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Elements of differential geometryR.Beig (Univ. Vienna)
ESI-EMS-IAMP School on Mathematical GR, 28.7. - 1.8. 2014
1. tensor algebra
2. manifolds, vector and covector fields
3. actions under diffeos and flows
4. connections
5. pseudo-Riemannian manifolds
6. geodesics
7. curvature
1
Tensor algebra
Let T be an n-dimensional vector space over R and T∗ its dual.
Elements u, v of T are called vectors, elements ω, µ of T∗ are called
covectors. In a basis {ei} the vector u has the form u =∑n
1 uiei =
uiei (note summation convention!) and the covector ω reads
ω = ωiei in the dual basis given by ei(ej) = δij . The action of ω on
the v reads ω(v) = ωivi. Although vi and ωi depend on the choice
of basis, ωivi does not. Reading ωiv
i ’from right to left’ gives the
identification T ∼= T∗∗. The spaces Trs consist of multilinear forms on
T∗ × ...T∗ × ..T (r copies of T∗, s copies of T). They have r upper
and s lower indices: U = U i1...irj1..js ei1 ⊗ ...eir ⊗ ej1 ⊗ ..ejs . In
particular T ∼= T1 and T∗ ∼= T1 and T11∼= L(T,T).
2
A scalar product γ on T is given by γ(u, v) = γijuivj with
γij = γji non-degenerate. γ of signature (++, ..+) is positive
definite, γ with (−,+,+, ...+) a Lorentzian metric. γ gives rise to a
unique quadratic form on V∗ given by γij where γijγjk = δik. The
quantities γij and γij yield an isomorphism between elements v ∈ Tand ω ∈ T∗ by means of ’raising and lowering of indices’, e.g.
vi(ω) = γijωj =: ωi. γ Lorentzian: a non-zero vector v ∈ T is
• timelike: γ(v, v) = γijvivj = −(v0)2 +
∑n−11 (vi)2 < 0
• null: γ(v, v) = γijvivj = −(v0)2 +
∑(vi)2 = 0
• spacelike γ(v, v) = γijvivj = −(v0)2 +
∑(vi)2 > 0
Null vectors form a double cone (’past and future light cone’) C,
timelike inside.
3
Subspaces W ⊂ T are
• spacelike: γ(., .)|W pos.def.
• null: γ(., .)|W degenerate, same as T tangent to C
• timelike: γ(., .)|W Lorentzian
Fundamental inequalities:
• ’reverse C-S inequality’: (γ(u, v))2 ≥ γ(u, u)γ(v, v), provided
that u or v (or both) are causal
• ’reverse ∆ inequ.: |u+ v| ≥ |u|+ |v], where
|u| =√
−γ(u, u) and u, v are causal, both future or both past
directed. Is essence behind the ’twin paradox’
4
5
Manifolds
Stated somewhat informally a (C∞, n-dimensional manifold) M is a
topological space (Hausdorff, 2nd countable), equipped with a set of
coordinate charts (U, xi), i.e. U open and xi map U bijectively into
an open set in Rn. These charts should cover M , s.th on overlapping
charts (U, xi), (U , xi), U ∩ U = {0} they are smoothly related:
xi = F i(xj), F i ∈ C∞(Rn,R), i = 1, ...n. Smoothness of
functions f :M → R is defined ’chartwise’, likewise smoothness of
maps between more general manifolds.
(T, γ), viewed as an affine space, is a manifold, namely Minkowski
spacetime, the realm of Special Relativity. Lorentzian manifolds, see
later, are the realm of General Relativity.
6
Let p ∈M . The tangent space Tp(M) at p can be defined as the
vector space of derivations (see below) acting on smooth functions
defined near p. Elements v ∈ Tp(M) can be shown to be the same,
in local coordinates (U, xi) with p ∈M , as directional derivatives,
i.e. v(f) = vi ∂f∂xi |p. Thus the ’coordinate vectors’ ∂
∂xi |p = ∂i|p form
a basis of Tp(M). It follows that, under a change of chart:
vi = (∂jxi)vj .
Example for tangent vector: a smooth curve γ : I →M with
γ(0) = p gives an element in Tp(M) via its tangent vector defined
by γ′(0)(f) = ddtf ◦ γ(t)|0. By the chain rule γ′(0) = dxi
dt(0)∂i|p.
All tangent vectors can be gotten in this way.
7
A vector field v on M is a smooth assignment to each p ∈M of a
vector vp ∈ Tp(M). Or, v maps C∞(M) into itself subject to
• v(af + bg) = av(f) + bv(g) (a, b ∈ R, f, g ∈ C∞(M))
• v(fg) = fv(g) + gv(f) ’Leibniz rule’
Here v(f)(p) = vp(f). The set of smooth vector fields is denoted by
X(M). It is a module over C∞(M), addition and scalar
multiplication being defined in the obvious way. In local coordinates
v ∈ X(M) can be written as v = vi(x)∂i or v(f) = vi(x)∂if .
Thus
vi(x) =∂xi
∂xj(x(x))vj(x(x)) (∗)
8
Given v, w ∈ X(M), the map f ∈ C∞(M) 7→ v(w(f)) does not
define a vector field, but the Lie bracket
[v, w] = vw − wv = (vj∂jwi − wj∂jv
i)∂i
does. Note [∂i, ∂j] = 0.
Jacobi identity: [v, [w, z]] + [z, [v, w]] + [w, [z, v]] = 0
9
Covectors: a covector at p is an element ωp of T ∗p (M). A covector
field or 1-form ω is defined in the obvious way. It is smooth if
ω(v)(p) = ωp(vp) is smooth for all v ∈ X(M). Let f ∈ C∞(M).
The 1-form df is defined by df(v) = v(f). In particular
dxi(∂j) = δij ...dual basis. ω = ωi(x)dxi, where ωi = ω(∂i).
E.g. df = ∂if dxi.
Under change of chart: ωi(x) =∂xj
∂xi (x)ωj(x(x)).
Higher order tensors (tensor fields) are defined in the obvious way,
e.g. the (1, 1)-tensor t = tij ∂i ⊗ dxj . Contraction, in a basis, is by
summation over a pair of up-and downstairs indices.
10
The operation d sending functions to 1-forms is a special case of an
operation d, sending p-forms, i.e. covariant, totally antisymmetric
tensors ωi1...ip , p < n into p+ 1- forms. E.g. when p = 1 we define
(check this is a 2-tensor!)
dω(u, v) = u(ω(v))− v(ω(u))− ω([u, v])
i.e. dωij = ∂iωj − ∂jωi. We have that ddf = 0, and dω = 0
implies ω = df when M is simply connected.
11
Action under flows
Let Φ :M → N be a diffeomorphism, i.e. a (smooth) mapping with
smooth inverse. The push-forward Φ∗v ∈ X(N) of v ∈ X(M) is
defined by (f ∈ C∞(N))
(Φ∗v)(f)(p) = v(f ◦ Φ)(Φ−1(p))
Locally yA = ϕA(xi) and
(Φ∗v)A(y) =
∂ϕA
∂xj(ϕ−1(y))vj(ϕ−1(y))
Next let Φ :M → N be smooth and ω a 1-form on N . Then the
pull-back ϕ∗ω on M is defined as (Φ∗ω)(v)(p) = ω(Φ∗v(Φ(p)).
(Note: does not require Φ invertible.)
12
In coordinates
(Φ∗ω)i(x) =∂ϕA
∂xi(x)ωA(ϕ(x))
Pull-back on higher covariant tensor fields analogous. Pull-back
Φ∗f ∈ C∞(N) of functions f ∈ C∞(N) is simply Φ∗f = f ◦ Φ.
For mixed tensors, say on N , their pull-back to M is defined by
’pull-back under Φ w.r. to the downstairs indices’ and ’pull-back under
Φ−1 w.r. to the upstairs indices’.
Vector fields define a local(-in-t) 1-parameter family Ψt of maps
M →M via their flow, i.e.
dψit
dt= vi(ψt), Ψ0(p) = id
13
The Ψt’s are local diffeomorphisms of M into itself in that they map
small neighbourhoods of each p ∈M diffeomorphically onto their
image. This is enough in order for the Lie derivative of w w.r. to v, i.e.
Lvw = ddt|t=0(Ψ−t)∗w to be defined. It turns out that
Lvw = [v, w] or
(Lvw)i = vj∂jw
i − wj∂jvi
Next Lvω is defined by Lv ω = ddt|t=0(Ψt)
∗ω. It turns out that,
locally,
(Lvω)i = vj∂jωi + ωj∂ivj, Lvf = v(f) = vi∂if
Similarly, for a 2-tensor gij ,
(Lvg)ij = vk∂kgij + gik ∂jvk + gkj ∂iv
k
14
Geometrically, the equation Lvt = 0 means that the structure defined
by the object t is invariant under the flow generated by v. E.g. for gij
a symmetric tensor of Riemannian or Lorentz signature the Killing
vector field v satisfying Lvg = 0 generates a flow leaving the
Riemannian (Lorentzian) structure invariant.
The operations d and Lv are ’natural’ in that they, appropriately,
commute with general diffeomorphisms. This means they require no
structure an M . In contrast, ∇v, defined presently, is not natural.
15
Pseudo-Riemannian manifolds
A manifold is called pseudo-Riemannian if it is provided with a
symmetric (0, 2)- tensor field g = gijdxi ⊗ dxj with
gij = g(∂i, ∂j) = gji non-degenerate. It is called Riemannian if g is
furthermore positive definite and Lorentzian if it has Lorentzian
signature at each p ∈M . Note that, e.g. in the Lorentzian case, there
is in general no chart near p ∈M , for which gij(x) = ηij = const.
This phenomenon is related to the presence of curvature.
16
Linear connections
A linear connection ∇ on M is an R-bilinear map
∇ : X(M)× X(M) → X(M)
(u, v) 7→ ∇uv with (f ∈ C∞(M))
• ∇fuv = f∇uv
• ∇u fv = u(f)v + f∇uv
So ∇uv is tensorial w.r. to u, i.e. defines a (1, 1) tensor. In local
coordinates (xi), ∇uv = uj(∇jvi)∂i, where
∇ivj = vj ;i = ∂iv
j + Γjikv
k , ∇∂j∂k = Γijk∂i
Note ∇Φ∗uΦ∗v = Φ∗∇uv except if Φ leaves connection invariant.
17
∇ can be naturally extended to act on tensor fields as follows:
∇uf := u(f) for f ∈ C∞(M). Then, for 1-forms ω, require that
∇u(v(ω)) = (∇uv)(ω) + u(∇uω), finally that ∇ satisfy the
Leibniz rule w.r. to ⊗ and be linear under addition. E.g.
(∇ut)ji∂j ⊗ dxi = uk(∂kt
ji + Γj
kltli − Γl
kitjl︸ ︷︷ ︸
∇ktji
)∂j ⊗ dxi
∇ is symmetric (torsion-free) if [u, v] = ∇uv −∇vu for all
u, v ∈ C∞(M). This in a local chart means that Γijk = Γi
kj .
Let Cijk = Ci
kj be a globally defined (1,2)-tensor field. Then, given ∇,
∇′uv = ∇uv + Ci
jkujvk∂i is also a symmetric connection.
18
∇ on pseudo-Riemannian manifold (M, g) is called metric when
∇u g(v, w) = g(∇uv, w) + g(v,∇uw)
Given g, there ∃ unique symmetric linear (’Levy-Civita’)connection
which is metric. It is given by
Γijk =
1
2gil(∂jgkl + ∂kgjl − ∂lgjk)
Henceforth ∇ will be Levy-Civita.
19
Geodesics
Let γ : I →M be a smooth curve on M and v a vector field along
γ. The covariant derivative of v along γ is defined as
Dv
Dt= ∇γ′v =
(dvi
dt+ Γi
jk
dxj
dtvk)∂i
The curve t 7→ γ(t) is geodesic when Dγ′
Dtis zero, i.e.
d2xi
dt2+ Γi
jk
dxj
dt
dxk
dt= 0
Prop.: Given p ∈M and v ∈ Tp(M), there ∃ interval I about t = 0
and a unique geodesic γ : I →M , s.th. γ(0) = p and γ′(0) = v.
20
Because ofd g(u, v)
dt= g(
Du
Dt, v) + g(u,
Dv
Dt)
the causal character of the geodesic, in the Lorentzian case, is
preserved. Timelike geodesics model freely falling pointlike bodies,
null geodesics play the role of light rays.
21
Curvature
Curvature, i.e. deviation from flatness, can be measured by the
degree of non-commutativity of ∇ acting on tensor fields, which in
turn is measured by the Riemann tensor. The basic observation is the
Prop.: The vector field given by (u, v, w ∈ X(M))
R(u, v)w := ∇u∇vw −∇v∇uw −∇[u,v]w
is tensorial in (u, v, w), i.e. defines a (1, 3)- tensor field Rijkl:
Rijkl = 2 ∂[iΓ
kj]l + 2Γk
m[iΓmj]l
22
Rijkl = gkmRijm
l can be shown to have the following symmetries
• Rijkl = −Rjikl = −Rijlk
• Rijkl +Rkijl +Rjkil = 0 ’1st Bianchi identity’
• Rijkl = Rklij
Here the last property follows from the other ones. For n = 4 the
number of algebraically independent components of Rijkl is 20.
Furthermore there is the following (’2nd Bianchi’) differential identity
∇iRjklm +∇kRijlm +∇jRkilm = 0
23
One can infer the Bianchi identities from the equivariance (see
Kazdan, 1981)
Rijkl[Φ∗g] = (Φ∗R)ijkl[g]
where Φ is a diffeomorphism M →M .
The identities fulfilled by Rijkl imply that the Ricci tensor
Rij = Rkikj satisfies Rij = Rji. Furthermore the Einstein tensor
Gij = Rij − 12gij g
klRkl is divergence-free, i.e.
gij∇iGjk = 0
24