+1 − · · · − vp+1wp+1.)

The cases we will be most interested in are:a). p = n, q = 0, in which case g is a Riemannian metric;b). p = n− 1, q = 1, in which case g is a Lorentzian metric.

By a semi-Riemannian manifold, we will mean a pair (M,g) consisting of a manifold Mand a semi-Riemannian metric, g, on M1.

Remark 0.1. Given a point x ∈M we may pick an orthonormal basis for TM on a neighbourhood,U , of x i.e. a basis ei(y)ni=1 for TyM , for each y ∈ U , such that

gy(ei(y), ej(y)) =

εi i = j,

0 i 6= j,∀y ∈ U,

where ε1 = · · · = εp = +1, εp+1 = . . . εp+q = −1 for a semi-Riemannian metric of signature (p, q).

Definition 0.2. An affine connection, ∇, on (M,g) is metric if ∇g = 0. It is torsion-free ifT = 0.

The following result is fundamental:

Theorem. Let (M,g) be a semi-Riemannian manifold. Then there exists a unique affine connec-tion, ∇, that is torsion-free and metric.

Sketch of Proof. From the explicit form of the torsion-free and metric conditions, we deduce theKoszul formula

2g(∇XY,Z) = X (g(Y,Z)) + Y (g(X,Z))− Z (g(X,Y))

+ g([X,Y] ,Z)− g([X,Z] ,Y)− g([Y,Z] ,X),

for all X,Y,Z ∈ X(M). We then show that this uniquely determines the connection, and checkthat the connection defined by this formula is metric and torsion-free.

The torsion-free metric connection given by this theorem is called the Levi-Civita connectionfor the metric g. For the rest of this course, unless explicitly stated otherwise, the only affineconnection that we will use on a semi-Riemannian manifold will be the Levi-Civita connection.

In the case of a semi-Riemannian manifold, it is also useful to define a (4, 0) version of thecurvature tensor, also denoted R ∈ T 0

4 (M), by

R(W,X,Y,Z) = g(R(W,X)Z,Y).

(Note the change in the order of vector fields.)We also define the Ricci curvature tensor (or simply Ricci tensor). This is a symmetric (0, 2)

tensor field on M defined as the trace Ric(X,Y) := tr [Z → R(Z,X)Y]. In this case, for X,Y ∈X(U), the Ricci tensor is defined on U by

Ric(X,Y) =n∑i=1

εi g(ei,R(ei,X)Y).

1For brevity, we will often abbreviate the term “semi-Riemannian metric” to just “metric”.

4 JAMES D.E. GRANT

Finally, we define the scalar curvature (or Ricci scalar), s, as the trace of Ric:

s =n∑i=1

εiRic(ei, ei).

Exercise 0.3. Prove that the Ricci tensor as defined in the last equation is symmetric. (You willneed to use the fact that the Levi-Civita connection is torsion-free.)

Maps between manifolds. Let M , N be smooth manifolds, and ϕ : M → N be a smoothmap. The differential of this map is the tangent map Dpϕ : TpM → Tϕ(p)N , and we define thecorresponding bundle map Dϕ : TM → TN by Dϕ(vp) := Dpϕ(vp), for vp ∈ TpM , p ∈M .

Given a (0, s) tensor field on N , S ∈ T 0s (N), the formula

(ϕ∗S)|p (v1, . . . ,vs) = Sϕ(p) (Dpϕ(v1), . . . , Dpϕ(vs)) , ∀v1, . . . ,vs ∈ TpM, p ∈M

defines a (0, s) tensor field on M , the pull-back of S by ϕ, which we denote by ϕ∗S ∈ T 0s (M).


Part 2. Riemannian geometry

We begin by studying some global properties of Riemannian manifolds2. Therefore, for theremainder of this part of the course, we will assume that (M,g) is a Riemannian manifold, sog ∈ T 0

2 (M) defines an inner product on TxM for each x ∈M .

1. Examples

Example 1.1. Let M = Rn, and take a chart U = M , ϕ = Id, and local Cartesian coordinates(x1, . . . , xn). The flat metric on Rn is given by

g =n∑i=1

(dxi)2,

where we have introduced the notation(dxi)2 := dxi⊗ dxi. In this coordinate system, gij = 1 for

i = j and 0 otherwise. The Christoffel symbols and the components of the Riemann tensor arezero, so the metric is flat. Lecture 2

In the case n = 2, we introduce polar coordinates (r, θ) ∈ [0,∞) × [0, 2π) by x1 = r cos θ,x2 = r sin θ. In terms of these coordinates, the metric takes the form

g = dr2 + r2dθ2.

An example of an orthonormal basis would then be

e1 =∂

∂r, e2 =

1r

∂

∂θ.

More generally, the n-dimensional flat metric may be written in the form

g = dr2 + r2gSn−1 .

where gSn−1 is the induced metric on the unit (n− 1) sphere in Rn.

Example 1.2. Let M = S2 with the metric induced by viewing M as a sphere of radius a in flatR3. In spherical polar coordinates (θ, φ) ∈ [0, π]× [0, 2π), the metric takes the form

gS2(a) = a2(dθ2 + sin2 θdφ2

).

An example of an orthonormal basis would be

e1 =1a

∂

∂θ, e2 =

1a sin θ

∂

∂φ.

Letting K := 1/a2, we may define a new coordinate r := θ/√K with 0 ≤ r ≤ π/

√K, in terms of

which the metric is

gS2(a) = dr2 +(

1√K

sin(√

Kr))2

dφ2.

More generally, if M = Sn, with the metric induced by viewing M as a sphere of radius a in flatRn+1, then

gSn(a) = dr2 +(

1√K

sin(√

Kr))2

gSn−1(1)

where, again, gSn−1 denotes the induced metric on the unit (n− 1) sphere in Rn.

Example 1.3. Let K ∈ R. We define the functions

snK(r) :=

1√K

sin(√

Kr)

K > 0,

r K = 0,1√|K|

sinh(√

|K|r)

K < 0(1.1)

2The majority of the material in this part of the course can be found in the books on Riemannian geometry byGallot et al [GHL] and Chavel [Cha]. Our exposition and notation largely follows that in the first chapter of Cheegerand Ebin [CE], and our treatment of the comparison theorems is partly based on some sections of Petersen [P].

6 JAMES D.E. GRANT

where r ∈ [0, π/√K] for K > 0, and r ∈ [0,∞) for K ≤ 0. We then define the metric3

gK := dr2 + snK(r)2gSn−1 . (1.2)

These metrics have the property that they are of constant curvature. In particular, the (0, 4) formof the curvature tensor takes the form

R(W,X,Y,Z) = K (gK(W,Y)gK(X,Z)− gK(W,Z)gK(X,Y)) . (1.3)

For later use, we note that (1.3) implies that the Ricci tensor of the metric gK takes the form

RicgK= K (n− 1)gK .

Although we will not use the concept of sectional curvature extensively, we introduce it here forthe sake of completeness. Given a point p ∈M , and a two-dimensional plane σ ⊆ TpM , we definethe sectional curvature of σ at p to be

Kp(σ) :=Rp(X,Y,X,Y)

gp(X,X)gp(Y,Y)− gp(X,Y)2,

where X,Y ∈ TpM are any two vectors at p that span the plane σ. (It is straightforward to showthat Kp(σ) depends only on the plane σ ⊆ TpM , and is independent of the vectors X,Y chosento span σ in the above expression.) Although, in general, the sectional curvature will depend onthe point p and the plane σ ⊆ TpM , in the case of our metrics gK , it follows from (1.3) thatKp(σ) = K for all p ∈M and all two-planes σ ⊆ TpM .

A consequence of the Gauss Lemma (see later) is that, given any Riemannian metric, we canfind local coordinates where the metric takes the form

g = dr2 + gn−1,

with gn−1 is a metric on the (n−1) sphere, depending on (r, θ), where θ are coordinates on Sn−1.One of the topics that we will investigate is that if we can put bounds on the curvature of such ametric on a manifold, then we can compare geometrical properties of the corresponding manifold(e.g. diameter, volume of metric balls) with those of the model metrics defined in equations (1.1)and (1.2).

2. Metric space structure on Riemannian manifold

Given a vector field X ∈ X(M), we define the function

|X| :=√

g(X,X) ∈ C(M),

which is smooth if X 6= 0.A curve in M is a smooth map c : [a, b] →M . The tangent vector to c is the vector field along

c (see Remark 2.2) defined by

c′(t) := Dtc

(∂

∂t

)∈ Tc(t)M, t ∈ [a, b].

A curve is regular if |c′(t)| 6= 0 (equivalently, c′(t) 6= 0), for all t ∈ [a, b]. The length (or arc-length)of the curve c is given by

L[c] :=∫ b

a

|c′(t)|dt =∫ b

a

√g(c′(t), c′(t))dt.

Remark 2.1. We have restricted ourselves to the case of smooth curves. It is straightforward togeneralise our discussion to the case of, for example, piecewise smooth curves, at the expense ofintroducing extra terms into the first and second variation formulae, which we will derive later.

Remark 2.2. Note that c′ is not a vector field on M , so objects such as, for example, ∇c′c′ are

not really well-defined on M . For this reason, we introduce the following concept.

3If you find the gSn−1 a little intimidating, just think of the case with n = 2, where gS1 = dθ2 with θ ∈ [0, 2π).


Definition 2.3. Let M , N be smooth manifolds and ϕ : N → M be a smooth map. A vectorfield along ϕ is a map X : N → TM , such that X(p) ∈ Tϕ(p)M , for all p ∈ N . The space of vectorfields along ϕ will be denoted Γ(ϕ, TM).

In the case where M has an affine connection, ∇, we may define an induced covariant derivative,∇ : X(N)×Γ(ϕ, TM) → Γ(ϕ, TM) (see Differential Geometry II), which allows us to define objectssuch as ∇c′c

′. We will implicitly use this concept when discussing geodesics and variation of arc-length but, for convenience, we will suppress the notation ∇ and proceed as if vector fields alongϕ were actually defined on M .

Remark 2.4. It follows from the Chain Rule that L[c] does not depend on the choice of parametri-sation of the curve c. In particular, if we have a regular curve c : [a, b] → M , and a C1 mapγ : [a, b] → [a′, b′] with γ′ > 0 then, defining c := c γ−1 : [a′, b′] →M , we find that

L[c] =∫ b′

a′|c′(s)|ds =

∫ b

a

|c′(t)|dt = L[c].

Given a regular curve c, we may use this invariance to find a reparametrisation with the propertythat c := c γ−1 : [a′, b′] → M has the property that |c′| = constant. Such a curve is said to beparametrised proportional to arc-length.

Definition 2.5. The distance function d : M ×M → R is given by

(p, q) 7→ d(p, q) := infL[c]

∣∣∣ c : [a, b] →M smooth with c(a) = p, c(b) = q.

Proposition 2.6. Let (M,g) be a connected Riemannian manifold. Then the distance functiond : M ×M → R defines a metric on M .

Proof. See Differential Geometry II.

Since we now know that a Riemannian manifold is automatically a metric space, we may definesuch concepts as convergence, Cauchy sequences, and completeness as for general metric spaces.One of our interests will be how geometric properties of Riemannian manifolds are related totheir metric space structures. For example, the Hopf-Rinow theorem relates completeness as aRiemannian manifold with completeness as a metric space.

3. First variation of arc-length

Definition 3.1. Let c : [a, b] →M be a smooth curve. A smooth variation of c is a smooth mapα : [a, b]× (−ε, ε) →M , for some ε > 0, with the property that α| [a, b]× 0 = c : [a, b] →M .

Lecture 3Let (t, s) be coordinates on Q := [a, b] × (−ε, ε). We wish to calculate how the length of the

curves cs : [a, b] → M defined by cs := α| [a, b]× s varies with s ∈ (−ε, ε). To this end, wedefine the vector fields along α

T := Dα

(∂

∂t

), V := Dα

(∂

∂s

).

In this case, the vector field (along cs) T(t, s) is the tangent vector to the curves cs, while V iscalled the variation vector field of α. We also define the vector fields along c:

t := Dc

(∂

∂t

)= T|s=0 , v := Dc

(∂

∂s

)= V|s=0 .

We will assume that the curve c is parametrised proportional to arc-length, with |c′(t)| = |t(t)| = l,for all t ∈ [a, b], where l > 0 is a constant.

Lemma 3.2. Let c : [a, b] → M be a smooth curve, parametrised such that |c′(t)| = l > 0 fort ∈ [a, b]. If α : [a, b]× (−ε, ε) →M is a smooth variation of c then

d

dsL[cs]

∣∣∣∣s=0

=1l

[g(t,v)

∣∣∣ba−∫ b

a

g(∇tt,v) dt

]. (3.1)

This expression is called the first variation formula.

8 JAMES D.E. GRANT

Proof. First note that (t, s) are coordinates on [a, b] × (−ε, ε), so [∂t, ∂s] = 0 on [a, b] × (−ε, ε).Therefore, Dα ([∂t, ∂s]) = [Dα(∂t), Dα(∂s)] = [T,V] = 0. Hence ∇TV = ∇VT. Therefore,

d

dsL[cs] =

d

ds

∫ b

a

|c′s| dt =∫ b

a

V(g(T,T)1/2

)dt =

∫ b

a

∇V

(g(T,T)1/2

)dt

=∫ b

a

12|T|

∇V (g(T,T)) dt =∫ b

a

1|T|

g(T,∇VT)dt =∫ b

a

1|T|

g(T,∇TV)dt

=∫ b

a

T(

1|T|

g(T,V))dt−

∫ b

a

[T(

1|T|

)g(T,V) +

1|T|

g(∇TT,V)]dt.

Setting s = 0, integrating the first term and using the fact that |t| = l is constant, we deduce that

d

ds

∫ b

a

|c′s| dt

∣∣∣∣∣s=0

=(

1lg(t,v)

)∣∣∣∣ba

− 1l

∫ b

a

g (∇tt,v) dt

as required.

Remark 3.3. If the variation α has fixed endpoints (i.e. α(a, s) = p, α(b, s) = q for all s ∈ (−ε, ε))then v(a) = v(b) = 0, and the first term in equation (3.1) vanishes.

Definition 3.4. Let M be a manifold with affine connection ∇. A curve c : [a, b] → M is ageodesic if its tangent vector c′ := Dc

(∂∂t

)∈ Γ(c, TM) satisfies

∇c′c′ = 0.

In the case where (M,g) is a semi-Riemannian manifold, and c is a geodesic (with respect to theLevi-Civita connection), then

∇c′ (g(c′, c′)) = 2g(c′,∇c′c′) = 0,

therefore g(c′, c′), and hence |c′|, is constant along c. An affinely parametrised geodesic with|c′| = 1 is said to be parametrised by arc-length.

Corollary 3.5. A smooth curve segment c of constant speed |c′| = l > 0 is a critical point of thearc-length functional under fixed end-point variations if and only if it is a geodesic.

Proof. In the case where the variation has fixed endpoints, the first variation formula becomes

d

dsL[cs]

∣∣∣∣s=0

= −1l

∫ b

a

g(∇tt,v) dt. (3.2)

If c is a geodesic then ∇tt = 0, so ddsL[cs]|s=0 = 0.

The non-trivial direction is to assume ddsL[cs]

∣∣s=0

= 0 for any variation with fixed endpoints. Inparticular, the expression on the right-hand-side of (3.2) must vanish for any vector field along c,v, that is generated by a smooth variation and has v(a) = v(b) = 0. If we assume, for the moment,that any vector field v ∈ Γ(c, TM) can arise from a smooth variation, then let v(t) = f(t) (∇tt) (t),for f a smooth function with f(a) = f(b) = 0. We then have

d

dsL[cs]

∣∣∣∣s=0

= −1l

∫ b

a

f(t)|∇tt|2 dt.

In order that the right-hand-side vanish for any function f , we require ∇tt = 0 on (a, b) i.e. c isa geodesic.

It remains to be shown that any vector field along c, v, arises from a smooth variation of c.However, given v, we simply define α(t, s) := expc(t) (sv(t)) (the exponential map will be discussedin the Section 4), and check that D(t,0)α

(∂∂s

)= v(t).

As mentioned earlier, we could have considered piecewise smooth curves instead of smoothcurves. We then have the more general result:

Proposition 3.6. A piecewise smooth curve segment c of constant speed |c′| = l > 0 is a criticalpoint of the arc-length functional under fixed end-point variations if and only if it is a geodesic.


Proposition 3.7. Let N be a submanifold of M without boundary and p ∈ M (p /∈ N). Letc : [a, b] →M be a geodesic such that c(a) ∈ N , c(b) = p, and such that c is a shortest curve fromN to p. Then c′(a) is perpendicular to Tc(a)N (i.e. gc(a)(c′(a),v) = 0, for all v ∈ Tc(a)N .)

Proof. If c′(a) is not perpendicular to Tc(a)N , then let x ∈ Tc(a)N be such that gc(a)(c′(a),x) > 0.Let γ be a curve in N with γ(0) = c(a), γ′(0) = x. Then construct any variation α : [a, b] ×(−ε, ε) →M such that

α| [a, b]× 0 = c : [a, b] →M ;

α(a, s) = γ(s), s ∈ (−ε, ε);α(b, s) = p, s ∈ (−ε, ε).

Let cs := α| [a, b]× s : [a, b] →M . Then the first variation formula (3.1) implies that

d

dsL[cs]

∣∣∣∣s=0

= −1lg(c′(a),x) < 0.

Hence, for sufficiently small s > 0, we have L[cs] < L[c]. This contradicts the fact that c is theshortest curve from N to p.

Remark 3.8. Note that the curves cs that we construct in the proof of the previous propositionneed not be geodesics. As long as they are smooth, our argument shows that L[cs] < L[c].

4. The exponential map and normal coordinates

Proposition 4.1. Let p ∈ M and v ∈ TpM . Then there exists a unique geodesic γv in M suchthat

(1) the initial velocity of γv is v; i.e. ddtγv(t)

∣∣t=0

= v;(2) the domain Iv of γv is the largest possible. Hence if α : J → M is a geodesic with initial

velocity v then J ⊆ Iv and α = γv|J .The geodesic γv is said to be maximal or geodesically inextendible.

Proof. See Differential Geometry II for details. The main idea is to write the geodesic equationsin a local coordinate chart (U,ϕ) on a neighbourhood of p. The geodesic equations then becomethe collection of ordinary differential equations

d2

dt2(xi c

)+∑j,k

(Γijk c

)(d (xj c)dt

)(d(xk c

)dt

)= 0,

for maps xi c : R → R, i = 1, . . . , n, with(xi c

)(0) = 0, d

dt

(xi c

)∣∣t=0

= vi, where vi are thecomponents of v in this coordinate system. The result then follows from the standard existenceand uniqueness results for ODEs.

Definition 4.2. Let p ∈ M . We define Dp ⊆ TpM to be the set of vectors v ∈ TpM such that Lecture 4the inextendible geodesic γv is defined at least on the interval [0, 1]. The exponential map of Mat p is the map

expp : Dp →M, v 7→ γv(1).

Proposition 4.3. Let (M,g) be a Riemannian manifold. For each p ∈ M there exists a neigh-bourhood, U , of 0 ∈ TpM and a neighbourhood, V , of p in M such that the map expp

∣∣U

: U → V

is a diffeomorphism (i.e. a smooth homeomorphism).

Let ei be a basis for TpM , orthonormal with respect to gp (i.e. gp(ei, ej) = δij , where δij = 1if i = j and 0 otherwise.) We may then define a coordinate system on the neighbourhood V byassigning the point expp

(∑ni=1 x

iei)

the coordinates (x1, . . . , xn). In particular, let q ∈ V . Thenexp−1

p q ∈ U , and therefore may be written in the form∑ni=1 x

iei, for some real numbers x1, . . . , xn,which we define to be the coordinates of q. Such coordinates are called normal coordinates at p.Note that xi : U → R are functions on U ⊆ TpM . We may also define the maps yi := xi exp−1

p :V → R so that, given q ∈ V , the yi(q) are the coordinates of the point q.

10 JAMES D.E. GRANT

Gauss Lemma. Let p ∈M , x ∈ Dp \ 0. Let vx,wx ∈ Tx (TpM) with vx radial. Then

gp(vx,wx) = gexpp x

((Dx expp

)(vx) ,

(Dx expp

)(wx)

). (4.1)


Remark 4.4. A remark may be in order at this point concerning the left-hand-side of equation (4.1).Since TpM is a vector space, given any x ∈ TpM , we may identify the tangent space Tx(TpM)with TpM itself. More precisely, let v ∈ TpM , then d

dt (x+ tv)∣∣t=0

defines a tangent vector byvx ∈ Tx(TpM). This map defines an isomorphism ψx : TpM → Tx(TpM). We may, in effect, nowturn TpM into a Riemannian manifold, by defining a metric G ∈ T 0

2 (TpM) by

Gx(vx,wx) = gp(v,w), ∀v,w ∈ TpM.

By the left-hand side of (4.1), we really mean Gx(vx,wx) or, equivalently, gp(v,w).We can then identify the vector fields x → ψx(ei) on TpM with the coordinate derivatives

∂/∂xi. (Proof: Look at their actions as directional derivatives at any point x ∈ TpM .) Therefore,we find that

G(

∂

∂xi,∂

∂xj

)= δij . (4.2)

If we now switch to polar coordinates (ρ, ψi) on U \ 0 ⊆ TpM , where ρ :=√∑

(xi)2 and ψi arecoordinates on the (n− 1)-spheres ρ = constant, then we find from (4.2) that

G(∂

∂ρ,∂

∂ρ

)= 1, G

(∂

∂ρ,∂

∂ψi

)= 0.

If we perform the same coordinate transformation on V \ p, with r :=√∑

(xi)2 and θi thecorresponding coordinates on the spheres r = constant, then we have

ρ = r expp, ψi = θi expp .

In particular, dρ =(expp

)∗dr, dψi =

(expp

)∗dθi. Since expp : U → V is a diffeomorphism, we

therefore have (D expp

)( ∂

∂ρ

)=

∂

∂r,

(D expp

)( ∂

∂ψi

)=

∂

∂θi.

Since ∂/∂ρ is a radial vector field, we may apply the Gauss Lemma to deduce that

g(∂r, ∂r) = G(∂ρ, ∂ρ) = 1, g(∂r, ∂θi) = G(∂ρ, ∂ψi) = 0.

In particular, in terms of these coordinates, the metric takes the form

g|V \p = dr2 + gSn−1(r),

on V \p, where gSn−1(r) is the metric induced, by g, on the hypersurface S(r) := q ∈ V \p :r(q) = r.

Definition 4.5. Let (M,g) be a Riemannian manifold, and f ∈ C∞(M). We define the vectorfield4 grad f ∈ X(M), gradient of f by

g(X, grad f) = 〈X, df〉 = X(f), ∀X ∈ X(M).

Corollary 4.6 (to Remark 4.4). On V \ p we have

grad r =∂

∂r.

4Some people denote this vector field by ∇f . However, given our use of ∇ for connections, this would, perhaps,be slightly confusing.


Proof. Given the form of the metric, g|V \p = dr2 + gSn−1(r), we deduce that

g(∂

∂r, ·)

= dr

on V \ p. Hence, for any vector field, X, on V \ p we have

g(X, grad r) = 〈dr,X〉 = g(∂

∂r,X).

Hence, grad r = ∂/∂r, as required.

Remark 4.7. Let q ∈ V \p, then there exists a unique v ∈ TpM such that q = expp v. Moreover,r(q) = |v| :=

√gp(v,v). If we consider the geodesic γv : [0, 1] → M with γ(0) = p, γ′(0) = v,

then γ(1) = q, and the length of this curve is

L[γv] =∫ 1

0

|γ′v(t)|dt =∫ 1

0

|v|dt = |v|,

where the second inequality follows from the fact that |γ′v| is constant, since γv a geodesic. Hence

r(q) = |v| = L[γv].

5. Hopf-Rinow theorem

Hopf-Rinow theorem. Let (M,g) be a Riemannian manifold. The following are equivalent:(a) The metric space (M,d) is complete;(b) For some p ∈M , expp is defined on all of TpM (i.e. M is geodesically complete at p);(c) For all p ∈M , expp is defined on all of TpM (i.e. M is geodesically complete);(d) Every closed, bounded subset of M is compact.


Corollary 5.1. Let (M,g) be a complete, connected Riemannian manifold. Given any two pointsp, q ∈M , there exists a geodesic from p to q the length of which is equal to d(p, q).


6. Bishop-Gromov comparison theoremLecture 5

Let (M,g) be a Riemannian manifold and f ∈ C∞(M). We have already defined the vectorfield grad f ∈ X(M) by

g(X, grad f) = 〈X, df〉 = X(f), ∀X ∈ X(M).

The Hessian of f is the (0, 2) tensor field on M defined by

Hess f (X,Y) = 〈∇Xdf,Y〉, ∀X,Y ∈ X(M).

Proposition 6.1.

Hess f (X,Y) = Hess f (Y,X) , ∀X,Y ∈ X(M).

Proof.

Hess f (X,Y)−Hess f (Y,X) = 〈∇Xdf,Y〉 − 〈∇Ydf,X〉= ∇X〈df,Y〉 − 〈df,∇XY〉 − ∇Y〈df,X〉+ 〈df,∇YX〉= X (Y (f))−Y (X (f)) + 〈df, [Y,X]〉= 0.

12 JAMES D.E. GRANT

We define the (1, 1) tensor field S by

S(X) := ∇X (grad f) , ∀X ∈ X(M),

in which case we haveHess f (X,Y) = g(S(X),Y) = g(X,S(Y)),

so S is symmetric with respect to the metric g. The trace of the operator S : X(M) → X(M) isthe Laplacian of f :

∆f := trS =n∑i=1

g(ei,S(ei)) =n∑i=1

Hess f(ei, ei),

where eini=1 is any orthonormal basis with respect to g.

Definition 6.2. Let (M,g) be a Riemannian manifold. A distance function on an open setU ⊆ M is a smooth map r : U → R with the property that |grad r| = 1. For a distance function,r, we will denote the vector field grad r ∈ X(U) by ∂r.

Example 6.3. Fix a point p ∈ M . Then the function r(q) := d(p, q) is a distance function, forq in a sufficiently small neighbourhood of p. This is closely related to the discussion of geodesiccoordinates in Section 4 and, in fact, the function r defined here coincides with that used ingeodesic polar coordinates. This is due to the fact (which we have not proved) that for sufficientlysmall neighbourhoods, U , of 0 ∈ TpM the geodesic γv will be a minimising curve between p andexpp v, and hence d(p, expp v) = |v| = r(expp v).

Proposition 6.4. Let U ⊆M be an open set and r : U → R be a distance function. Then

∇∂r∂r = 0

on U .

Proof. Let X ∈ X(U), then

g(∇∂r∂r,X) = g(S(∂r),X) = g(∂r,S(X)) = g(∂r,∇X∂r) =12∇Xg(∂r, ∂r) =

12∇X(1) = 0.

Non-degeneracy of g then implies that ∇∂r∂r = 0, as required.

Proposition 6.5. Let r be a distance function on an open set U ⊆M . Then, on U ,

∂r (∆r) +1

n− 1(∆r)2 ≤ ∂r (∆r) + tr(S2) = −Ric(∂r, ∂r). (6.1)


Proof. Let p ∈ U , and eini=1 be an orthonormal basis on a neighbourhood of p. Then

∂r (∆r) = ∇∂r

(n∑i=1

g (ei,∇ei∂r)

),

=n∑i=1

g (∇∂rei,∇ei

∂r) +n∑i=1

g (ei,∇∂r∇ei

∂r)

=n∑i=1

g (∇∂rei,∇ei

∂r) +n∑i=1

g(ei,∇ei∇∂r∂r +∇[∂r,ei]∂r + R(∂r, ei)∂r

)=

n∑i=1

g (∇∂rei,∇ei

∂r) + 0 +n∑i=1

g (ei,S([∂r, ei]))−Ric(∂r, ∂r)

= −Ric(∂r, ∂r) +n∑i=1

g (∇∂rei,∇ei

∂r) +n∑i=1

g (S(ei), [∂r, ei])

= −Ric(∂r, ∂r) +n∑i=1

g (∇∂rei,∇ei

∂r) +n∑i=1

g (S(ei),∇∂rei −∇ei

∂r)

= −Ric(∂r, ∂r)−n∑i=1

g (S(ei),∇ei∂r) + 2

n∑i=1

g (∇∂rei,∇ei

∂r)

= −Ric(∂r, ∂r)−n∑i=1

g (S(ei),S(ei)) + 2n∑i=1

g (∇∂rei,∇ei

∂r)

= −Ric(∂r, ∂r)−n∑i=1

g(ei,S2(ei)

)+ 2

n∑i=1

g (∇∂rei,∇ei∂r)

= −Ric(∂r, ∂r)− trS2 + 2n∑i=1

g (∇∂rei,∇ei∂r) . (6.2)

To calculate the final term on the right-hand-side, we note that, since ∇∂rei ∈ X(U), there

exist functions φij on U such that

∇∂rei =n∑j=1

φijej .

Since eini=1 is an orthonormal basis, it follows that g(ei, ej) are constant, and therefore

0 = ∇∂r(g (ei, ej)) = g (∇∂r

ei, ej) + g (ei,∇∂rej)

= g

(∑k

φikek, ej

)+ g

(ei,∑k

φjkek

)= φij + φji.

Hence φij = −φji. Therefore, the final term in (6.2) takes the form

2n∑

i,j=1

φijg (ej ,S(ei)) = −2n∑

i,j=1

φjig (ej ,S(ei)) (skew-symmetry of φ)

= −2n∑

i,j=1

φjig (S(ej), ei) (S symmetric w.r.t. g)

= −2n∑

i,j=1

φijg (S(ei), ej) (relabelling summation indices)

= −2n∑

i,j=1

φijg (ej ,S(ei)) (symmetry of g).

14 JAMES D.E. GRANT

Since the first line equals (−1) times the last line, we deduce that∑ni,j=1 φijg (ej ,S(ei)) = 0,

so (6.2) reduces to∂r (∆r) = −Ric(∂r, ∂r)− trS2,

as required.To deduce the required inequality in (6.1), we note that S is symmetric with respect to g and

therefore, at any fixed point p ∈M , we may choose an orthonormal basis for TpM in which S|p isdiagonal, with the diagonal entries being the eigenvalues of S|p. Noting that S(∂r) = ∇∂r

∂r = 0,we deduce that S|p has a zero eigenvalue, and therefore can be put in the form diag[0, λ1, . . . , λn−1].We define an inner product on (n− 1)× (n− 1) matrices by the formula

〈a, b〉 := tr (ab) , a, b ∈ R(n− 1).

Viewing S|p as a n×n matrix of the form(

0 00 s

), where s = diag[λ1, . . . , λn−1], we deduce, from

the Cauchy-Schwartz inequality that

|tr S|p | = |tr s| = |〈1n−1, s〉| ≤ ‖1‖ · ‖s‖ =√n− 1

√tr (s2) =

√n− 1

√tr (S|2p).

Since p is arbitrary, we deduce that

tr (S2) ≥ 1n− 1

(trS)2 =1

n− 1(∆r)2 .

The inequality in (6.1) then follows.

Remark 6.6. An equation, like (6.1), of the general form ∂rΘ + Θ2 + f(r) = 0, with f a fixedsmooth function, is referred to as a Riccati equation. Equations of this type will be useful in bothRiemannian and Lorentzian geometry.

Remark 6.7. More generally, on a Riemannian manifold (M,g) a formula of Bochner asserts thatfor any smooth function u on M we have the identity

12∆|gradu|2 = trS2 + g (gradu, grad (∆u)) + Ric(gradu, gradu).

The derivation of this formula is similar to that of equation (6.1), but a little more involved.Lecture 6

We now apply the formula (6.1) to the metric in normal coordinates that we derived earlier.In particular, let (M,g) be a Riemannian manifold and p ∈ M . We know that there exists aneighbourhood, U ⊆ TpM , of 0 ∈ TpM and a neighbourhood, V ⊆ M , of p in M such thatexpp : U → V a diffeomorphism, and such that the metric, g, restricted to V \ p takes the form

g|V \ p = dr2 + h(r, θ), (6.3)

where r ∈ (0, R), for some R > 0, θ denotes dependence on the (n − 1)-sphere S(r) := exppv ∈TpM : |v| = r (r < R) and h(r, θ) is the induced metric on S(r). (In particular, h(·, ∂r) = 0.)Since grad r = ∂r and g(∂r, ∂r) = 1, it follows that r is a distance function on V \p. (Note that,for notational simplicity, we drop the restriction to V \ p on g.)

Aside on integration: In order to define volumes on an oriented manifold, we need to definea volume form i.e. dvol ∈ Ωn(M). Given a Riemannian metric, g, on M there is a naturalvolume form associated to it, which we denote by dvolg, defined as follows. In local coordinatesx1, . . . , xn, we have the matrix of components of the metric gij := g(∂/∂xi, ∂/∂xj). We thendefine

dvolg :=√

det (gij)dx1 ∧ · · · ∧ dxn.

(Note that I will sometimes drop the g on dvolg if it is clear what metric we are working with.)If we wish to integrate a function f ∈ C∞(M) over a subset S ⊆ M , which we assume to lie inthe subset of M covered by coordinates xi, then we define∫

S

f(x)dvolg =∫S

f(x)√

det (gij)dx1 ∧ · · · ∧ dxn,


which we define to be equal to the multiple integral∫x(S)

f(x1, . . . , xn)√

det (gij)dx1 . . . dxn

over the relevant subset of Rn.

Example 6.8.

(1) Let (M,g) be Rn with the flat metric. In Cartesian coordinates, we have g =∑ni=1

(dxi)2,

so√

det (gij) = 1 and dvolg = dx1 ∧ · · · ∧ dxn. In this case,∫Sf(x)dvolg is simply the

multiple integral of the function f over the set S ⊆ Rn.(2) Let (M,g) two-sphere S2 with the metric induced from embedding S2 as the unit sphere

in flat R3. In polar coordinates x1 = θ, x2 = φ, we have

g = dθ2 + sin2 θdφ2.

Therefore the components of the metric are

gij =(

1 00 sin2 θ

),

so√

det (gij) = sin θ and dvolg = sin θdθ ∧ dφ. In this case,∫Sf(x)dvolg is the double

integral ∫S

f(θ, φ) sin θdθdφ,

over the relevant subset S ⊆ S2. A particular case would be the calculation of Vol (S2),where we take f(θ, φ) = 1 and integrate over S = S2, in which case we have

Vol (S2) =∫ 2π

0

∫ π

0

1 sin θdθdφ = 4π.

(3) Let (M,g) be Rn with the flat metric, written in polar coordinates, so

g = dr2 + r2gSn−1 ,

where gSn−1 is the standard round metric on the (n − 1)-sphere of radius 1. If we adoptcoordinates x1 = r, x2 = θ1, . . . , xn = θn−1, where θ1, . . . , θn−1 are coordinates on Sn−1,then the metric takes the local form

g = dr2 + r2n−1∑i,j=1

aij(θ1, . . . , θn−1)dθi ⊗ dθj ,

for some functions aij . In this case, the matrix of components of the metric g decomposesin block form

gij =(

1 00 r2a

),

and we have √det (gij) = rn−1

√det (aij).

Since√

det (aij)dθ1 ∧ · · · ∧ dθn−1 = dvolgSn−1 , we deduce that

dvolg = rn−1dr ∧ dvolgSn−1 .

(4) More generally, if we consider the constant curvature metrics gK introduced in Section 1,then an almost identical calculation to that in the previous example shows that the volumeelement takes the form

dvolg = snK(r)n−1dr ∧ dvolgSn−1 .

16 JAMES D.E. GRANT

Consider our general metric in normal coordinates centred at p ∈M of the form given in (6.3),where h(r, θ) is a general metric on the (n− 1)-sphere. In this case, a similar calculation to thatin Example 3 implies that

dvolg = dr ∧ dvolh(r,θ) =√

det (hij)dr ∧ dθ1 ∧ · · · ∧ dθn

=

√det (h)√

det (gSn−1)dr ∧ dvolgSn−1 .

Therefore, if we define a function a(r, θ) by

a(r, θ)n−1 =

√det (h)√

det (gSn−1),

then the volume element of the metric (6.3) takes the form

dvol = a(r, θ)n−1dr ∧ dvolSn−1 , (6.4)

where dvolSn−1 denotes the volume element of the standard sphere of radius 1.As r → 0, we require that the metric (6.3) approaches the flat metric on Rn (so that the metric

g is not singular at r = 0), and therefore

g → dr2 + r2gSn−1 as r → 0.

In terms of the volume element, this implies that

dvol → rn−1dr ∧ dvolSn−1 as r → 0.

In particular,a(r, θ) → r as r → 0,

which implies that the function a obeys the boundary conditions

a(0, θ) = 0,∂a

∂r(0, θ) = 1.

Finally, in order to apply equation (6.1), we need to calculate ∆r in the above metric. In localcoordinates (r, θ) we have

∆r =∑i,j

gij

(∂2r

∂xi∂xj−∑k

Γkij∂r

∂xk

)where Γijk denote the Christoffel coefficients of the metric g. Since x1 = r, the first term vanishes,and only the k = 1 term survives in the second term, so we have

∆r = −∑i,j

gijΓrij = −12

∑i,j,k

gijgrk [∂igkj + ∂jgki − ∂kgij ]

= −12

∑i,j,k

gijgrr [∂igrj + ∂jgri − ∂rgij ]

=12

∑i,j

gij∂rgij ≡12tr(g−1∂rg

)=

12∂r (det g)

det g=∂r (det g)1/2

(det g)1/2.

Here, we have used the fact that grr = 1, ∂igrj = 0 for all i, j and the fact that, given any invertible,symmetric matrix Λ(x), we have tr (Λ−1 dΛ

dx ) = d det Λdx /det Λ. (If you didn’t know this last fact, it

can easily be deduced by fixing a basis in which Λ is diagonal.) Since dvol = (det g)1/2 dnx, wededuce from (6.4) that (det g)1/2 = a(r, θ)n−1 · (deth)1/2, where deth is the determinant of themetric on Sn−1. Since deth is independent of r, we therefore have

∆r =∂r (a(r, θ))n−1

(a(r, θ))n−1 =(n− 1)a

∂a

∂r. (6.5)


Bishop-Gromov Comparison Theorem (Version 1). Let (M,g) be a complete Riemannianmanifold with the property that there exists a constant K ∈ R such that

Ric(X,X) ≥ K (n− 1)g(X,X), ∀X ∈ X(M). (6.6)

Let p ∈ M and B(p, r) := expp(v ∈ TpM : |v| < r), where we restrict to r ∈ [0, R) such thatB(p,R) ⊆ V . Then the map

r 7→ Vol (B(p, r),g)Vol (B(p, r),gK)

is non-increasing, where gK denotes the metric of constant curvature K (see Section 1). Inparticular,

Vol (B(p, r),g) ≤ Vol (B(p, r),gK)for r ∈ [0, R).

Remark 6.9. The statement that Vol (B(p, r),g) ≤ Vol (B(p, r),gK) is the original result proved byBishop, who proved this result for r less than the injectivity radius5 of the manifold (M,g). Gro-mov [Gr] later generalised this result to show that the map r → Vol (B(p, r),g)/Vol (B(p, r),gK)is non-increasing and also noted that, with suitable modifications, the result holds for all valuesof r. (We will extend the following proof to cover Gromov’s more general result after we havestudied the second variation formula and conjugate points etc.)

Lecture 7

Proof of Theorem 6. Let the metric g satisfy the curvature condition (6.6). The function r is adistance function on the set V , and therefore satisfies

∂r (∆r) +1

n− 1(∆r)2 ≤ −K(n− 1).

Given that ∆r = (n− 1)a′/a (from now on, ′ will denote partial differentiation with respect to r)we deduce that a satisfies

1a

∂2a

∂r2≤ −K.

Since a(r, θ) ∼ r, for sufficiently small r, it follows that a(r, θ) > 0, for sufficiently small r > 0.Therefore,

∂2a(r, θ)∂r2

+Ka(r, θ) ≤ 0.

We now wish to use this equation to compare the properties of the metric g with those of theconstant curvature metric gK . To do this, we compare the function a with the function, snK(r),defined in equation (1.1) of Section (1). This function satisfies

∂2snK(r)∂r2

+KsnK(r) = 0, snK(0) = 0, sn′K(0) = 1.

We then have

∂r (snK(r)a′(r)− sn′K(r)a(r)) = snK(r)a′′(r)− sn′′K(r)a(r)

≤ −KsnK(r)a(r) +KsnK(r)a(r)= 0.

Hence snK(r)a′(r) − sn′K(r)a(r) is non-increasing. Since, evaluated at r = 0, it equals zero, wededuce that

snK(r)a′(r)− sn′K(r)a(r) ≤ 0.Since snK , a are positive for sufficiently small r > 0, we deduce that(

a(r)snK(r)

)′≤ 0. (6.7)

We therefore deduce that

the map r 7→ a(r)snK(r)

is non-increasing. (6.8)

5This term will be defined later.

18 JAMES D.E. GRANT

Moreover, a(r), snK(r) ∼ r as r → 0, so limr→0(a(r)/snK(r)) = 1. Thereforea(r)

snK(r)≤ 1 for r > 0.

Finally, we consider the volumes

Vol (B(p, r),g) :=∫ r

0

∫Sn−1

1dvolg =∫ r

0

∫Sn−1

a(t, θ)n−1dt ∧ dvolgSn−1

Vol (B(p, r),gK) :=∫ r

0

∫Sn−1

1dvolgK=∫ r

0

∫Sn−1

snK(t)n−1dt ∧ dvolgSn−1

= Vol (Sn−1)∫ r

0

snK(t)n−1dt

and the ratio (abbreviating dvolgSn−1 with just dθ)

Vol (B(p, r),g)Vol (B(p, r),gK)

=

∫ r0

∫Sn−1 a(t, θ)n−1dθdt∫ r

0

∫Sn−1 snK(t)n−1dθdt

=1

Area(Sn−1)

∫ r0

∫Sn−1 a(t, θ)n−1dθdt∫ r0snK(t)n−1dt

.

Taking the limit as r → 0, we see that Vol (B(p, r),g)/Vol (B(p, r),gK) → 1, since a(r, θ), snK(r) ∼r. To prove that the above ratio is non-increasing, we take is derivative:d

dr

[Vol (B(p, r),g)

Vol (B(p, r),gK)

]=

1Vol (Sn−1)

1(∫ r0snK(t)n−1dt

)2×[∫

Sn−1a(r, θ)n−1dθ ×

∫ r

0

snK(t)n−1dt− snK(r)n−1

∫ r

0

∫Sn−1

a(t, θ)n−1dθdt

]=

1Vol (Sn−1)


)2 ∫ r

0

snK(t)n−1snK(r)n−1

×∫Sn−1

[(a(r, θ)snK(r)

)n−1

−(a(t, θ)snK(t)

)n−1]dθdt. (6.9)

From (6.8), we know that a(t,θ)snK(t) , and hence

(a(t,θ)snK(t)

)n−1

, is a non-increasing function of t. Hence,in the integral in (6.9), since 0 ≤ t ≤ r, we have(

a(r, θ)snK(r)

)n−1

−(a(t, θ)snK(t)

)n−1

≤ 0.

Since snK(r) ≥ 0 for sufficiently small r, it follows from (6.9) that

d

dr

[Vol (B(p, r),g)

Vol (B(p, r),gK)

]≤ 0.

Therefore the ratio of volumes is non-increasing. Since we know that Vol (B(p, r),g)/Vol (B(p, r),gK) →1 as r → 0, we also deduce that Vol (B(p, r),g) ≤ Vol (B(p, r),gK).

7. Second variation of arc-length

Definition 7.1. Let c : [a, b] → M be a smooth curve. A (two parameter) smooth variationof c is a smooth map α : [a, b] × (−δ, δ) × (−ε, ε) → M , for some ε, δ > 0, with the propertythat α| [a, b]× 0 × 0 = c : [a, b] → M . For (r, s) ∈ (−δ, δ) × (−ε, ε), we define the curvecr,s := α| [a, b]× r × s : [a, b] →M .

In this case, taking coordinates (t, r, s) on [a, b] × (−δ, δ) × (−ε, ε), we define the vector fieldsalong α:

T := Dα

(∂

∂t

), V := Dα

(∂

∂r

), W := Dα

(∂

∂s

),

and the vector fields along c:

t := T|r=s=0 , v := V|r=s=0 , w := W|r=s=0 .


We will assume that the curve c is parametrised proportional to arc-length, with |c′(t)| = |t(t)| = l,for all t ∈ [a, b], where l > 0 is a constant.

Lemma 7.2. Let c : [a, b] → M be a geodesic with |c′| = l > 0. If α is a variation of c as abovethen

∂2L

∂r∂s

∣∣∣∣r=s=0

=1lg(t,∇wv)

∣∣∣∣ba

+1l

∫ b

a

[g((∇tv)⊥ , (∇tw)⊥) + g(t,R(w, t)v)

]dt, (7.1)

where, given any vector field X along c, we define

X⊥ := X− g(X, t)|t|2

t,

the component of X perpendicular to t ≡ c′. The expression in (7.1) is called the second variationformula.

Proof. As in the proof of the first variation formula, note that ∇TV = ∇VT, ∇TW = ∇WT and∇VW = ∇WV. We therefore have

∂2

∂r∂sL[cr,s] =

∂2

∂r∂s

∫ b

a

∣∣c′r,s∣∣ dt =∫ b

a

∇V∇W (g(T,T))1/2 dt =∫ b

a

∇V

(1|T|

g(T,∇WT))dt.

=∫ b

a

∇V

(1|T|

g(T,∇TW))dt.

=∫ b

a

[− 1|T|3

g(T,∇VT)g(T,∇TW) +1|T|

g(∇VT,∇TW) +1|T|

g(T,∇V∇TW)]dt

=∫ b

a

[− 1|T|3

g(T,∇TV)g(T,∇TW) +1|T|

g(∇TV,∇TW) +1|T|

g(T,∇V∇TW)]dt.

Lecture 8We may rewrite the final term in the integrand using the following

g(T,∇V∇TW) = g(T,∇T∇VW +∇[T,V]W + R(V,T)W

)= ∇T (g(T,∇VW))− g(∇TT,∇VW) + g(T,R(V,T)W).

Setting r = s = 0, and using the fact that ∇tt = 0 and |t| = l is constant, we deduce that

∂2

∂r∂sL[cr,s]

∣∣∣∣r=s=0

=∫ b

a

[− 1l3

g(t,∇tv)g(t,∇tw) +1lg(∇tv,∇tw)

+1l

(∇t (g(t,∇vw)) + g(t,R(v, t)w))]dt

=1lg(t,∇vw)

∣∣∣∣ba

+1l

∫ b

a

[g((∇tv)⊥ , (∇tw)⊥) + g(t,R(v, t)w)

]dt.

We now note that• ∇vw = ∇wv;• g(t,R(v, t)w) = R(v, t, t,w) = R(t,w,v, t) = R(w, t, t,v) = g(t,R(w, t)v)

Therefore,

∂2

∂r∂sL[cr,s]

∣∣∣∣r=s=0

=1lg(t,∇wv)

∣∣∣∣ba

+1l

∫ b

a

[g((∇tv)⊥ , (∇tw)⊥) + g(t,R(w, t)v)

]dt.

as required.

Remarks 7.3.

(1) Note that, by the argument at the end of the preceding proof, it follows that the right-hand-side of (7.1) is symmetric under interchange of v and w.

(2) The endpoint contribution to the above expression vanishes if ∇wv vanishes at a and b.In particular, if v and/or w vanishes at c(a) and c(b), then this term is zero.

20 JAMES D.E. GRANT

Let c : [a, b] → M be a geodesic, with |c′(t)| = l for t ∈ [a, b]. Let Γ0(c, TM) be the space ofvector fields, X, along c such that X(a) = X(b) = 0 that are orthogonal to c i.e.

Γ0(c, TM) :=

X ∈ Γ(c, TM)

∣∣∣∣∣∣∣X(a) = 0

X(b) = 0

gc(t)(X(t), c′(t)) = 0,∀t ∈ [a, b]

.

Definition 7.4. We define the symmetric bilinear form I : Γ0(c, TM) × Γ0(c, TM) → R, calledthe index form, by

I(X,Y) :=1l

∫ b

a

[g((∇tX)⊥ , (∇tY)⊥) + g(t,R(X, t)Y)

]dt,

for X,Y ∈ Γ0(c, TM).

Remark 7.5. Note that if g(X, t) = 0 along c then

0 = ∇t (g(X, t)) = g(∇tX, t).

Therefore ∇tX is orthogonal to t, so

∇tX = (∇tX)⊥ .

Therefore, the index form may be written in the simpler form

I(X,Y) :=1l

∫ b

a

[g(∇tX,∇tY) + g(t,R(X, t)Y)] dt, X,Y ∈ Γ0(c, TM).

Proposition 7.6. Given X,Y ∈ Γ0(c, TM),

I(X,Y) := −1l

∫ b

a

g(X,∇t∇tY + R(Y, t)t) dt. (7.2)

Proof. We haveg(∇tX,∇tY) = ∇tg(X,∇tY)− g(X,∇t∇tY).

Upon integration, this yields∫ b

a

g(∇tX,∇tY) dt = g(X,∇tY)|ba −∫ b

a

g(X,∇t∇tY) dt = −∫ b

a

g(X,∇t∇tY) dt,

since X(a) = X(b) = 0. Using the symmetries of the (0, 4) version of the Riemann tensor, we have

g(t,R(X, t)Y) ≡ R(X, t, t,Y) = R(t,Y,X, t) = −R(Y, t,X, t) ≡ −g(X,R(Y, t), t)

Hence

I(X,Y) := −1l

∫ b

a

[g(X,∇t∇tY) + g(X,R(Y, t)t)] dt,

as required.

Corollary 7.7. The bilinear form I : Γ0(c, TM) × Γ0(c, TM) → R is degenerate if and only ifthere exists Y ∈ Γ0(c, TM), not identically zero, such that

∇t∇tY + R(Y, t)t = 0.

Proof. For I to be degenerate implies that there exists Y ∈ Γ0(c, TM), not identically zero, suchthat I(X,Y) = 0 for all X ∈ Γ0(c, TM). The result then follows from equation (7.2) by a standardcalculus of variations argument.

Definition 7.8. Let c : [a, b] → M be a geodesic parametrised proportional to arc-length. Avector field along a geodesic c, J ∈ Γ(c, TM), that satisfies the Jacobi equation

∇t∇tJ + R(J, t)t = 0

is called a Jacobi field along c.


Remark 7.9. In local coordinates, xi, a Jacobi field along a geodesic γ has components J i(t)that obey the second order ordinary differential equations

d2

dt2J i(t) +

∑j,k,l

Rklij(γ(t))Jk(t)

d(xl γ

)dt

d(xj γ

)dt

= 0.

where the geodesic γ. It follows that if we fix initial data J(0),J′(0) ∈ Tγ(0)M , then there willexist a unique Jacobi field along γ with this initial data.

Proposition 7.10. If cs : [a, b] →M are geodesics, then V is a Jacobi field along cs.

Proof. If cs are geodesics, then we have ∇TT = 0. Hence

∇T∇TV = ∇T∇VT = ∇V∇TT + R(T,V)T = R(T,V)T.

Hence ∇T∇TV + R(V,T)T = 0, so V is a Jacobi field along cs.

Definition 7.11. A point q ∈ M is conjugate to p ∈ M along a geodesic c : [0, 1] → M withc(0) = p, c(1) = q if there exists a non-zero Jacobi field, J, along c such that J(0) = J(1) = 0.

Proposition 7.12. A point q ∈ M is conjugate to p ∈ M along a geodesic c if and only if q isa singular value of expp : TpM → M (i.e. q = expp t for t ∈ TpM and Dt expp : TpM → TqMhas non-trivial kernel). The order of a conjugate point is the dimension of the kernel of theDt expp : Tt(TpM) → Texpp tM .

Proof. Let J be a non-zero Jacobi field along c with J(0) = 0 and J(1) = 0 and u := J′(0) ∈ TpM .The smooth variation

α(t, s) := expp [t (t + su)] .

of c has variation vector field v(t) := D(t,s)α(∂∂s

)∣∣s=0

. Since the curves cs : [0, 1] → M aregeodesics, Proposition 7.10 implies that v is a Jacobi vector field along c. Since v(0) = 0 andv′(0) = u = J′(0), it follows that v = J. We then have

J(1) = D(t,s)α

(∂

∂s

)∣∣∣∣s=0,t=1

=(Dt expp

)(J′(0)) = 0.

Therefore, Dt expp has non-trivial kernel J′(0).Reversing the above construction then implies that, given u ∈ Tt(TpM) such that Dt expp(u) =

0, the Jacobi field J with J(0) = 0, J′(0) = u will obey J(1) = 0, so p and q are conjugate.

Proposition 7.13. Let c : [a, b] → M be a geodesic, c′ = t, and let J be a Jacobi field thatvanishes at c(a) and c(b). Then g(t,J) = g(t,J′) = 0.

Proof. Note thatt (g(t,J′)) = g(t,J′′) = g(t,−R(J, t)t) = 0.

Therefore g(t,J′) is constant on c. However, g(t,J′) = t (g(t,J)), so g(t,J) (as a function oft) has constant derivative. Since g(t,J)|c(a) = g(t,J)|c(b) = 0, it follows that g(t,J) = 0 and,therefore, g(t,J′) = 0.

Lecture 9

Proposition 7.14. Let c : [a, b] →M be a geodesic, with conjugate point at t = t0 ∈ (a, b). Thenthere exists a vector field along c, X, such that I(X,X) < 0. (In general, X is continuous but notdifferentiable.)

Proof. Let Y be a Jacobi field along c with Y(a) = Y(t0) = 0, but Y not identically zero. By theprevious proposition, Y and ∇tY are orthogonal to t. We then define the vector field along c:

Y(t) :=

Y(t) t ∈ [a, t0],0 t ∈ [t0, b].

Then Y ∈ Γ0(c, TM) and I(Y, Y) = 0. Now let Z ∈ Γ0(c, TM) be the vector field along c definedby the conditions

Z(t0) = − (∇tY) (t0), ∇tZ = 0.

22 JAMES D.E. GRANT

Now let ϕ : [a, b] → R be a smooth function with ϕ(a) = ϕ(b) = 0 and ϕ(t0) = 1. Letting λ ∈ Rbe a real parameter, we then consider

Xλ := Y + λϕZ ∈ Γ0(c, TM).

We then have

I(Xλ,Xλ) = I(Y, Y) + 2λI(Y, ϕZ) +O(λ2)

= 2λ∫ t0

a

[g(∇tY,∇t (ϕZ)) + g(t,R(Y, t)Z)] dt+O(λ2)

= 2λ∫ t0

a

∇t [g(∇tY, ϕZ)] dt− 2λ∫ t0

a

ϕ [g(Z,∇t∇tY + R(Y, t)t)] dt+O(λ2)

= 2λ g(∇tY, ϕZ)∣∣∣t0a

+O(λ2)

= 2λ g(∇tY, ϕZ)∣∣∣t0

+O(λ2)

= −2λ∣∣∣ (∇tY) (t0)

∣∣∣2 +O(λ2).

Hence, for sufficiently small λ > 0, we have I(Xλ,Xλ) < 0.

Corollary 7.15 (to Proposition 7.14). A geodesic cannot be minimal after its first conjugatepoint.

Proof. Assuming, in the proof of the Proposition 7.14, that c(t0) is the first point conjugate toc(0) along c, consider the variation generated by Xλ:

α(t, s) := expc(t) (sXλ(t)) .

It then follows thatd

dsL[cs]

∣∣∣∣s=0

= 0,d2

ds2L[cs]

∣∣∣∣s=0

= I(Xλ,Xλ) < 0

for sufficiently small λ > 0. Therefore, for sufficiently small s > 0, we have L[cs] < L[c], so c isnot minimising6.

Recall that a geodesic, c : [a, b] →M , is minimising between points c(a) and c(b) if

L[c] = d(c(a), c(b)).

A basic result is that a geodesic is initially minimising. For example, let M be complete, p ∈M ,v ∈ TpM , then there exists a T > 0 (T can be ∞) such that the geodesic c : [0,∞) → M withc(0) = p, c′(0) = v satisfies

d(c(0), c(t)) = L[c|[0,t]], ∀t ∈ [0, T ].

(A similar result is true if M is not complete, but we have to then take care concerning the domainof c.)

Remark 7.16. Note that, on a complete manifold, the interval on which a geodesic is minimising isclosed. In particular, let c : [0,∞) →M be a geodesic with L[c[0,t]] = d(c(0), c(t)) for all t ∈ [0, b),for some b > 0. Since c is continuous, it follows that both L[c[0,t]] and d(c(0), c(t)) are continuousfunctions of t. Hence, taking the limit as t → b, we deduce that d(c(0), c(b)) = L[c[0,b]], so c isminimising on the interval [0, b].

Let (M,g) be a complete Riemannian manifold. For v ∈ TpM , we defined γv to be the geodesicsatisfying γv(0) = p, γ′v(0) = v. Let

Iv := t ∈ R : γv minimal on [0, t].

6Stricly speaking, we are assuming that the path cs is contained in M for sufficiently small values of s. Since,generally, we will be applying such results on complete manifolds, this condition is automatically satisfied.


As mentioned in Remark 7.16, Iv is closed, so we let Iv = [0, ρ(v)], where ρ is a map TpM →(0,+∞]. If ρ(v) <∞, then γv(ρ(v)) is the cut point of p along γv(or cut point of γv with respectto p.

Remarks 7.17.

• If M is compact, then for any v ∈ TpM with |v| = 1, we have ρ(v) ≤ diamM , where

diamM := supp,q∈M

d(p, q).

By the Hopf-Rinow theorem, diamM <∞.• If we restrict to the unit sphere Sp := v ∈ TpM : |v| = 1, then one can show that the

map Sp → (0,+∞];v → ρ(v) is upper semi-continuous. If (M,g) is complete, then thismap is continuous7.

The question of whether this is the only way that a geodesic can cease to be minimising isanswered, negatively, by the following.

Proposition 7.18. Let (M,g) be a complete Riemannian manifold and c : [a, b] → M a min-imising geodesic from c(a) to c(b). If there exists a distinct geodesic from c(a) to c(b) of the samelength, then c is not minimal on any larger interval [a, b+ ε].

Proof. If c and γ are geodesics from c(a) to c(b) of the same length, then, given ε > 0, we definethe curve

ϕ(t) :=

γ(t) t ∈ [a, b]c(t) t ∈ (b, b+ ε]

.

This curve joins the points c(a) and c(b+ε). By the Hopf-Rinow theorem, there exists a minimisinggeodesic, s, from ϕ(a) to ϕ(b+ ε). Since ϕ is not a smooth geodesic (it is not smooth at ϕ(b)), itfollows from the results of first variation of arc-length that ϕ cannot be minimising. Therefore

L[s] < L[ϕ] = L[γ|[a,b]] + L[c|[b,b+ε]] = L[c|[a,b]] + L[c|[b,b+ε]] = L[c|[a,b+ε]].

Hence c is not minimising from c(a) to c(b+ ε).

Remark 7.19. Note that if c : [a, b] →M is minimising up to c(b), and there exists another geodesicγ : [a, b] →M from c(a) to c(b), then we know from Remark 7.16 that d(c(a), c(b)) = L[c|[a,b]], sowe must have L[γ] ≥ L[c].

We can deduce this result in an alternative fashion. If L[γ] < L[c], then we can define a curveϕ : [a, b+ ε] →M given by

ϕ(t) =

γ(t) t ∈ [a, b]c(b− (t− b)) t ∈ [b, b+ ε]

.

For sufficiently small ε > 0, this defines a curve from c(a) to c(b − ε) with length less thanL[c|[a,b−ε]], contradicting the fact that c is minimal up to c(b).

Lecture 10From now on, assume that (M,g) is a complete Riemannian manifold

Proposition 7.20. Let (M,g) be a complete Riemannian manifold, p ∈ M , and c : [0,∞) → Ma geodesic with c(0) = p and |c′(t)| = 1. Given T ∈ (0,∞), the point c(T ) is the cut point of palong c if and only if one (or both) of the following holds for t = T , and neither holds for anysmaller value of t:

(1) there exists a geodesic γ 6= c from p to c(T ) with L[γ] = L[c];(2) c(T ) is conjugate to p along c.

7See [Cha], Theorem III.2.1 for a proof.

24 JAMES D.E. GRANT

Proof. Firstly note that by Corollary 7.15 and Proposition 7.18, if either of the above conditionsholds at t = T , then c will not be minimising for t > T .

To prove the converse statement, let v := c′(0). Then c = γv : [0,∞) →M and is minimising fort ∈ [0, T ], but not minimising for t > T , where T ≡ ρ(v) <∞. Given any decreasing sequence εiwith εi → 0 as i→∞, the Hopf-Rinow theorem implies the existence of geodesics σi : [0, Li] →Mwith Li := d(p, γ(T + εi)) with the property that σi(0) = p, σi(Li) = γv(T + εi) and |σ′i(0)| = 1(i.e. σ′i(0) ∈ Sp). Since Sp is compact, there exists a convergent subsequence of σ′i(0), whichwe again denote by σ′i(0), with σ′i(0) → w ∈ Sp as i → ∞. We then have a limiting geodesic,σ := γw : [0,∞) →M . By continuity of the exponential map,

limi→∞

expp(Liσ′i(0)) = expp( lim

i→∞Liσ

′i(0)) = expp(Tw)

However, we also have

limi→∞

expp(Liσ′i(0)) = lim

i→∞σi(Li) = lim

i→∞c(T + εi) = c(T ) = γv(T ) = expp(Tv).

Hence v,w ∈ Sp and expp(Tw) = expp(Tv). There are now two possibilities:

• w 6= v,• w = v.

In the first case, γw and γv are distinct geodesics from p to γv(T ), both of length T .In the second case, we wish to show that DTv expp : Tv(TpM) → Tγv(T )M is singular. Assume,

to the contrary, that DTv expp is non-singular. Then the inverse function theorem implies thatexpp defines an injective mapping from a neighbourhood of Tv in TpM , U , to a neighbourhood,V , of γv(T ) in M . For sufficiently large i, we have σi(Li) ∈ V and σ′i(0) ∈ U , and therefore therelation

expp ((T + εi)v) = γv(T + εi) = σi(Li) = expp Li (σ′i(0))

implies that(T + εi)v = Liσ

′i(0).

Therefore, σ′i(0) = v and Li = T + εi. Hence

d(p, γv(T + εi)) = T + εi = L[γv|[0,T+εi]].

This contradicts the fact that γv is not minimising beyond the point γv(T ). Hence D expp issingular at Tv ∈ TpM , so γv(T ) is conjugate to p along γv.

8. Cut locus

LetUp := tv ∈ TpM : v ∈ Sp, t ∈ [0, ρ(v)) .

Remarks 8.1.

(1) Up is an open neighbourhood of 0 ∈ TpM .(2) expp : Up → expp(Up) is injective8. In particular, if q ∈ expp Up then there exists a unique

v ∈ Up with expp v = q and d(p, q) = |v| which, in our earlier notation, is r(q). Hencer(q) = d(p, q) for all q ∈ expp Up.

(3) expp : Up → expp(Up) is non-singular everywhere i.e. Dv expp is non-singular for allv ∈ Up. (Otherwise expp v is conjugate to p along γv and thus cannot lie in expp Up.)

8Proof : If q ∈ expp(Up) is the image of two distinct vectors, v,w, in Up, then there are two distinct geodesics

from p to q. If |v| 6= |w|, in which case one of these geodesics is non-minimising, which contradicts the fact thatboth v and w lie in Up. Otherwise, |v| = |w|, in which case there exist two distinct geodesics between p and q ofthe same length. However, in this case, the geodesics cannot be minimising after the point q, again contradictingthe fact that v, w lie in Up.


Definition 8.2. Let p ∈M . The cut-locus of p in TpM is the set

C(p) := ρ(v)v : v ∈ Sp such that ρ(v) <∞ ⊂ TpM.

The cut-locus of p is the set

Cut(p) := γv(ρ(v)) : v ∈ Sp such that ρ(v) <∞ ≡ expp C(p)

i.e. the union of the cut-points of p. The injectivity radius of (M,g) at p, inj(p), is

inj(p) = infv∈Sp

ρ(v).

Remark 8.3.

(1) Equivalently,

inj(p) = d(p,Cut(p))

= supR ∈ (0,∞] : v ∈ TpM : |v| < R ⊆ Up

where we define d(p,Cut(p)) = ∞ if Cut(p) is empty.(2) It can be shown that, for any p ∈M , the set Cut(p) is a subset of M of measure zero (see,

e.g., [GHL, §3.96]).

Given points p, q ∈ M , the Hopf-Rinow theorem implies the existence of a minimal geodesic,γv : [0,∞) → M with |v| = 1 such that γv(0) = p, γv(T ) = q. Since γv is minimising, it followsthat T ≤ ρ(v). Hence either T = ρ(v), in which case q ∈ Cut(p), or T < ρ(v), in which caseq ∈ expp Up.

We therefore have proved the first part of the following result:

Proposition 8.4. Let (M,g) be a complete Riemannian manifold. Let p ∈M . Then

M =(expp (Up)

)∪ Cut(p). (8.1)

The above union is disjoint.

Proof. To prove that the union is disjoint, let x ∈(expp (Up)

)∩ Cut(p). Since x ∈ expp (Up),

there exists a geodesic c with c(0) = p, c(1) = x that is minimal on the larger interval [0, 1 + ε].However, x ∈ Cut(p) implies that there exists a geodesic, γ, with γ(0) = p, γ(1) = x that is notminimal for t > 1. Since both of these geodesics are minimal, they must have the same length.Since c remains minimal after x whereas γ does not, then c and γ must be distinct geodesics fromp to x of the same length. However, Proposition 7.18 then tells us that c cannot be minimal afterthe point x, and we have a contradiction. Hence

(expp (Up)

)∩Cut(p) = ∅, and the union in (8.1)

is disjoint.

Remark 8.5. The above result implies that if we remove the point p from M , then the flow alongradial geodesics from p defines a map M \ p → Cut(p). This map turns out to be a retraction,and Cut(p) is a deformation retract of M \ p9. In the case where M is compact, the aboveresult implies that M is the disjoint union of an open ball in Rn and the continuous image ofan (n − 1)-sphere. Such information allows one to deduce various topological properties of M(e.g. certain homotopy groups) in terms of those of the cut locus Cut(p). (See, e.g., [K] for moreinformation on this topic.)

Lecture 11

Remark 8.6. Since the exponential map is smooth, it follows that the map expp : Up → expp(Up)is smooth10 Therefore, the exponential coordinates yi : expp(Up) → R introduced in Section 4 aresmooth. As such, the distance function r : expp(Up) → R; q → | exp−1

p q| ≡ d(p, q) will be smoothon expp(Up) \ p.

9See, e.g., [HY] for a definition of these terms, if they are unfamiliar.10In particular, the Hopf-Rinow theorem and definition of Up imply that we are in the domain where the geodesic

equations with initial conditions c(0) = p, c′(0) = v ∈ Up have a unique solution for t ∈ [0, ρ(v). As such, within

this domain, we can then deduce smooth dependence on initial conditions.

26 JAMES D.E. GRANT

Bishop-Gromov Comparison Theorem (Final Version). Let (M,g) be a complete Riemann-ian manifold of dimension n with

Ric(X,X) ≥ K (n− 1)g(X,X), ∀X ∈ X(M)

for some K ∈ R. Let VK(r) denote the volume of the ball of radius r in the complete, simply-connected space of constant curvature K and, given p ∈M , let Bg(p, r) := q ∈M : dg(p, q) < r.Then the map

r 7→ Vol (Bg(p, r))VK(r)

is non-increasing. In particular,Vol (Bg(p, r)) ≤ VK(r).

Proof. The proof follows that of Version 1, until we get to the point where we deduce that

a(r, θ)snK(r)

is non-increasing as a function of r. For fixed θ ∈ Sp := v ∈ TpM : |v| = 1, this holds up tor = ρ(v) i.e. up to the point where the geodesic γθ intersects the cut-locus of p. To extend theresult beyond the cut-locus, we define

a+(r, θ) :=

a(r, θ) 0 ≤ r ≤ ρ(θ),0 r > ρ(θ).

We then see that

the map r 7→ a+(r, θ)snK(r)

is non-increasing for r > 0, for each θ ∈ Sp.

With this definition, we see that

Vol (Bg(p, r)) =∫ r

0

∫Sn−1

1dvolg =∫ r

0

∫Sn−1

a+(t, θ)n−1dt ∧ dvolgSn−1 .

Equation (6.9) is now replaced by

d

dr

[Vol (Bg(p, r))

VK(r)

]=

1Vol (Sn−1)


)2 ∫ r

0

snK(t)n−1snK(r)n−1

×∫Sn−1

[(a+(r, θ)snK(r)

)n−1

−(a+(t, θ)snK(t)

)n−1]dθdt.

Since a+(t,θ)snK(t) is a non-increasing function of t, we have

d

dr

[Vol (Bg(p, r))

VK(r)

]≤ 0.

Remark 8.7. Where we said “complete, simply-connected space of constant curvature K” in thestatement of this theorem, we simply mean metrics gK and the corresponding manifolds that weintroduced in Section 111. Also,

VK(r) =∫ r

0

∫Sn−1

snK(t)n−1dθdt.

11The reason I stated the final version of the Bishop-Gromov result as above (which, admittedly, is inconsistentwith the way I have presented material previously) is that it is the way you are likely to find the result stated intextbooks.


Corollary 8.8 (Myers’s theorem). Let M be a complete Riemannian manifold of dimension n,such that the Ricci curvature of M obeys the inequality

Ric(X,X) ≥ (n− 1)Kg(X,X), ∀X ∈ X(M)

for some constant K > 0. Then the diameter of M satisfies diam (M) ≤ π/√K.

Proof. Assume, to the contrary, that diam (M) > π/√K. Then there exist points p, q ∈ M such

that d(p, q) > π/√K. Let r(x) := d(p, x), for x ∈ M . From the comparison results earlier, we

know that

∆r = (n− 1)∂ra

a≤ (n− 1)

∂rsnK(r)snK(r)

= (n− 1)√K cot

(√Kr), (8.2)

as long as x remains away from the cut-locus of p. Let c be a minimising geodesic from p to q,parametrised by arc-length (i.e. |c′| = 1). Since c is minimal, we know that c(π/

√K) is not in the

cut-locus of p, and hence r is smooth at that point. However, letting t→ π/√K from below, we

deduce from (8.2) that ∆r → ∞. This is a contradiction, since the left-hand-side must be finiteaway from the cut-locus. Hence diamM ≤ π/

√K.

Lecture 12

Corollary 8.9. Let M obey the conditions of Myers’s Theorem. Then M is compact.

Proof. By Theorem 8.8, given a point p ∈ M then M = q ∈ M : d(p, q) ≤ diamM, anddiamM ≤ π/

√K < ∞. Since this is a closed ball of finite radius in a complete manifold, the

Hopf-Rinow theorem implies that this ball, and hence M , is compact.

Rearranging this result a little, we obtain a result that anticipates some of the ideas inherentin the proof of the singularity theorems in Lorentzian geometry:

Corollary 8.10. Let M be a non-compact Riemannian manifold of dimension n, such that theRicci curvature of M obeys the inequality

Ric(X,X) ≥ (n− 1)Kg(X,X), ∀X ∈ X(M)

for some constant K > 0. Then M is incomplete.

28 JAMES D.E. GRANT

Part 3. Lorentzian geometry

9. Inner products of Lorentzian signature

In Riemannian geometry, one has an inner product on the tangent space, TpM , for each p ∈M .Each tangent space is isomorphic to Rn, and we may choose an orthonormal basis such that theinner product takes the form

〈v,w〉R = v1w1 + v2w2 + · · ·+ vnwn,

where (v1, . . . , vn), (w1, . . . , wn) are the components of the vectors v, w with respect to the chosenbasis. We then define a norm, ‖ · ‖R on Rn ∼= TpM , with the property that

‖v‖2R = 〈v,v〉R =

(v1)2

+(v2)2

+ · · ·+ (vn)2 .

For any r > 0, the surface v ∈ TpM : ‖v‖R = r is an (n − 1)-sphere and, in particular, iscompact.

On a Lorentzian manifold, we have an inner product of Lorentzian signature on each tangentspace. It is conventional to take the dimension of our manifold to be (n + 1), so TpM ∼= Rn+1,and to choose a basis e0, e1, . . . , en such that inner product takes the form

〈v,w〉L := −v0w0 + v1w1 + · · ·+ vnwn,

where (v0, v1, . . . , vn) are the components of a vector v ∈ TpM , etc. We also define the Lorentzian“norm-squared” of a vector by

‖v‖2L = 〈v,v〉L = −

(v0)2

+(v1)2

+(v2)2

+ · · ·+ (vn)2 .

In contrast to the Riemannian case, a non-zero vector v ∈ TpM may have norm-squared that isnon-positive. A non-zero vector v is defined to be

• space-like if ‖v‖2L > 0;

• time-like if ‖v‖2L < 0;

• null if ‖v‖2L = 0.

In physical terms, the 0-direction corresponds to the time direction. Bearing this in mind, we saythat a non-space-like vector12 v is future-directed if v0 > 0 and past-directed if v0 < 0.

In Lorentzian geometry, the level sets v ∈ Rn+1 with ‖v‖2L = a, for a ∈ R, are non-compact.

These are of different character, depending on the value of a:(1) a = +k2 > 0: In this case, we need v :

∑ni=1(v

i)2 = k2 + (v0)2, which is a non-compacthyperboloid. For n > 1, this set is connected.

(2) a = −k2 < 0: We require v : (v0)2 = k2 +∑ni=1(v

i)2. This set consists of two hyper-boloids, corresponding to whether v0 is positive or negative. Each of these hyperboloidsis connected, but non-compact.

(3) a = 0: We require v : (v0)2 =∑ni=1(v

i)2. If we include the zero vector, then this set isthe union of two cones, each with vertex at 0. This set is called the null cone/light cone(of 0). The subset with v0 > 0 is the future null/light cone, and the subset with v0 < 0 isthe past null/light cone. The light-cone and its future/past subsets are non-compact.

10. Lorentzian manifolds

A Lorentzian metric on a manifold M is a section, g, of T 02 (M) that is symmetric, non-

degenerate and defines an inner product of Lorentzian signature on each tangent space TxM ,for each x ∈ M . A Lorentzian manifold will mean a pair (M,g) consisting of a manifold M anda Lorentzian metric, g, on M .

Standard material from Riemannian geometry concerning the Levi-Civita connection, the cur-vature tensor and the Ricci tensor carry over, unchanged, to the case of Lorentzian manifolds. TheRiemannian concept of sectional curvature is not, generally, well-defined for Lorentzian metrics,and is infrequently used.

12i.e. time-like or null. We will also refer to such vectors as causal .


A vector field, X ∈ X(M), is defined as being time-like, space-like or null if Xp ∈ TpM istime-like, space-like or null with respect to the Lorentzian inner product induced on TpM , foreach p ∈ M . We will generally assume that our manifolds are time-orientable, in the sense thatit is possible to define, continuously, a division of non-space-like vectors into two classes, referredto as future-directed and past-directed. This is equivalent [O’N, Chapter 5, Lemma 32] to theexistence of a non-vanishing time-like vector field on M . Although the existence and uniquenesstheorem for geodesics carries over directly, we must now consider geodesics that are

• space-like i.e. ‖c′‖2 > 0;• time-like i.e. ‖c′‖2 < 0;• null i.e. ‖c′‖2 = 0

separately. In the case of non-space-like geodesics, we also characterise geodesics as future-directedand past-directed according to whether their tangent vectors are future or past-directed.

As general references on Lorentzian geometry, the book closest to the general approach that wewill take is [HE], although there are some points at which we will follow [O’N]. Lecture 13

10.1. Examples.

Example 10.1 (Minkowski space). M = R4, and we adopt local coordinates (t, x, y, z). It isconventional to also write these as x0 = t, x1 = x, x2 = y, x3 = z

g = −dt2 + dx2 + dy2 + dz2. (10.1)

Generally, we denote Rn+1 with a Lorentzian metric of the above form by Rn,1. More generallystill, we denote Rp+q with the flat metric of signature (p, q)

g =p∑a=1

(dxa)2 −q∑i=1

(dti)2 (10.2)

by Rp,q.If we adopt spherical coordinates, (r, θ, φ) in the spatial (i.e. x, y, z) directions, then the met-

ric (10.1) takes the formg = −dt2 + dr2 + r2gS2 ,

where gS2 denotes the standard round metric on the unit 2-sphere. More generally, the (n + 1)-dimensional Minkowski metric takes the form

g = −dt2 + dr2 + r2gSn−1 . (10.3)

Example 10.2 (de Sitter space). The (n+ 1)-dimensional de Sitter metric may be looked on asthe metric induced on the hyperboloid −t2 + r2 = 1, in Rn+1,1 with the metric (10.3). Notingthat r cannot be zero on the surface −t2 + r2 = 1, we may adopt new coordinates (T,R) with

r = R coshT, t = R sinhT,

on a neighbourhood of this surface, which is the set R = 1. In terms of which the Minkowskimetric takes the form

g = dR2 −R2dT 2 +R2 cosh2 TgSn

on this neighbourhood. The de Sitter metric is then the metric induced on the hyper-surfaceR = 1:

gdS = −dT 2 + cosh2 TgSn , T ∈ R.This metric is of constant curvature, in the sense that the (0, 4) version of the curvature tensor

takes the form

R(U,V,W,X) = K (g(U,W)g(V,X)− g(U,X)g(V,W)) , ∀U,V,W,X ∈ X(M)

with K = 1 in the current case. The Ricci tensor of g is then

Ric = Kng.

30 JAMES D.E. GRANT

Example 10.3 (Anti-de Sitter space). The corresponding metric with constant negative curva-ture, K = −1, is the anti-de Sitter metric. The (n + 1)-dimensional anti-de Sitter metric mayconstructed as the induced metric on a hyper-surface in Rn+2 with the flat metric of signature(n, 2). In particular, we write

g = −du2 − dv2 +n∑i=1

(dxi)2, u, v, xi ∈ R

= −(dT 2 + T 2dt2

)+ dR2 +R2gSn−1 , T,R ∈ [0,∞), t ∈ [0, 2π),

= −dT 2 + dR2 − T 2dt2 +R2gSn−1 , T,R ∈ [0,∞), t ∈ [0, 2π),

where u = T cos t, v = T sin t, R = |x|. The anti-de Sitter metric is then the metric induced onthe component of the submanifold −T 2 +R2 = −1. Rewriting the metric above with T = ρ cosh r,R = ρ sinh r with ρ ∈ [0,∞), r ∈ R, we have

g = −dρ2 + ρ2dr2 − ρ2 cosh2 r dt2 + ρ2 sinh2 r gSn−1 .

The metric induced on the surface ρ = 1 is then

gAdS = − cosh2 r dt2 + dr2 + sinh2 r gSn−1 .

Although the construction from Rn+2 leads to the coordinate t having period 2π, this leads to theabove metric having closed time-like curves. We avoid this problem by “de-identifying” the timevariable t, and taking the above metric to be defined for

t ∈ R, r ∈ R.

In this case, the topology of the manifold on which the anti-de-Sitter metric lies is simply Rn+1.

A feature of the anti-de Sitter metric that will be of interest to us is that it is geodesicallycomplete, but there exist points in it that are not connected by any geodesic. Consider, forsimplicity, the case n = 1, in which case we have the two-dimensional Lorentzian metric

gAdS = − cosh2 r dt2 + dr2, (t, r) ∈ R2.

As indicated above, this may be viewed as the induced metric on the hyperboloid Σ := (u, v, x) ∈R2,1 : −u2 − v2 + x2 = −1 in R1,2 with the metric g = −du2 − dv2 + dx2. The geodesics of theanti-de Sitter metric are then the intersection of the hyperboloid Σ with planes in R1,2 that passthrough the origin. In particular, letting x := (u, v, x) ∈ R1,2 be the position vector in R1,2 thena plane through the origin is the set of points x that satisfy an equation of the form

g(a,x) = −au− bv + cx = 0,

where a = (a, b, c) is a (constant) vector in R1,2. A geodesic on Σ (viewed simply as a set, ratherthan a parametrised curve c : R → Σ) is then the intersection

x ∈ R1,2 : g(x,x) = −1,g(a,x) = 0.In terms of the coordinates introduced above, this implies that we need

au+ bv = cx

=⇒ aT cos t+ bT sin t = cR

=⇒ aρ cosh r cos t+ bρ cosh r sin t = cρ sinh r.

Since Σ = ρ = 1, the geodesics on Σ are lines of the form

a cosh r cos t+ b cosh r sin t = c sinh r

i.e.a cos t+ b sin t = c tanh r,

where a, b, c are real constants. If we consider geodesics through the point t = r = 0, then wededuce that we must have a = 0, and therefore

b sin t = c tanh r.Lecture 14


Case 1: c = 0 In this case, the geodesic equations, ∇vv = 0 imply that drds = const, where s is

an affine parameter along the geodesic. Parametrising by arclength implies that the constant is±1, in which case the geodesic curves take the form

t(s) = 0, r(s) = ±s.

As such, the geodesics simply travel along the r axis in the positive and negative directions.

Case 2: c 6= 0 For c 6= 0, we then deduce that the geodesic takes the form

r(t) = tanh−1 (λ sin t) , (10.4)

where λ := b/c ∈ R. If we now consider t as a parameter on the geodesic (i.e. view the geodesicas a map c : t 7→ (t, r(t)) for t ∈ I, where I is some sub-interval of R), then we find that∥∥∥∥dxdt

∥∥∥∥2

≡(dr

dt

)2

− cosh2 r = · · · = λ2 − 1(1− λ2 sin2 t)2

.

(Note that the right-hand-side of this expression is not constant, implying that t is not an affineparameter along the geodesics. In particular, if we let v = dx

dt , then ∇vv is equal to a non-zero multiple of v, rather than 0. We can, however, find a reparametrisation of the curve ass 7→ (t(s), r(x)) for which v = dx

ds satisfies ∇evv = 0.) There are now three separate cases toconsider

Case 2a: λ > 1 In this case, the geodesics have∥∥dxdt

∥∥2> 0, and so are space-like. From (10.4),

we see that r → ∞ as t → arcsinλ−1. As such, the space-like geodesics reach space-like infinityat a finite value of t. The affine distance along such a curve,

∫ ∥∥dxdt

∥∥ dt, is, however, infinite.

Case 2b: λ = 1 In this case, the geodesics have∥∥dxdt

∥∥2= 0, and so are null. From (10.4), we see

that r →∞ as t→ π2 .

Case 2a: λ < 1 In this case, the geodesics have∥∥dxdt

∥∥2< 0, and so are time-like. From (10.4), we

see that r(t+ 2π) = r(t) and r(t+ π) = −r(t). Therefore all of the time-like geodesics starting att = r = 0 pass through the points t = π, r = 0 and t = 2π, r = 0.

A consequence of this discussion is that, if we consider any point t = π, r 6= 0, then thereexists no geodesic between the point t = 0, r = 0 and that point. More generally, if we considerany points contained in the region enclosed by the null geodesics that pass through the pointt = π, r = 0 (i.e. points on any space-like geodesic through that point) then there is no geodesicconnecting the point t = 0, r = 0 to that point.

Since anti-de Sitter space is geodesically complete, this implies that the corollary of the Hopf-Rinow theorem that states that, for Riemannian manifolds, completeness implies the existence ofminimising geodesics between points, is strictly false in the Lorentzian case.

Definition 10.4. For sets A,U ⊆ M we define the chronological future of A relative to U to bethe set

I+(A,U) := q ∈ U : ∃p ∈ A and a future-directed time-like curve from p to q in U

and the causal future of A relative to U

J+(A,U) := (A ∩ U) ∪ q ∈ U : ∃p ∈ A and a future-directed causal curve from p to q in U .

When U = M , we write I+(A) and J+(A), respectively.Similarly, we define I−(A,U), J−(A,U).

Remarks 10.5.

(1) By a curve we mean one of non-zero extent, so we can have S * I+(S,U).(2) A case that will be of particular interest to us is when the set A consists of a point p ∈M .

In this case, we use the notation I±(p, U), J±(p, U), I±(p), J±(p).(3) We will see later that the set I±(A) is open. The set J±(A) need not be closed, though.

32 JAMES D.E. GRANT

(4) In the case of anti-de Sitter space, if we take the point p := t = r = 0 and q := t =π, r = 0, then the set J+(p) ∩ J−(q) is non-compact. The fact that there exist points inM that are time-like separated that cannot be joined by a time-like geodesic turns out tobe related to this fact.

Lecture 15

Example 10.6 (The Schwarzschild solution). In physics, this is the metric that describes a non-rotating, un-charged black hole. It depends on a parameter m ∈ R which, asymptotically, can beidentified with the Newtonian mass. Therefore, as far as physics is concerned, it is best to assumethat m ≥ 0, although the solution does make sense for m < 0.

Let (t, r, θ, φ) be coordinates on R4, with (r, θ, φ) the usual spherical polar coordinates on R3.The exterior Schwarzschild solution is defined on the subset r > 2m of R4, which is topologicallyR2 × S2. In this region, the metric takes the form

g = −(

1− 2mr

)dt2 +

(1− 2m

r

)−1

dr2 + r2(dθ2 + sin2 θdφ2

),

This metric becomes badly-behaved as r → 2m, but this is simply due to the fact that (t, r) arenot a good choice of coordinates on this region. We may rewrite the metric in the form

g = −(

1− 2mr

)[dt−

(1− 2m

r

)−1

dr

][dt+

(1− 2m

r

)−1

dr

]+ r2

(dθ2 + sin2 θdφ2

),

and define a new coordinate, u ∈ R, by

du = dt+(

1− 2mr

)−1

dr = · · · = d[t+ r + log |r − 2m|],

sou = t+ r + log |r − 2m|+ const.

Rewriting the metric in terms of the coordinates (u, r, θ, φ), it then takes the form

g = −(

1− 2mr

)[du− 2

(1− 2m

r

)−1

dr

]du+ r2

(dθ2 + sin2 θdφ2

),

= −(

1− 2mr

)du2 + 2drdu+ r2

(dθ2 + sin2 θdφ2

). (10.5)

We may use this expression to define a new metric, g′, given by the above coordinate expressionon the manifold, M ′, defined by

u ∈ R, r ∈ (0,∞), (θ, φ) ∈ S2.

On restriction to the original manifold M , where r > 2m, the metric g′|M is isometric to theoriginal metric g.

Definition 10.7. In general, a Lorentzian manifold (M ′,g′) is said to be an extension of aLorentzian manifold (M,g) if there exists an isometric embedding i : M → M ′. A Lorentzianmanifold (M,g) for which any extension (M ′,g′) satisfies i(M) = M ′ is said to be inextendible.

Remark 10.8. It turns out that the extension, (M ′,g′) of the external Schwarzschild solution is notinextendible. See [HE, Chapter 5.5] for the full extension of the external Schwarzschild solution.

Calculating the norm-squared of the curvature tensor

|R|2 :=∑i,j,k,l

R(ei, ej , ek, el)R(ei, ej , ek, el),

where ei is an orthonormal basis:

g′(ei, ej) =

−1 i = j = 0,1 i = j = 1, 2, 3,0 otherwise,


then we find that

|R|2 = constant× m2

r6.

Therefore, the curvature of the Schwarzschild solution is singular at r = 0. As such, we cannotextend the metric in a smooth way to include the set r = 0.

In order to determine whether the Schwarzschild solution is geodesically complete or not, weconsider time-like geodesics that are ingoing (i.e. dr

ds ≤ 0, where s is an affine parameter alongthe geodesic) and radial (i.e. dθ

ds = dφds = 0, where (θ, φ) are coordinates on S2). The geodesic

equations then imply that(1− 2m

r

)= E,

−1 = − E2(1− 2m

r

) +(

1− 2mr

)−1(dr

ds

)2

,

where E is a constant. The second equation implies that(dr

ds

)2

= (E2 − 1) +2mr.

In the case E2 − 1 > 0, this implies that

dr

ds= −

((E2 − 1) +

2mr

)1/2

.

Therefore, letting r(s = 0) := r0 > 0, we have

s = −∫ r(s)

r0

√rdr(

(E2 − 1) + 2mr

)1/2 .Since the denominator is greater than or equal to

(E2 − 1

)1/2 and hence bounded away from zero,and the numerator is bounded, it follows that r(s0) = 0, where

s0 = −∫ 0

r0

√rdr(

(E2 − 1) + 2mr

)1/2 <∞.

As such, the radially ingoing geodesic c : [0, s0) → M ′ cannot be extended (continuously) to theinterval [0, s0], and is therefore incomplete.

As such, the Schwarzschild solution contains incomplete time-like geodesics and, hence, isgeodesically incomplete.

For more information concerning global properties of several important exact solutions of theEinstein equations, see [HE, Chapter 5].

11. GeodesicsLecture 16

Proposition 11.1. Let p ∈M and v ∈ TpM . Then there exists a unique geodesic γv in M suchthat

(1) the initial velocity of γv is v; i.e. ddtγv(t)

∣∣t=0

= v;(2) the domain Iv of γv is the largest possible. Hence if α : J → M is a geodesic with initial

velocity v then J ⊆ Iv and α = γv|J .The geodesic γv is said to be maximal or geodesically inextendible.

Definition 11.2. Let p ∈ M . We define Dp ⊆ TpM to be the set of vectors v ∈ TpM such thatthe inextendible geodesic γv is defined at least on the interval [0, 1]. The exponential map of Mat p is the map

expp : Dp →M, v 7→ γv(1).

34 JAMES D.E. GRANT

Gauss Lemma. Let p ∈M , x ∈ Dp \ 0. Let vx,wx ∈ Tx (TpM) with vx radial. Then

gp(vx,wx) = gexpp x

((Dx expp

)(vx) ,

(Dx expp

)(wx)

).

Proposition 11.3. Let (M,g) be a Lorentzian manifold. For each p ∈M there exists a neighbour-hood, Up, of 0 ∈ TpM and a neighbourhood, Vp, of p in M such that the map expp

∣∣Up

: Up → Vp

is a diffeomorphism.

Definition 11.4. A subset S of a vector space is star-shaped about 0 if v ∈ S implies tv ∈ Sfor all t ∈ [0, 1]. If Up and Vp are as above and Up is star-shaped, then Vp is called a normalneighbourhood of p. An open set C ⊆ M is convex if C is a normal neighbourhood of each of itspoints.

We wish to consider smooth causal (i.e. time-like and null) curves in a convex normal neigh-bourhood Vp of M with expp : Up → Vp as above. (Most of our considerations may be generalisedto the case of piece-wise smooth curves where, if there is a discontinuity in the tangent vector tat time t, then we require that

g(t(t−), t(t+)) < 0,so that the tangent vector lies in the same half of the null cone (i.e. both are future-directed, orboth are past-directed).)

The following result is, essentially, a corollary to the Gauss Lemma:

Lemma 11.5. Let Vp be a convex normal neighbourhood of p ∈ M . The time-like geodesicsthrough p are orthogonal to the surfaces of constant σ (with σ < 0) where, for q ∈ Vp, σ(q) :=gp(exp−1

p q, exp−1p q).

Sketch of Proof. This is similar to the proof of the Gauss Lemma. Let X(s), s ∈ (−ε, ε) be a smoothcurve in TpM with g(X(s),X(s)) = −1, and α(t, s) := expp(tX(s)). Then σ(α(t, s)) = −t2. Weneed to show that g(T,V) = 0. However,

Tg(T,V) = ∇Tg(T,V) = g(T,∇TV) =12∇Vg(T,T) = 0.

Remark 11.6. This implies that the surfaces σ = constant, where constant < 0, are space-like. Ahyper-surface with normal vector n is

• space-like if its normal is time-like (i.e. g(n,n) < 0);• time-like if its normal is spac-like (i.e. g(n,n) > 0);• null if its normal is null (i.e. g(n,n) = 0).

Example 11.7. Consider the surfaces r = constant in the Schwarzschild solution in the coordinatesystem (u, r, θ, φ). The normal one-form to the surface r = constant can be taken as σ = dr, forwhich the corresponding normal vector field is n =

(1− 2m

r

)∂r + ∂u (i.e. g′(∂u, ·) = dr). Then

g(n,n) = 〈dr,n〉 =(1− 2m

r

). Therefore the surface r = r0 is time-like for r0 > 2m, space-like for

r0 < 2m and null for r0 = 2m.

For a convex normal neighbourhood Vp ⊆ M the intrinsic causality is the same as that ofMinkowski space. In particular, we have

Proposition 11.8. Let Vp be a convex normal neighbourhood of p ∈M . Then the points that canbe reached from p by time-like curves in Vp are those of the form expp v where v ∈ TpM obeysg(v,v) < 0 .

Sketch of Proof. Let Cp ⊂ TpM denote the set of time-like vectors at p. (i.e. the interior of thenull cone in TpM with vertex 0 ∈ TpM .) Let q ∈ Vp be a point that can be reached from p by atime-like curve, γ, with γ(0) = p. Let γ := exp−1

p γ be the corresponding curve in TpM . Thenγ′(0) ∈ TpM , γ′(0) ∈ T0(TpM). Identifying the tangent spaces Tx(TpM) with TpM in the usualway and using the fact that D0 expp is then the identity map, we have

γ′(0) = γ′(0).


Therefore, γ′(0) is a time-like vector at 0 ∈ TpM . Hence the curve γ enters the set Cp.Lecture 17Note that

(expp Cp

)∩Vp is the subset of Vp consisting of points r for which σ(r) < 0. Moreover,

the surfaces σ = constant are spacelike. Since γ is time-like, it therefore cannot be tangent tothe sets σ = constant, so σ must decrease monotonically along γ. Therefore σ(q) < 0, and henceq ∈ expp Cp. Therefore there exists v ∈ Cp such that q = expp v.

A limiting argument gives the following, stronger result:

Proposition 11.9. Let Vp be a convex normal neighbourhood of p ∈ M . Then the points thatcan be reached from p by time-like (resp. causal) curves in Vp are those of the form expp v wherev ∈ TpM obeys g(v,v) < 0 (resp. g(v,v) ≤ 0).

Sketch of Proof. The only case not covered by the previous result is where q ∈ Vp can be reachedfrom p by a causal curve, but no time-like curve. In this case, we perturb the causal curve (λ,say), and define an approximating family of time-like curves λi with endpoints p and qi, whereqi → q as i → ∞. Using continuous dependence of solutions of the geodesic equations on initialconditions (or just continuity of the exponential map), it then follows that q ∈ expCp = expCp.Since Cp = v ∈ TpM : g(v,v) ≤ 0, we are done.

Remark 11.10. In particular, if q ∈ Vp can be reached from p by a causal curve but not by anytime-like curve, then we must have q = expp v where v ∈ TpM is null (i.e. g(v,v) = 0). In thiscase q = expp v lies on a null geodesic t 7→ expp(tv) through p. To summarise, any point in Vpthat can be reached from p by a causal curve but not by any time-like curve must lie on a nullgeodesic through p.

12. Lengths of causal curves

Definition 12.1. Given a causal curve, c : [a, b] → M , from p to q, we define the length (orarc-length) of c to be

L[c] :=∫ b

a

(−g(c′(t), c′(t)))1/2 dt. (12.1)

(If the curve is only piece-wise smooth, we take the integral over the differentiable sections of thecurve.)

Remark 12.2. More generally, given a piecewise smooth curve c : [a, b] → M , we can define thearc-length of c to be

L[c] :=∫ b

a

|c′(t)| dt.

where |c′(t)| := |g(c′(t), c′(t))|. If we restrict to curves that are either (piecewise) causal (i.e.g(c′(t), c′(t)) ≤ 0 for all t ∈ [a, b]) or (piecewise) spacelike (i.e. g(c′(t), c′(t)) > 0 for all t ∈ [a, b]),then we can define |c′(t)| := (εg(c′(t), c′(t)))1/2, where ε = signg(c′(t), c′(t)).

Since we are interested only in the causal case, we simply work with the the definition (12.1).

Example 12.3. In flat Minkowski space, M := Rn,1, let p = t = 0, x1 = · · · = xn = 0, q =t = 1, x1 = · · · = xn = 0. Then there exists a timelike geodesic c : [0, 1] →M : s→ (s, 0, . . . , 0)from p to q of length

L[c] =∫ 1

0

√−1 · −1dt = 1.

If we allow piecewise smooth curves, then there exists a null curve also joining p to q of the form

c(s) =

(s, s, 0, . . . , 0) s ∈ [0, 1

2 ],(s, 1− s, 0, . . . , 0) s ∈ ( 1

2 , 1].

In this case, since the curve is null, its length is

L[c] =∫ 1/2

0

√−1 · 0dt+

∫ 1

1/2

√−1 · 0 = 0 < L[c].

36 JAMES D.E. GRANT

Therefore, unlike in the Riemannian case, geodesics are generally not curves of minimal length inLorentzian geometry. The following proposition shows that essentially the opposite is true.

Proposition 12.4. Let p ∈ M and q lie in a convex normal neighbourhood Vp ⊆ M of p. Then,if p and q can be joined by a timelike curve in Vp, the longest such curve is the (unique) timelikegeodesic from p to q in Vp.

Proof. Let q ∈ Vp be such that there exists a timelike curve from p to q. By the previous result,q = expp v with v ∈ TpM timelike, and the length of this curve is

L[γv] =∫ 1

0

√−g(γ′v, γ′v)dt =

√−g(exp−1

p (q), exp−1p (q)) =

√−σ(q).

Now let λ : [α, β] → Vp; s 7→ λ(s) be any timelike curve from p to q that lies in Vp, with length

L[λ] =∫ β

α

√−g(λ′(s), λ′(s))ds

Given X : [α, β] → TpM with g(X(s),X(s)) = −1, we define

α(t, s) := expp (tX(s)) .

Then, given the curve λ, there exists such an X and a function s 7→ f(s) such that

λ(s) = α(f(s), s).

We then haveλ′(s) = f ′(s)T + V.

Since T is tangent to time-like geodesics, and V is tangent to surfaces σ = constant, it fol-lows from Lemma 11.5 that g(T,V) = 0. Moreover, the surfaces σ = constant are space-like, sog(V,V) ≥ 0, with equality if and only if V = 0. Finally, T is tangent to (affinely-parametrised)geodesics, therefore g(T,T) = gp(X(s),X(s)) = −1. Putting these relations together, we there-fore find that

−g(λ′, λ′) = −g(f ′(s)T + V, f ′(s)T + V)

= −f ′(s)2g(T,T)− g(V,V)

= −g(V,V) + f ′(s)2

≤ f ′(s)2,

with equality if and only if V = 0. Hence

L[λ] =∫ β

α

√−g(λ′(s), λ′(s))ds ≤

∫ β

α

f ′(s)ds = f(β)− f(α).

However,σ(λ(s)) = σ(expp(f(s)X(s))) = ‖f(s)X(s)‖2 = −f(s)2,

so f(s) =√−σ(λ(s)). Hence f(α) = 0 and f(β) =

√−σ(q), so

L[λ] ≤√−σ(q) = L[γv].

As remarked above, for equality we require V = 0. In this case, λ is also a geodesic. Sinceexpp : Up → Vp is a diffeomorphism, the geodesic from p to q lying in Vp is unique (i.e. it is γv),and therefore, up to reparametrisation, λ must coincide with γv.

Lecture 18

Corollary 12.5. Let p ∈M and q lie in a convex normal neighbourhood Vp ⊆M of p. Then, if pand q can be joined by a causal curve in Vp, the longest such curve is the (unique) causal geodesicfrom p to q in Vp.

Proof. The previous result covers the case when p and q can be joined by a timelike curve. If pand q can be joined by a causal curve, but not any timelike curve, then q lies on a (unique) nullgeodesic, c, through p. Then c is the only causal curve from p to q, so L[c] = 0 is maximal.


13. Variation of arc-length

Since the initial results here are, up to some minus signs, the same as in the Riemannian case,we simply give a brief summary of how these results are adapted to the Lorentzian setting.

The notation is as before: given a smooth time-like curve c : [a, b] → M we define the (one-parameter) smooth variation α : [a, b] × (−ε, ε) → M where we now assume that the curvescs : [a, b] →M ; t→ cs(t) := α(t, s) are time-like. We define the vector fields along α:

T := (Dα)(∂

∂t

), V := (Dα)

(∂

∂s

),

in terms of which we now require that g(T,T) < 0, for all (t, s) ∈ [a, b]× (−ε, ε).

13.1. First variation of arc-length.

Lemma 13.1 (First variation formula (Lorentzian)). Let c : [a, b] → M be a smooth, timelikecurve segment, parametrised by arc-length (i.e. |c′| ≡

√−g(c′(t), c′(t)) = 1, for all t ∈ [a, b]). Let

α : [a, b]× (−ε, ε) →M be a smooth variation of α as above. Then

d

dsL[cs]

∣∣∣∣s=0

= − g(t,v)∣∣∣ba

+∫ b

a

g(∇tt,v) dt.

Proof. As in Riemannian case, but remember the minus sign in |T|.

Corollary 13.2. A smooth time-like curve c : [a, b] → M of constant speed |c′| = 1 is a criticalpoint of the arc-length functional under fixed end-point variations if and only if it is a geodesic.

We may also consider a timelike curve c : [a, b] → M from a spacelike three-surface S to apoint p. In this case, we consider variations with the property that α(a, s) ∈ S, α(b, s) = p for alls ∈ (−ε, ε). Therefore the variation vector field v(a) lies in Tc(a)S.

Corollary 13.3. Let S be a space-like submanifold of M without boundary and p ∈ M \ S suchthat there exists a time-like curve from S to p. A necessary condition for a curve c : [a, b] → Mwith c(a) ∈ S, c(b) = p to be the longest time-like curve from S to p is that c is a time-like geodesic,and c′(a) is orthogonal to Tc(a)S.

Proof. The proof is essentially the same as that of Proposition 3.7 in the Riemannian case. In theLorentzian case, let x ∈ Tc(a)S with gc(a)(c′(a),x) > 0. The corresponding variation has

d

dsL[cs]

∣∣∣∣s=0

= g(c′(a),x) > 0.

Hence, for sufficiently small s > 0, we have L[cs] > L[c], so c cannot be maximising.

13.2. Second variation of arc-length. As before, for a two-parameter smooth variation of c,α : [a, b]× (−δ, δ)× (−ε, ε) →M , we define the vector fields along α:

T := Dα

(∂

∂t

), V := Dα

(∂

∂r

), W := Dα

(∂

∂s

).

We still assume that the variation is time-like i.e. g(T,T) < 0, for all (t, r, s) ∈ [a, b] × (−δ, δ) ×(−ε, ε).

Lemma 13.4. Let c : [a, b] → M be a time-like geodesic parametrised by arc-length (i.e. |c′| ≡√−g(c′(t), c′(t)) = 1, for all t ∈ [a, b]). If α is a variation of c as above then

∂2L

∂r∂s

∣∣∣∣r=s=0

= − g(t,∇wv)|ba −∫ b

a

[g((∇tv)⊥ , (∇tw)⊥) + g(t,R(w, t)v)

]dt,

= − g(t,∇vw)|ba −∫ b

a

[g((∇tv)⊥ , (∇tw)⊥) + g(t,R(v, t)w)

]dt,

38 JAMES D.E. GRANT

where, given any vector field X along c, we define

X⊥ := X− g(X, t)g(t, t)

t = X + g(X, t)t,

the component of X orthogonal to t ≡ c′.

Remark 13.5. Note that

(∇tv)⊥ = ∇tv −g(∇tv, t)g(t, t)

t = ∇t

(v − g(v, t)

g(t, t)t)

= ∇t

(v⊥).

Therefore the integrand in the second variation depends only on v⊥ and w⊥, the components ofv and w orthogonal to t.

As in the Riemannian case, we define the index form by

I(v,w) := −∫ b

a

[g(∇tv,∇tw) + g(t,R(v, t)w)] dt =∫ b

a

[g(v,∇t∇tw + R(w, t)t)] dt

for v,w ∈ Γ0(c, TM) :=X ∈ Γ(c, TM)

∣∣X(a) = 0,X(b) = 0,gc(t)(X(t), c′(t)) = 0,∀t ∈ [a, b].

As in the Riemannian case, the bilinear form I : Γ0(c, TM)×Γ0(c, TM) → R is degenerate if andonly if there exists a Jacobi field along c that vanishes at the endpoints of c.

Definition 13.6. A timelike geodesic, c, from p to q is maximal if the index form is negativesemi-definite (i.e. I(v,v) ≤ 0, for all v ∈ Γ0(c, TM)).

Remarks 13.7.

(1) Equivalently, c is maximal if d2

ds2L[cs]∣∣∣s=0

≤ 0 for any variation, α, of c.

(2) A geodesic, c, is not maximal if there exists a small variation of c that yields a longercurve between p and q.

Proposition 13.8. A timelike geodesic curve, c, from p to q is maximal if and only if there is nopoint conjugate to p along c between p and q.

Proof. In the Riemannian case, we showed that if there exists a point along c, before q, that isconjugate to p along c, then the index form at q is not positive semi-definite (cf. Proposition 7.14).Adapting the same proof to the Lorentzian case, we deduce that if c : [a, b] → M with c(a) = p,c(b) = q and there exists t0 ∈ (a, b) such that c(t0) conjugate to p along c, then the index format q is not negative semi-definite. Taking the contra-positive, if the index form at q is negativesemi-definite (i.e. c is maximal) then there exists no point along c conjugate to p before q.

In the reverse direction, we first prove that if there are no points conjugate to p up to andincluding q, then the index form at q is negative definite.

Therefore, assume that any (non-trivial) Jacobi field along c that is zero at c(a) is non-zeroat c(t), for t ≤ b. Let e1, . . . , en be a basis for the orthogonal complement, c′(a)⊥, of c′(a) inTc(a)M . We then define Ei to be the Jacobi fields along c with initial conditions

Ei(a) = 0, E′i(a) = ei.

(′ will denote ∇t, where t := c′ is the tangent vector to c.) Since these are (non-trivial) Jacobifields along c that vanish at p, they are then non-zero at c(t) for all t ∈ (a, b].

Claim 1: g(t,Ei) = 0 along c.

Proof. Taking derivatives, we have

∇2tg(t,Ei) = g(t,∇2

tEi) = −g(t,R(Ei, t)t) = 0.

Therefore ∇tg(t,Ei) = g(t,∇tEi) = constant = 0, since g(t,∇tEi) = 0 at c(a). Thereforeg(t,Ei) is constant. Since Ei(a) = 0, we have g(t,Ei) = 0 along c


Hence the Ei are orthogonal to c′ along c. As such Ei form a basis for vector fields along c thatvanish at p and are orthogonal to c′. Given any vector field, V ∈ Γ0(c, TM), there exist functionsf i (with f i(q) = 0) with the property that

V =n∑i=1

f iEi.

To calculate I(V,V), we need to calculate

g(∇tV,∇tV) + g(t,R(V, t)V).

We write∇tV = A + B,

whereA :=

∑(f i)′

Ei, B :=∑

f iE′i.

Claim 2:g(∇tEi,Ej) = g(Ei,∇tEj)

along c.

Proof.

∇t (g(∇tEi,Ej)− g(Ei,∇tEj)) = −g(R(Ei, t)t,Ej) + g(R(Ej , t)t,Ei) = 0.

Therefore g(∇tEi,Ej) − g(Ei,∇tEj) is constant along c. Since Ei vanishes at p, the resultfollows.

Then

g(B,B) =∑i,j

f if jg(E′i,E

′j)

=∑i,j

f if j (g(E′i,Ej)

′ − g(E′′i ,Ej))

=

∑i,j

f if jg(E′i,Ej)

′

−∑i,j

(f if j

)′g(E′

i,Ej) +∑i,j

f if jg(R(Ei, t)t,Ej)

= (g(B,V))′ −∑i,j

((f i)′f jg(Ei,E′

j) + f i(f j)′g(E′i,Ej)

)+ g(R(V, t)t,V)

= (g(B,V))′ − 2g(A,B) + g(R(V, t)t,V),

where we have used Claim 2 in the fourth equality. Lecture 19Hence

g(∇tV,∇tV) + g(t,R(V, t)V) = g(A,A) + 2g(A,B) + g(B,B) + g(t,R(V, t)V)

= g(A,A) + (g(B,V))′ .

We therefore have

I(V,V) = − g(B,V)|ba −∫

g(A,A)dt = −∫

g(A,A)dt,

since V(p) = V(q) = 0. Since A is space-like, it follows that I(V,V) ≤ 0. For equality, we requireA = 0, which implies f i are constant, hence zero. Therefore I(V,V) ≤ 0 with equality if andonly if V = 0. Therefore I is negative definite.

Finally, if q is the first conjugate point to p along c, then the above implies that the index formon the interval [a, t] is negative definite for all t ∈ (a, b). Given V ∈ Γ0(c, TM) and a sequenceεi > 0 with εi → 0, we construct the vector field Vi along the curve c by defining

Vi(t) :=

(b−εi)−tb−t V(t) t ∈ [a, b− εi],

0 t ∈ (b− εi, b].

40 JAMES D.E. GRANT

Then I(Vi,Vi) < 0 and converge to I(V,V). Hence I(V,V) ≤ 0 for all V ∈ Γ0(c, TM), so I isnegative semi-definite.

Let S ⊂ M be a space-like hyper-surface (i.e. submanifold of codimension 1.) We will denoteits unit (time-like) normal vector field by n ∈ Γ(S, TM), with g(n,n) = −1. Given vector fieldstangent to S (i.e. orthogonal to n), we define the second-fundamental form/extrinsic curvature13

χ(v,w) := g(∇vn,w).

Since S is a submanifold, it follows that [v,w] is tangent to S which, in turn, implies that χ issymmetric. We also define the trace of χ:

κ := trχ =n∑i=1

χ(ei, ei),

where ei is any orthonormal basis for TpS.Given that a geodesic c from space-like hypersurface S to point q is a critical point of the

arc-length functional if (and only if) if it is normal to S (i.e. c′(a) is orthogonal to Tc(a)S), weonly consider the second variation for such geodesics.

Lemma 13.9. Let c : [a, b] → M be a time-like geodesic, parametrised by arc-length, normalto spacelike three-surface S. If α is a variation of c with α(a, r, s) ∈ S, α(b, r, s) = c(b) for all(r, s) ∈ (−δ, δ)× (−ε, ε) and the variation vectors v, w orthogonal to t ≡ c′, then

∂2L

∂r∂s

∣∣∣∣r=s=0

= −χ(v(a),w(a))−∫ b

a

[g(∇tv,∇tw) + g(t,R(v, t)w)] dt,

where χ is the second fundamental form of S.

Proof. We simply need to calculate the endpoint term in the second variation formula. We have

− g(t,∇vw)|ba = g(t,∇vw)|a = ∇v(a)g(t(a),w(a))− g(∇v(a)t(a),w(a)) = 0− χ(v(a),w(a))

= −χ(v(a),w(a)),

as required.

Definition 13.10. We define the index form by

IS(v,w) := −χ(v(a),w(a))−∫ b

a

[g(∇tv,∇tw) + g(t,R(v, t)w)] dt

for v,w ∈ ΓS0 (c, TM) :=X ∈ Γ(c, TM)

∣∣X(b) = 0,gc(t)(X(t), c′(t)) = 0,∀t ∈ [a, b].

Definition 13.11. A timelike geodesic curve from S to p normal to S is maximal if IS(v,v) ≤ 0,for all v ∈ ΓS0 (c, TM). (Equivalently, d2L

ds2

∣∣∣s=0

≤ 0 for all variations with cs(a) ∈ S, c(b) = q.)

Given a geodesic, c : [a, b] → M , with c(a) ∈ S, c(b) = q, we wish to consider variations, α,of c with initial endpoints in S. As such, we restrict to variations with V(a, 0, 0) tangent to S.Moreover, we wish to have T(a, r, s) = n(α(a, r, s)) the unit normal to S. Since ∇TV = ∇VT,this implies that we require

∇TV|r=s=0,t=a = v′(a) = ∇VT|r=s=0,t=a = ∇v(a)n.

Hence, given any tangential vector w ∈ Tc(a)S, we have

g(w,v′(a)) = g(w,∇v(a)n) = χ(w,v(a)). (13.1)

As such, if we are considering variations through geodesics normal to S, then the natural boundaryconditions to impose are v(a) ∈ Tc(a)S and (13.1).

13In more general contexts, one defines the second fundamental form of a submanifold S ⊂ M to be the mapII : X(S)× X(S) → X(S)⊥ defined by

II(v,w) = normal component of ∇vw.

In the current context, this carries the same information as our object χ.


Definition 13.12. A S-Jacobi field along normal geodesic c is a Jacobi field along c, J, with(1) J(a) is tangent to S;(2) g(w,J′(a)) = χ(w,J(a)) for all w ∈ Tc(a)S.

A point c(t), for t ∈ (a, b], is conjugate to S along c (or a focal point of S along c) if there existsa non-zero S-Jacobi field, J, along c with J(t) = 0.

Proposition 13.13. A timelike normal geodesic, c, from S to q is maximal if and only if thereis no point conjugate to S along c before q.

Proof. As for 13.8. Lecture 20

Proposition 13.14. Let (M,g) be a Lorentzian manifold such that Ric(V,V) ≥ 0 for all time-like vector fields V. Let S be a spacelike hypersurface in M and p ∈ S. Let c be the geodesicthrough p with c(0) = p, c′(0) = n(p). Then, if κ(p) < 0, there will be a point conjugate to S alongc within a distance b(p) := −n/κ(p) of S, provided that the geodesic c can be extended that far.

Remark 13.15. From now on, we will often explicitly mention that results hold along geodesicswith the proviso that the geodesic “can be extended that far”. We are aiming to prove thatvarious Lorentzian manifolds are geodesically incomplete. The logic of the proofs is to assumethat (usually time-like or causal) geodesics can be extended arbitrarily far (i.e. that the manifoldis geodesically complete in some sense) and show that, when combined with various geometricalconditions (e.g. curvature restrictions), this assumption leads to a logical contradiction. As such,we will assume that the relevant geodesics may be extended to the point where results such asProposition 13.14 become relevant.

If, on the other hand, we were to assume from the outset that there exist geodesics that cannotbe extended to the required value of the affine parameter, then our manifold would then, bydefinition, be incomplete and there would be nothing to prove.

Proof of Proposition 13.14. We show that, under the stated conditions, the index form is notnegative semi-definite. Let ei, i = 1, . . . , n be an orthonormal basis for TpS and parallel transportthese along c to give vector fields Ei along c. Then let f(t) := 1 − t/b(p) for t ∈ [0, b(p)]. ThenfEi ∈ ΓS0 (c, TM) and

−IS(fEi, fEi) = χ(f(0)Ei(0), f(0)Ei(0)) +∫ b(p)

0

[g(∇t (fEi) ,∇t (fEi)) + g(t,R(fEi, t)fEi)] dt

= χ(ei, ei) +∫ b(p)

0

[f ′(t)2 + f(t)2g(t,R(Ei, t)Ei)

]dt.

Summing over i = 1, . . . , n givesn∑i=1

−IS(fEi, fEi) = κ(p) +∫ b(p)

0

[nf ′(t)2 − f(t)2Ric(t, t)

]dt.

= κ(p) +∫ b(p)

0

n1

b(p)2dt−

∫ b(p)

0

f(t)2Ric(t, t) dt

≤ κ(p) +n

b(p)= κ(p)− κ(p) = 0.

Thereforen∑i=1

IS(fEi, fEi) ≥ 0,

so there exists an i such that IS(fEi, fEi) ≥ 0. Therefore there exists v ∈ ΓS0 (c, TM) withIS(v,v) ≥ 0 (i.e. v := fEi), so I is not negative definite. Therefore c is not maximal, so theremust be a conjugate point to S along c at, or before, c(b(p)).

Corollary 13.16. Let S be a compact spacelike hypersurface in M without boundary. If Ric(V,V) ≥0 for all time-like vectors V and κ(p) < 0 for all p ∈ S. Then there exists q ∈ S such thatmaxp∈S κ(p) = κ(q) < 0. Defining b := −n/κ(q) then, along each future-directed normal geodesic

42 JAMES D.E. GRANT

to S, there will exist a point conjugate to S along that geodesic within a distance b of S, providedthat the geodesics can be extended that far.

Proof. The existence of q follows from the compactness of S. From the previous proposition, givenp ∈ S there exists a conjugate point along the corresponding normal geodesic within distance b(p)of S. Since

b(p) = − n

κ(p)=

n

|κ(p)|≤ n

|κ(q)|= − n

κ(q)≡ b,

it follows that, given any p ∈ S, there exists a conjugate point along the corresponding normalgeodesic within distance b of S.

Example 13.17 (Misner space). Consider flat R1,1 with the metric g = −dt2 +dx2. We considerthe interior of the past null cone of 0, with coordinates (T, θ) such that t = T cosh θ, x = T sinh θwith T < 0, θ ∈ R. If we now identify points (T, θ) with (T, θ + 2nπ), then the space-likehyper-surfaces T = constant < 0 are compactified to become topologically S1. The metric on theresulting space is simply

gMisner = −dT 2 + T 2dθ2, T < 0, θ ∈ [0, 2π).Lecture 21

The surfaces Sa := T = a < 0 have unit future-directed time-like normal

n = ∂T .

To calculate the extrinsic curvature of Sa, we require

χ(∂θ, ∂θ) ≡ g(∂θ,∇∂θ∂T ) = T 2〈dθ,∇∂θ

∂T 〉.

In local coordinates (T, θ), we therefore need to calculate

〈dθ,∇∂θn〉 ≡ 〈dθ,

∑a=r,θ

∂θna +∑b=r,θ

Γaθbnb

∂a〉 = ΓθθT =12gθθ∂T gθθ =

1T.

Henceχ(∂θ, ∂θ) = T.

Taking the trace,

κ = gθθχ(∂θ, ∂θ) =1T.

Note that, since T < 0, if follows that κ < 0. Note that, in this case, b(p) = 1/|T (p)| and thatthe normal geodesics from the surface Sa all converge to the point T = 0, which is at parametervalue |a| from Sa. In the current example the metric is flat, so Ric = 0, so this is consistent withCorollary 13.16.

14. Causality

14.1. Causality relations. Recall from previously:

Definition 14.1. For sets A,U ⊆ M we define the chronological future of A relative to U to bethe set

I+(A,U) := q ∈ U : ∃p ∈ A and a future-pointing time-like curve from p to q in U

and the causal future of A relative to U

J+(A,U) := (A ∩ U) ∪ q ∈ U : ∃p ∈ A and a future-pointing causal curve from p to q in U .

When U = M , we write I+(A) and J+(A), respectively. Similarly, we define I−(A,U), J−(A,U).

Remark 14.2. By a curve we mean one of non-zero extent, so we can have S * I+(S,U).


Definition 14.3. The time separation d : M ×M → [0,∞] is defined by

d(p, q) =

0 q /∈ J+(p)supL[γ] q ∈ J+(p),

for p, q ∈M , where the supremum is taken over future-directed piecewise causal curves from p toq. Similarly, if A,B are subsets of M then we define

d(A,B) = supd(p, q) : p ∈ A, q ∈ B.

Remark 14.4. Note that d(p, q) 6= d(q, p) in general.

Definition 14.5. Let c : I → M be a future-directed, causal curve (We will assume that allcurves are piece-wise smooth.) defined on domain I ⊆ R. Let a := inf I, b := sup I, where weallow a = −∞ and/or b = +∞. A point p ∈ M is a future endpoint of c if for all sequences biwith bi → p then c(bi) → c(p). A causal curve is future-inextendible (in a set S) if it has no futureendpoint (in S). More generally, given a subset S ⊆M , a causal curve is future-inextendible in Sif it has no future endpoint in S.

We define the corresponding notions for past endpoints, etc, by considering sequences ai withai → a.

Intuitively speaking, a causal curve without a future endpoint either extends indefinitely intothe future, or must reach some finite parameter value at which it does not admit a continuousextension within M . The latter occurs, for example, if we consider flat Rn,1 and remove a point.

Definition 14.6. Let S be a closed subset of M . The future Cauchy development of S, D+(S),is the set of points p ∈ M such that every past-inextendible causal curve through p intersects S.Similarly, we define D−(S), and D(S) := D+(S) ∪D−(S).

Lecture 22

Definition 14.7. A set S is achronal if S ∩ I+(S) is empty (i.e. no two points of S have time-like separation). A set S is acausal if S ∩ J+(S) is empty (i.e. no two points of S have causalseparation).

A closed achronal set S is a future Cauchy hypersurface if J+(S) ⊆ D+(S).

Proposition 14.8. Let S be a closed, achronal, spacelike hypersurface in M . Given any q ∈D+(S), there exists a maximising geodesic from S to q.

Proof. Later.

The reason that we are interested in such a result is because it allows us to deduce the followingrudimentary singularity theorem:

Corollary 14.9. Let (M,g) be a Lorentzian manifold such that Ric(V,V) ≥ 0 for all time-like vector fields V. If there exists a compact, achronal, spacelike future Cauchy surface withoutboundary, S, with κ ≤ κ0 < 0 on S, then M is time-like geodesically incomplete.

Proof. Let c be a normal, future-directed geodesic from S. Then, from Corollary 13.16, we knowthat there exists a conjugate point to S along c at c(t0) where t0 ≤ b := −n/κ0. Letting t1 > b,then there are two possibilities. Firstly, it is possible that c cannot be extended to this valueof t. In this case, by definition, M is geodesically incomplete and we are finished. The secondpossibility is that there exists q := c(t1) ∈ M . Since c(t0) is conjugate to S along c, and t0 < t1,it follows that c cannot be a maximising geodesic from S to q. Since S is a future Cauchy surfaceand q ∈ J+(S), we deduce that q ∈ D+(S) and therefore, from 14.8, we know that there exists amaximising geodesic, c, from S to q. From first variation of arclength, we know that c must benormal to S. However, since c is maximising, we must have L[c] ≥ L[c] = t1 > b, therefore weagain deduce from Corollary 13.16 that there must be a point conjugate to S along c prior to q.Therefore c cannot be maximising up to q. This contradicts the definition of c.

Remark 14.10. Note that, in the previous proof, we have actually shown the stronger result thatevery time-like geodesic normal to S has length less than or equal to b.

44 JAMES D.E. GRANT

The conditions in the above theorem are rather strong and, with some work, can be significantlyrelaxed. The most general result along these lines is the following.

Theorem 14.11. Let (M,g) be a (four-dimensional) Lorentzian manifold such that the followingconditions hold:

(1) Ric(v,v) ≥ 0 for every causal vector v;(2) there exists a compact space-like three-surface S without edge;(3) the unit normals to S are everywhere converging on S (i.e. κ < 0 on S).

Then the space-time (M,g) is time-like geodesically incomplete.

Remark 14.12. The conditions on the surface S (i.e. condition (2)) are much weaker than in 14.9.In this case, however, only the existence of an incomplete time-like geodesic is proved, rather thanthe stronger conclusion mentioned in Remark 14.10.

For the rest of the course, we will concentrate on outlining the proof of Proposition 14.8, whichimplies the weaker result 14.9.

End of examinable material


Appendix A. Proof of Proposition 14.8 (Non-examinable)Lecture 23

The following is an outline of the main arguments leading to the proof of the following result.

Proposition A.1. Let S be a closed, achronal, spacelike hypersurface in M (without boundary).Given any q ∈ D+(S), there exists a maximising geodesic from S to q.

For full details of the proof, see either Chapter 14 of [O’N] or Chapter 6 of [HE].

Definition A.2. The strong causality condition holds at p ∈ M if given any neighbourhood, U ,of p there exists a neighbourhood V ⊆ U of p such that any causal curve c : [a, b] → M withc(a), c(b) ∈ V lies in U i.e. c(t) ∈ U for all t ∈ [a, b].

A set N ⊆M is globally hyperbolic if the strong causality condition holds on N and if, for anyp, q ∈ N , the set J+(p) ∩ J−(q) is compact and contained in N .

Remarks A.3.

(1) Intuitively speaking, this means that a causal curve that starts arbitrarily close to p (i.e.in the set V ) and leaves a fixed neighbourhood of p (i.e. U) cannot later return arbitrarilyclose to p (i.e. reenter the set V ).

(2) Global hyperbolicity, along with suitable convexity conditions, implies that every neigh-bourhood of p contains a neighbourhood of p that no (future-inextendible) causal curveintersects more than once. (See [O’N, §14, Exercise 10(b)].) This is the definition of strongcausality used by Hawking and Ellis [HE].

Lemma A.4. The map d : M ×M → [0,∞] is lower semi-continuous.

Proof. The only non-trivial case is when d(p, q) > 0. Therefore let q ∈ I+(p) where, for themoment, we assume 0 < d(p, q) < ∞. We want to show that, given any ε > 0, there exists aneighbourhood U of p and a neighbourhood V of q such that d(p′, q′) > d(p, q)− ε for all p′ ∈ U ,q′ ∈ V . Let γ be a time-like curve from p to q with L[γ] > d(p, q)− ε

3 . Let C be a convex normalneighbourhood of q and q1 a point of γ ∩ C prior to q. There then exists a neighbourhood, V , ofq such that if q′ ∈ V then the geodesic from q1 to q′ is causal and has length > L[γ|[q1,q]] − ε

3 .Similarly, we construct a neighbourhood U of p such that, given any p′ ∈ U and q′ ∈ V , they canbe joined by a causal curve of length > L[γ|[p,q]]− 2ε

3 > d(p, q)− ε, as required.Finally, if q ∈ I+(p) and d(p, q) = ∞, then the above construction shows that, given any N > 0,

there are neighbourhoods U and V such that d(p′, q′) > N for all p′ ∈ U and q′ ∈ V .

Lemma A.5. Let N ⊆ M be a globally hyperbolic open set. Then d(p, q) < ∞ for all p, q ∈ N ,and the map d : N ×N → [0,∞) is continuous.

Proof. Given p, q ∈ N with q ∈ I+(p), we first wish to prove that d(p, q) < ∞. To do this,note that strong causality and the fact that J+(p) ∩ J−(q) is compact implies that we can coverJ+(p) ∩ J−(q) by a finite number, k < ∞, of convex normal neighbourhoods such that each setcontains no causal curve of length greater than some bound ε > 0. Since any causal curve fromp to q can enter each such neighbourhood at most once, it must have finite length. In particular,d(p, q) ≤ kε <∞. Lecture 24

We prove that d is upper semi-continuous by contradiction. In particular, assume that d is notupper semi-continuous at (p, q) ∈ N × N . Then there exists an ε > 0 and sequences pn → p,qn → q with the property that d(pn, qn) ≥ d(p, q) + ε for all n. Since d(pn, qn) > 0, thereexists a causal curve cn from pn to qn with L[cn] > d(pn, qn) − 1

n . Since N is open, there existpoints p− ∈ I−(p) ∩N and q+ ∈ I+(q) ∩N and, without loss of generality, we may assume thatpn ∈ I+(p−) and qn ∈ I−(q+). In this case, the curves cn are contained in the set J+(p−)∩J−(q+).Since N is globally hyperbolic, this set is compact. In the next paragraph, we show that theseconditions imply the existence of a causal curve, c, from p to q of length L[c] ≥ d(p, q) + ε. Thiscontradicts the definition of d(p, q), giving the required result.

Finally, we sketch the proof of the fact that given causal curves, cn, contained in a compactsubset, K := J+(p−)∩J−(q+), of a globally hyperbolic set N with endpoints pn, qn that converge

46 JAMES D.E. GRANT

to points p, q, then we can construct a causal curve (in fact, a broken geodesic) from p to q withlength ≥ d(p, q) + ε. Beginning at the point p0 := p, let B0 be a convex normal neighbourhood ofp contained in K. There are two possibilities:

1). If q ∈ B0, then we take the (unique) geodesic, c, from p to q in B0. In this case, L[c] ≥limn→∞ L[cn] > limn→∞ d(pn, qn) ≥ d(p, q) + ε, so we are done.

2). If q /∈ B0, then for sufficiently large n, qn /∈ B0. Therefore, for such n, the curves cn mustleave the set B0. Let cn(sn) be the first points at which cn intersect ∂B0. Since ∂B0 is compact,there exist a subsequence of cn, which we again denote by c1n, with the property that c1n(s

1n)

converge to a point p1 ∈ ∂B0. Note that p1 ∈ J+(p) but, by construction, p1 6= p. Now let B1 bea convex normal neighbourhood of p1 and repeat the above procedure.

The above process produces a sequence of points p1, p2, . . . in K. Since each consecutive pairof points lie in a convex neighbourhood, we may form the broken geodesic curve, c, that we getby joining together the geodesics between the consecutive points in the sequence. If the aboveprocedure does not terminate, then the curve c will be an future-inextendible causal curve. Assuch, it must leave the compact set K after some finite time. This, however, implies that pk /∈ Kfor sufficiently large k, contradicting the fact that cn are contained in K. Therefore, the abovesequence must be finite p1, . . . , pk and ends at pk ≡ q. Forming the broken geodesic that weget by joining the points of the sequence p0, . . . , pk gives a broken geodesic from p to q, and asubsequence cn of cn, with points cn(sn,i) that converge to the pi as n → ∞, for each i =1, . . . , k. The length L[cn] will be less than or equal to the length, Ln, of the corresponding brokengeodesic that connects the points cn(0), cn(sn,1), . . . , cn(sn,k). Since Ln → L[c] as n → ∞, wededuce that L[c] ≥ limn→∞ L[cn] ≥ d(p, q) + ε.

Remark A.6. The sequence of points pi constructed from the sequence of future-directed causalcurves cn is called a limit sequence for cn. Given any sequence of future-directed causal curveswith cn(0) → p and such that there exists a neighbourhood of p that almost all of the cn leave,then cn has a limit sequence starting at p.

The broken geodesic, c, constructed from the limit sequence is a quasi-limit of the sequencecn. In the case where the set of points pi is infinite, then c is a future inextendible causalcurve. The existence of such a curve may often violate various causality conditions and, hence,allows us to deduce that, for a quasi-limit of a sequence cn, the set pi will be finite.

Proposition A.7. Let N ⊆ M be a globally hyperbolic open set, and p, q ∈ N with q ∈ J+(p).Then there exists a causal geodesic from p to q the length of which is equal to d(p, q).

Proof. Let p, q ∈ N , with q ∈ J+(p). Let U be a convex normal neighbourhood of p containedin N . (Such a U exists, since N is, by assumption, open.) If q ∈ U the result is automatic fromthe fact that U is a convex normal neighbourhood, so assume that q /∈ U . Now consider the map∂U ∩ (J+(p) ∩ J−(q)) → [0,∞) given by r 7→ d(p, r) + d(r, q). Note that, since p, q and r areelements of the globally hyperbolic set N , it follows from Lemma A.5 that this map is continuous.Also, since ∂U ∩ (J+(p) ∩ J−(q)) is a closed subset of a compact set (i.e. J+(p) ∩ J−(q)) it istherefore compact. Hence there will exist x ∈ ∂U ∩ (J+(p) ∩ J−(q)) at which the above functionachieves its maximal value.

Claim 1: d(p, x) + d(x, q) = d(p, q).

Proof of Claim 1: Let γ : [0, b] → N be any continuous, future-directed, causal curve from pto q, with γ(0) = p, γ(b) = q. Since M is assumed Hausdorff, there exists a ∈ (0, b) such thatγ(a) ∈ ∂U . Therefore

L[γ] = L[γ|[0,a]] + L[γ|[a,b]] ≤ d(p, γ(a)) + d(γ(a), q) ≤ d(p, x) + d(x, q),

where the last inequality follows from the definition of the point x ∈ ∂U . Therefore, any continu-ous, future-directed, causal curve from p to q has length L[γ] ≤ d(p, x)+d(x, q). Therefore d(p, q) =supγ L[γ] ≤ d(p, x) + d(x, q). The triangle inequality, however, yields d(p, q) ≥ d(p, x) + d(x, q).Therefore d(p, q) = d(p, x) + d(x, q), as required.


Given p and x ∈ ∂U , construct the future-directed, causal geodesic from p through x, which wedenote by c. We assume that p = c(0), x = c(a) for some a > 0. Assuming that M is geodesicallycomplete, we can assume that c : [0,∞) →M is defined.

Claim 2: For all y ∈ J−(q) ∩ c we have d(p, y) + d(y, q) = d(p, q).

Proof of Claim 2: Assume the contrary. The map t 7→ d(p, c(t)) + d(c(t), q) is continuous, sothe set t ∈ [0,∞) : d(p, c(t)) + d(c(t), q) = d(p, q) is true is a closed subset of [0,∞). This setis non-empty, since 0 and a are elements of it. Therefore, there will exist a T > 0 such that theclaim is true for t ≤ T , but false for t > T . Now construct a convex normal neighbourhood ofc(T ), V , contained in N . As above, we construct a point m ∈ ∂V ∩ J+(c(T )) ∩ J−(q) such thatd(c(T ),m) + d(m, q) = d(c(T ), q). We then construct the geodesic λ from c(T ) passing throughm. We then have

d(p, c(T )) + d(c(T ),m) + d(m, q) = d(p, c(T )) + d(c(T ), q) = d(p, q) ≥ d(p,m) + d(m, q).

Therefored(p, c(T )) + d(c(T ),m) ≥ d(p,m).

Along with the triangle inequality this implies that d(p, c(T )) + d(c(T ),m) = d(p,m). Thereforeλ c is a curve that maximises the distance from p to m. From the first variation formula, thisimplies that λ c must be a smooth geodesic. Therefore λ is a continuation of the geodesic curvec. This contradicts the definition of T as the final point on c at which our required equality holds.

Conclusion of proof of Proposition: Finally, since the set J+(p)∩J−(q) is compact, the curvec must leave the set J−(q) at some point y. If y 6= q then y must lie on a past-directed nullgeodesic, λ, from q to y. In this case, λc gives a causal curve from p to q of length d(p, q). (Note,also, that d(p, y) = d(p, q).) However, this curve is not a causal geodesic, and hence, from the firstvariation formula, can be varied to give a longer causal curve from p to q. This contradicts thefact that d(p, y) = d(p, q). As such, the assumption that y 6= q must be false i.e. y = q. Therefore,c is a causal geodesic from p to q with L[c] = d(p, q).

Remark A.8. Except for the last part, which is Lorentzian in nature, the above proof is just amodification of the usual proof of the fact that any two points on a complete Riemannian manifoldcan be joined by a minimising geodesic.

Remark A.9. More generally, Proposition A.7 holds for N any globally hyperbolic set, not neces-sarily open. (See, e.g., [HE, Proposition 6.7.1].)

Lemma A.10. Let A be an achronal set. Then int(D(A)) is globally hyperbolic.

Sketch of Proof. 14 To begin, we show that strong causality holds on int(D(A)). We do this intwo stages.

1). Let p ∈ int(D(A)) and assume that there exists a closed causal curve, c, through p. Then,traversing c over and over again defines an inextendible causal curve, c, through p. Such a curvenecessarily intersects I+(A) and I−(A). As such, there exist points q ∈ c ∩ I−(A), r ∈ c ∩ I+(A).Since r ∈ J−(q) (as there exists a past-directed causal curve from q to r i.e. the relevant partof c) it follows that r ∈ I−(A). Since r ∈ I+(A), this implies the existence of a time-like curvethat intersects A twice. This contradicts the fact that A is achronal. Therefore there can exist noclosed causal curve through any point p ∈ int(D(A)).

2). To show that int(D(A)) obeys strong causality, we assume, on the contrary, that there existsp ∈ int(A) and future-directed causal curves cn : [0, 1] → M such that cn(0) → p and cn(1) → p,but every cn leaves some fixed neighbourhood, U , of p. We then construct a quasi-limit of thecn with vertices pi. If this set is finite, then it must end at limn→∞ cn(1) = p. We have thenconstructed a closed causal curve through p, which violates the result of Part 1). If the set piis infinite then the corresponding quasi-limit, c, is a future-inextendible causal curve. As such, cmust enter I+(A) at some point and remain there. Hence there exists an i such that pi ∈ I+(A)

14For full details, see [O’N, §14, Theorem 38]

48 JAMES D.E. GRANT

and a subsequence cm of cn and an s ∈ [0, 1] such that cm(s) → pi. (WLOG, may assumethat cm(s) ∈ I+(A).) Since pi 6= p, we now consider the curves cm|[s, 1]. From these, wemay construct a past-directed quasi-limit with vertices pi starting from p (since we know thatlimn→∞ cn(1) = p). If pi is finite, then it must end at pi (since limm→∞ cm(s) = pi). Puttingthe quasi limits together then gives a causal curve from p to itself, which again violates the resultof Part 1). However, if pi is infinite, then the quasi-limit, c, is a past-inextendible causal curve,and hence must intersect I−(A). Hence some cm|[s, 1] must intersect I−(A) i.e. there exists s > swith cm(s) ∈ I−(A). Since cm(s) ∈ I+(A), we may therefore construct a time-like curve thatintersects A more than once, violating achronality of A.

Let p, q ∈ int(D(A)). To prove compactness of J+(p) ∩ J−(q), first assume that q ∈ J+(p)(otherwise the set is empty, in any case). If q = p then Part 1) above implies that J+(p)∩J−(p) =p, so we are done. Therefore, let q ∈ J+(p) \ p. Let xn be a sequence of points inJ+(p) ∩ J−(q). We may then construct future-directed causal curves, cn, that pass through p,xn and q. We then construct a limit-sequence pi from the cn. By similar methods to thoseused in Part 2), one can show that this sequence being infinite contradicts achronality of A, sothe sequence pi must be finite. It then follows that there exists intervals [sm,i, sm,i+1], wherecm(sm,i) → pi, cm(sm,i+1) → pi+1 such that the point xm lies on the segment cm|[sm,i, sm,i+1] forinfinitely many m. Passing to this subsequence, we may assume that the xm lie (for sufficientlylarge m) in a convex open neighbourhood C with compact closure. Therefore xm will have asubsequence converging to a point x which will satisfy x ∈ J+(pi) and x ∈ J−(pi+1). Hencex ∈ J+(p) ∩ J−(q).

Finally, given p, q ∈ int(D(A)), we need to show that J+(p)∩ J−(q) is contained in int(D(A)).Again, we may assume q ∈ J+(p) \ p. The main ideas are contained in the case where p, q ∈I+(A). We then let q+ ∈ I+(q) ∩D(A) ⊆ D+(A). Then, J+(p) ∩ J−(q) is contained in the openset I−(q+) ∩ I+(A), so it is enough to show that this set is contained in int(D(A)). Let γ be apast-directed time-like curve from q+ to a point y ∈ I−(q+)∩I+(A). If γ intersects A then y wouldlie in I−(A) (since γ would be a past-directed time-like curve from A to y) and I+(A), and wouldtherefore violate achronality of A. Hence γ cannot intersect A. Therefore, given any past-directedinextendible causal curve, c, from y, the curve c γ is a past-directed inextendible causal curvefrom q+. Since q+ ∈ D+(A), it follows that c γ must intersect A. Since γ does not intersect A,it follows that c must intersect A. Hence y ∈ D+(S). Hence I−(q+) ∩ I+(A) ⊆ D+(A). SinceI−(q+) ∩ I+(A) is open, it follows that I−(q+) ∩ I+(A) ⊆ int(D(A)). The proof for arbitraryp, q ∈ int(D(A)) involves no new ideas.

In a similar way, one can prove the following results15

Lemma A.11. Let A be an achronal set and p ∈ int(D(A)) \ I−(A), then the set J−(p)∩D+(A)is compact.

Lemma A.12. Let S be a closed, achronal, space-like hypersurface in M (without boundary)16.Then D(S) is globally hyperbolic.

Proof of Proposition A.1. By Lemma A.12,D(S) is a globally hyperbolic open set. By Lemma A.11,given q ∈ int(D(A)) \ I−(A), the set J−(q) ∩D+(S) is compact. Hence, since S is closed, the set(J−(q) ∩D+(S))∩S is compact i.e. J−(q)∩S is compact. By Lemma A.5, the map r 7→ d(r, q) iscontinuous on J−(q)∩S, and therefore takes a maximum at some point p ∈ S. By Proposition A.7,there exists a geodesic, c, from p to q of length d(p, q) = d(S, q).

15For proofs, see [O’N, §14, Lemma 40] and [HE, §6, Proposition 6.6.7, Part (2)], respectively.16Such an S is necessarily acausal


References

[CE] J. Cheeger, D. G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Publishing Co.,Amsterdam, 1975. 5

[Cha] I. Chavel, Riemannian geometry, A modern introduction, Cambridge Studies in Advanced Mathematics,vol. 98, Cambridge University Press, Cambridge, 2006. 2, 5, 23

[GHL] S. Gallot, D. Hulin, J. Lafontaine, Riemannian geometry, Springer-Verlag, Berlin, 2004. 2, 5, 25[Gr] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, vol.

152, Birkhauser Boston Inc., Boston, MA, 1999. 17[HE] S. W. Hawking, G. F. R. Ellis, The large scale structure of space-time, Cambridge University Press,

London, 1973. 29, 32, 33, 45, 47, 48[HY] J. G. Hocking, G. S. Young, Topology, second ed., Dover Publications Inc., New York, 1988. 25[K] S. Kobayashi, On conjugate and cut loci , Studies in Global Geometry and Analysis, Math. Assoc. Amer.

(distributed by Prentice-Hall, Englewood Cliffs, N.J.), 1967, pp. 96–122. 25[O’N] B. O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press Inc. 2,

29, 45, 47, 48[P] P. Petersen, Riemannian geometry, Graduate Texts in Mathematics, vol. 171, Springer-Verlag, New York,

1998. 5[W] F. W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics,

vol. 94, Springer-Verlag, New York, 1983. 2

Index

acausal set, 43

achronal set, 43

arc-length, 35

Bishop-Gromov Comparison Theorem, 17, 26

causal future of A relative to U , 31, 42

chronological future of A relative to U , 31, 42

conjugate point, 21

order, 21

connection

affine, 2

curvature, 2

torsion, 2

Levi-Civita, 3

metric, 3

on a vector bundle, 2

torsion-free, 3

convex, 34

covariant derivative, 2

curvature

of connection on vector bundle, 2

scalar, 4

sectional, 6

curve, 6

arc-length, 6

length, 6

regular, 6

tangent vector, 6

cut point, 23

cut-locus, 25

cut-locus of p in TpM , 25

distance function, 7, 12

exponential map, 9

first variation formula, 7

future Cauchy development, D+(S), 43

future Cauchy hypersurface, 43

future endpoint, 43

future-inextendible curve, 43

future-inextendible curve in S, 43

geodesic, 8

maximal, 38

minimising, 22

parametrised by arc-length, 8

geodesically inextendible, 9, 33

globally hyperbolic, 45

gradient

of a function, 10

Hessian, 11

index form, 20

injectivity radius

of (M,g) at p, 25

Jacobi equation, 20

Jacobi field, 20

Koszul formula, 3

Laplacian, 12

Lorentzian manifoldtime-orientable, 29

maximal, 9, 33metric

Lorentzian, 3Riemannian, 3semi-Riemannian, 3

Myers’s theorem, 27

normal coordinates, 9normal neighbourhood, 34

pull-backtensor field, 4

Riccati equation, 14Ricci curvature tensor, 3Ricci scalar, 4Ricci tensor, 3

scalar curvature, 4second variation formula, 19sectional curvature, 6smooth variation of c, 7

two parameter, 18star-shaped about 0, 34strong causality condition, 45

time separation, 43

variation vector field, 7vector

null, 28space-like, 28time-like, 28

future-directed, 28past-directed, 28

vector fieldalong a map, 7null, 29space-like, 29time-like, 29

vector fieldsLie bracket, 2

50

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GLOBAL LORENTZIAN GEOMETRYpersonal.maths.surrey.ac.uk/st/jg0032/teaching/GLG1/... · 2013-07-10 · GLOBAL LORENTZIAN GEOMETRY JAMES D.E. GRANT Contents Part 1. Preliminary material