(Wünsch's) nearly invariant calculus for parabolic geometries · Bibliography Parabolic...

Bibliography Parabolic Geometries and Weyl connections Nearly invariant calculus

(Wunsch’s) nearly invariant calculus for parabolicgeometries

Jan Slovak

Masaryk University, Brno, Czech Republicjoint work with Andreas Cap

(and others over years — M.G. Eastwood, V. Soucek, A.R. Gover)

August 2, 2006

Structure

1 Bibliography

2 Parabolic Geometries and Weyl connectionsCartan connections as analogies to affine geometry onmanifoldsWeyl structures, connections and Rho–tensors

3 Nearly invariant calculusInvariant and fundamenta differentials

Structure

1 Bibliography

Important sources

P. Gauduchon, Structures de Weyl-Einstein, espaces detwisteurs et varietes de type S1 × S3, J. reine angew. Math.469 (1995), 1–50.

A. Cap, J. Slovak, V. Soucek, Invariant operators onmanifolds with almost Hermitian symmetric structures, I.Invariant differentiation, Acta Math. Univ. Commenianae 66(1997), 33–69; II. Normal Cartan connections, Acta Math.Univ. Commenianae 66 (1997), 203–220; III. StandardOperators, Diff. Geom. Appl. (2000),

M.G. Eastwood, J. Slovak, Semi-holonomic Verma modules,J. of Algebra 197 (1997), 424–448

V. Wunsch, On conformally invariant differential operators,Math. Nachr. 129 (1986), 269–281.

T. Branson, Second order conformal covariants, Proc. Amer.Math Soc. 126 (1998), 1031–1042.

and many more ....

Important sources

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Important sources

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Important sources

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Important sources

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Important sources

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Important sources

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Structure

1 Bibliography

Cartan connection

Definition

Cartan geometries of type G/P are deformations of thehomogeneous space G → G/P with the Maurer–Cartan formω ∈ Ω1(G ; g):absolute parallelism ω ∈ Ω1(G, g) on a principal fiber bundleG → M with structure group P, enjoying suitable invarianceproperties with respect to the principal action of P:

ω(ζX )(u) = X for all X ∈ p, u ∈ G (the connectionreproduces the fundamental vertical fields)

(rb)∗ω = Ad(g−1) ω (the connection form is equivariantwith respect to the principal action)

ω|TuG : TuG → g is a linear isomorphism for all u ∈ G (theabsolute parallelism condition).

Cartan connection

Definition

Cartan connection

Definition

Cartan connection

Definition

Parabolic geometries

Definition

Parabolic geometries are Cartan connections with the choice ofparabolic P in semisimple real G .Morphisms of the parabolic geometries are principal fiber bundlemorphisms ϕ : G → G′ with ϕ∗(ω′) = ω.

Fixed grading g = g−k ⊕ . . . g−1 ⊕ g0 ⊕ g1 ⊕ · · · ⊕ gk on Liealgebra g, p = g0 ⊕ g1 ⊕ · · · ⊕ gk , g0 is the reductive part in p,

also write P+ = exp p+ and G− = exp g− for thecorresponding nilpotent subgroups.

Unique decomposition g = g0· expΥ1· . . . · expΥk , g ∈ P,g0 ∈ G0, and Υi ∈ gi .

Grading element is the unique E ∈ g0 with adE|gi= i· idgi for

all i = −k, . . . , k.

Definition

all i = −k, . . . , k.

Definition

all i = −k, . . . , k.

Definition

all i = −k, . . . , k.

Definition

all i = −k, . . . , k.

Curvature

Definition

The structure equations K = dω + 12 [ω, ω] define the curvature K

of the Cartan connection ω.

ω defines constant vector fields ω−1(X ), X ∈ g.

Trg ·ω−1(X )(u) = ω−1(Adg−1 ·X )(u·g).

Curvature function

κ(u)(X ,Y ) = K (ω−1(X )(u), ω−1(Y )(u))

= [X ,Y ]− ω(u)([ω−1(X ), ω−1(Y )]).

Curvature is a horizontal 2–form.

Curvature

Definition

Trg ·ω−1(X )(u) = ω−1(Adg−1 ·X )(u·g).

Curvature function

κ(u)(X ,Y ) = K (ω−1(X )(u), ω−1(Y )(u))

= [X ,Y ]− ω(u)([ω−1(X ), ω−1(Y )]).

Curvature

Definition

Trg ·ω−1(X )(u) = ω−1(Adg−1 ·X )(u·g).

Curvature function

κ(u)(X ,Y ) = K (ω−1(X )(u), ω−1(Y )(u))

= [X ,Y ]− ω(u)([ω−1(X ), ω−1(Y )]).

Curvature

Definition

Trg ·ω−1(X )(u) = ω−1(Adg−1 ·X )(u·g).

Curvature function

κ(u)(X ,Y ) = K (ω−1(X )(u), ω−1(Y )(u))

= [X ,Y ]− ω(u)([ω−1(X ), ω−1(Y )]).

Curvature

Definition

Trg ·ω−1(X )(u) = ω−1(Adg−1 ·X )(u·g).

Curvature function

κ(u)(X ,Y ) = K (ω−1(X )(u), ω−1(Y )(u))

= [X ,Y ]− ω(u)([ω−1(X ), ω−1(Y )]).

Weyl connections

Definition

Weyl structure on a parabolic geometry (G, ω) is a G0 equivariantsection σ : G0 = G/P+ → G.

θ = σ∗ω− = θ−k + · · ·+ θ−1 : TG0 → g−k ⊕ · · · ⊕ g−1

γ = σ∗ω0 : TG0 → g0

P = σ∗ω+ = P1 + · · ·+ Pk : TG0 → p+ = g1 ⊕ · · · ⊕ gk .

On the underlying manifold M we obtain:

θ is a soldering form for M

γ is a linear connection form on M, θ + γ is the affine Weylconnection

P is a one–form on M valued in T ∗M

Weyl connections

Definition

θ = σ∗ω− = θ−k + · · ·+ θ−1 : TG0 → g−k ⊕ · · · ⊕ g−1

γ = σ∗ω0 : TG0 → g0

P = σ∗ω+ = P1 + · · ·+ Pk : TG0 → p+ = g1 ⊕ · · · ⊕ gk .

Weyl connections

Definition

θ = σ∗ω− = θ−k + · · ·+ θ−1 : TG0 → g−k ⊕ · · · ⊕ g−1

γ = σ∗ω0 : TG0 → g0

P = σ∗ω+ = P1 + · · ·+ Pk : TG0 → p+ = g1 ⊕ · · · ⊕ gk .

Weyl connections

Definition

θ = σ∗ω− = θ−k + · · ·+ θ−1 : TG0 → g−k ⊕ · · · ⊕ g−1

γ = σ∗ω0 : TG0 → g0

P = σ∗ω+ = P1 + · · ·+ Pk : TG0 → p+ = g1 ⊕ · · · ⊕ gk .

Weyl connections

Definition

θ = σ∗ω− = θ−k + · · ·+ θ−1 : TG0 → g−k ⊕ · · · ⊕ g−1

γ = σ∗ω0 : TG0 → g0

P = σ∗ω+ = P1 + · · ·+ Pk : TG0 → p+ = g1 ⊕ · · · ⊕ gk .

Weyl connections

Definition

θ = σ∗ω− = θ−k + · · ·+ θ−1 : TG0 → g−k ⊕ · · · ⊕ g−1

γ = σ∗ω0 : TG0 → g0

P = σ∗ω+ = P1 + · · ·+ Pk : TG0 → p+ = g1 ⊕ · · · ⊕ gk .

Weyl connections

Definition

θ = σ∗ω− = θ−k + · · ·+ θ−1 : TG0 → g−k ⊕ · · · ⊕ g−1

γ = σ∗ω0 : TG0 → g0

P = σ∗ω+ = P1 + · · ·+ Pk : TG0 → p+ = g1 ⊕ · · · ⊕ gk .

Weyl connections

Definition

θ = σ∗ω− = θ−k + · · ·+ θ−1 : TG0 → g−k ⊕ · · · ⊕ g−1

γ = σ∗ω0 : TG0 → g0

P = σ∗ω+ = P1 + · · ·+ Pk : TG0 → p+ = g1 ⊕ · · · ⊕ gk .

Natural bundles and tractor bundles

Natural bundles

P–representation λ on a vector space V provides the homogeneousvector bundle G ×P V and,more generally, the associated vector bundles

V = G ×P V

with standard fiber V over all manifolds with a parabolic geometryof the type G/P. These are the natural bundles V.

Tractor bundles

The natural bundles with G–modules V are called the tractorbundles

Note: The adjoint representation G provides the adjoint tractorbundles A, the standard representation of a matrix group on Rn

provides the standard tractors T .

Natural bundles

V = G ×P V

Tractor bundles

Natural bundles

V = G ×P V

Tractor bundles

Transformations

Change of Weyl connections

If σ = σ· expΥ, and q = q1 + · · ·+ q` is a section of a naturalbundle, then:

q` =∑

‖i‖+j=`1i!(−1)iλ(Υ1)

i1 . . . λ(Υk)ik qj

∇ξs = ∇ξs+∑

‖i‖+j=01i!(−1)i

(ad(Υ1)

i1 . . .ad(Υk)ik (ξj))•s

Here i is a multiindex (i1, . . . , ik) with ij ≥ 0.We put i ! = i1! · · · ik ! and ‖i‖ = i1 + 2i2 + · · ·+ kik , while(−1)i = (−1)i1+···+ik

Transformations

q` =∑

‖i‖+j=`1i!(−1)iλ(Υ1)

∇ξs = ∇ξs+∑

‖i‖+j=01i!(−1)i

(ad(Υ1)

Transformations

q` =∑

‖i‖+j=`1i!(−1)iλ(Υ1)

∇ξs = ∇ξs+∑

‖i‖+j=01i!(−1)i

(ad(Υ1)

Transformations

q` =∑

‖i‖+j=`1i!(−1)iλ(Υ1)

∇ξs = ∇ξs+∑

‖i‖+j=01i!(−1)i

(ad(Υ1)

Tranformations – continued

Change of Rho–tensors

If σ = σ· expΥ, then:

Pi (ξ) =∑

‖j‖+`=i(−1)j

j! ad(Υk)jk · · · ad(Υ1)j1(ξ`) +∑k

∑‖j‖+m=i

j1=···=jm−1=0

(−1)j

j!(jm+1) ad(Υk)jk · · · ad(Υm)jm(∇ξΥm) +∑‖j‖+`=i

(−1)j

j! ad(Υk)jk · · · ad(Υ1)j1(P`(ξ)).

Structure

1 Bibliography

The derivatives

Invariant derivative

In analogy to the covariant derivatives we define ∇ω : C∞(G, V) →C∞(G, g∗− ⊗ V), ∇s(u)(X ) = ω−1(X )(u)·s ∈ V. This operationdoes not transform sections into sections, in general!

Fundamental derivative

The extension of the invariant derivative to argument X ∈ g

Dω : C∞(G, V) → C∞(G, g∗⊗V), ∇s(u)(X ) = ω−1(X )(u)·s ∈ Vis and invariant differential operator A∗ ⊗ V → V called thefundamental derivative.

Dωξ+ζs = ∇ξs + P(ξ) • s − ζ • s, ξ a vector in TM, ζ a vertical

vector on G0.

on all tractor bundles, the fundamental derivative is related tothe invariant linear connection ∇V

ξ s = ∇ξs + P(ξ) • s + ξ • s.

The derivatives

vector on G0.

ξ s = ∇ξs + P(ξ) • s + ξ • s.

The derivatives

vector on G0.

ξ s = ∇ξs + P(ξ) • s + ξ • s.

The derivatives

vector on G0.

ξ s = ∇ξs + P(ξ) • s + ξ • s.

Invariant jet operator

Both invariant and fundamental derivatives allow iteration!

1st order invariant jets

C∞(G, V) 3 s 7→ (s,∇ωs) ∈ C∞(G, V⊕ (g∗− ⊗ V))defines a natural operator (action given on J1V).

higher orders

C∞(G, V)P 3 s 7→ jkωs = (s,∇ωs, . . . , (∇ω)ks) ∈C∞(G, V⊕ · · · ⊕ (⊗kg∗− ⊗ V))P

is an invariant operator valued in semi–holonomic jets.

Note: Symmetrization provides similar formulae for holonomic jetsjkωs, but these are not equivariant!

higher orders

Bianchi and Ricci identities

The Bianchi identity in terms of the invariant differnetial is0 =∑

([κ(X ,Y ),Z ]+κ([X ,Y ],Z )−κ(κ(X ,Y ),Z )−∇ω

Zκ(X ,Y )

while the Ricci identity reads∇ω

X∇ωY s −∇ω

Y∇ωX s = ∇ω

[X ,Y ] −∇ωκ−(X ,Y )s + κ≥0(X ,Y ) • s

Note: also available in terms of the fundamental derivative.

([κ(X ,Y ),Z ]+κ([X ,Y ],Z )−κ(κ(X ,Y ),Z )−∇ω

Zκ(X ,Y )

)while the Ricci identity reads∇ω

X∇ωY s −∇ω

Y∇ωX s = ∇ω

[X ,Y ] −∇ωκ−(X ,Y )s + κ≥0(X ,Y ) • s

([κ(X ,Y ),Z ]+κ([X ,Y ],Z )−κ(κ(X ,Y ),Z )−∇ω

Zκ(X ,Y )

)while the Ricci identity reads∇ω

X∇ωY s −∇ω

Y∇ωX s = ∇ω

[X ,Y ] −∇ωκ−(X ,Y )s + κ≥0(X ,Y ) • s

The expansion

Proposition

Whatever expression we build of the invariant semiholonomic jetsof sections and the Cartan curvatures in terms of the invariantderivatives (or the fundamental derivatives) will expand inexpressions in the iterated covariant derivatives by means of theWeyl connections, the iterated covariant derivatives of thecurvature tensor of the Cartan connections and of the iteratedcovariant derivatives of the P tensors.

Example:∇ω

X∇ωY s(u) = (Dω)2s(u)(X ,Y ) = (∇X + P(v)(X ))(∇s(v))(Y ) =

(Tσ·γ−1(u)(X )− ω−1(P(v)(X ))(u))·(∇ω s)(u)(Y ) =∇X∇Y s(v) +∇[P(v)(X ),Y ]g−

s(v)− [P(v)(X ),Y ]p • s(v)

Main theorem

Proposition

The transformation properties of any expansion of a formula ininvariant derivatives and Cartan curvatures are algebraic in Υ’s.

Theorem

Every expression in affine invariants of the Weyl connections withalgebraic transformation rules in Υ’s is an expansion of someexpression in invariant derivatives and the Cartan curvature.

(Wünsch's) nearly invariant calculus for parabolic geometries · Bibliography Parabolic...

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