Short course On the isometry group of Lorentz manifolds

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Short course

On the isometry group of Lorentz manifolds

GELOGRA’11

VI International Meeting on Lorentzian Geometry Granada 2011

Paolo Piccione

Departamento de Matemática Instituto de Matemática e Estatística

Universidade de São Paulo

September 6–9, 2011

Paolo Piccione Isometry group of Lorentz manifolds

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Global geometry: Riemannian vs. Lorentzian

Compact Riemannian manifolds:

are complete and geodesically complete are geodesically connected

have compact isometry

Compact Lorentz manifolds:

may be geodesically incomplete

may fail to be

geodesically connected have possibly non compact isometry group

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Topics

Why is the set of Lorentzian isometries a (finite dimensional)Liegroup?

Classify the Lie groups that act isometrically (and faithfully) on (compact) Lorentzian manifolds.

Discuss compactness and non compactness issues. Can one obtain geometric information on the manifold when its isometry group is given?

Examples

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Topics

Discuss compactness and non compactness issues. Can one obtain geometric information on the manifold when its isometry group is given?

Examples

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Topics

Discuss compactness and non compactness issues.

Can one obtain geometric information on the manifold when its isometry group is given?

Examples

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Topics

Examples

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Topics

Examples

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Topics

Examples

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Some bibliography. —1

R. J. Zimmer, Inventiones 1986:

Let(M,g)be a compact manifold, and let G be a

connected Lie group acting by isometries locally faithfully on M. Then either of the two possibilities occur:

G is locally isomorphic toSL(2,R)×K , with K compact; G is amenable (equiv., compact modulo radical).

M. Gromov, 1988

If(M,g)is a Lorentz manifold with finite volume, and let G be a Lie group acting locally faithfully and isometrically on M. If the Zariski closure G^aofAd(G)⊂Aut(g)has no co-compact normal algebraic subgroup, then:

for almost every m∈M, the stabilizer G_m⊂G is discrete; eitherg=sl(2,R)⊕abelian, or there exists a

one-dimensional idealI₀⊂gsuch thatg/I₀is abelian.

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Some bibliography. —1

G is locally isomorphic toSL(2,R)×K , with K compact;

G is amenable (equiv., compact modulo radical).

M. Gromov, 1988

for almost every m∈M, the stabilizer G_m⊂G is discrete; eitherg=sl(2,R)⊕abelian, or there exists a

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Some bibliography. —1

G is locally isomorphic toSL(2,R)×K , with K compact;

G is amenable (equiv., compact modulo radical).

M. Gromov, 1988

for almost every m∈M, the stabilizer G_m⊂G is discrete;

eitherg=sl(2,R)⊕abelian, or there exists a

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Some Bibliography. —2

G. D’Ambra, Inventiones 1988

If(M,g)is a compact, simply connectedreal analytic Lorentz manifold, thenIso(M,g)is compact.

Example of compact simply connected

pseudo-Riemannian manifold of type(7,2)with non compact isometry group.

N. Kowalsky, Ann. Math. 1996

If G actsnon properlyon a compact Lorent manifold, and all G-stabilizers are discrete, then G is locally isomorphic to SL(2,R).

In general, if G acts non properly, then G is locally isomorphic to eitherSO(1,n)or toSO(2,n).

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Some Bibliography. —2

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Some Bibliography. —2

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Some Bibliography. —2

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Some Bibliography. —3

S. Adams, G. Stuck, Inventiones 1997, & A. Zeghib, 1997 Let G be a connected, simply connected Lie group. The following are equivalent:

G is the universal cover ofIso₀(M,g)for some compact Lorentz manifold M;

G is isomorphic to L×K ×R^d, where K is compact and semisimple, d ≥0, and L in the the following list:

SL(2,^R)

Heisenberg groupHeis²ⁿ⁺¹ an oscillator group.

A. Zeghig, 1998

Geometric description of compact Lorentz manifold with Iso0(M,g)non compact.

P.P., A. Zeghig, 2010

Geometric description of compact Lorentz manifold with Iso(M,g)

Iso₀(M,g)infinite.

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Some Bibliography. —3

G is isomorphic to L×K×R^d, where K is compact and semisimple, d ≥0, and L in the the following list:

SL(2,^R)

A. Zeghig, 1998

Geometric description of compact Lorentz manifold with Iso₀(M,g)non compact.

Iso0(M,g)infinite.

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Some Bibliography. —3

SL(2,^R)

A. Zeghig, 1998

Iso0(M,g)infinite.

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Some Bibliography. —3

SL(2,^R)

A. Zeghig, 1998

Iso0(M,g)infinite.

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Isometry group and covering spaces

(M,g)semi-Riemannian manifold π :Me →M covering,ge=π^∗(g) H =

ψe∈Iso(M,e eg) :π ψ(x)e

=π ψ(ye )

ifπ(x) =π(y) H 3ψe7−→^Φ ψ∈Iso(M,g) (ψ◦π =π◦ψ)e

Γ =Ker(Φ)group of covering automorphisms ofπ IfMe is theuniversal coverofMthen:

Φis onto

H=Nor(Γ) =⇒ Iso(M,g) =Nor(Γ) Γ.

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Isometry group and covering spaces

(M,g)semi-Riemannian manifold

π :Me →M covering,ge=π^∗(g) H =

=π ψ(ye )

Φis onto

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Isometry group and covering spaces

(M,g)semi-Riemannian manifold π :Me →M covering,ge=π^∗(g)

H =

=π ψ(ye )

Φis onto

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Isometry group and covering spaces

=π ψ(ye )

ifπ(x) =π(y)

H 3ψe7−→^Φ ψ∈Iso(M,g) (ψ◦π =π◦ψ)e Γ =Ker(Φ)group of covering automorphisms ofπ IfMe is theuniversal coverofMthen:

Φis onto

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Isometry group and covering spaces

=π ψ(ye )

Φis onto

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Isometry group and covering spaces

=π ψ(ye )

Γ =Ker(Φ)group of covering automorphisms ofπ

IfMe is theuniversal coverofMthen: Φis onto

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Isometry group and covering spaces

=π ψ(ye )

Φis onto

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Isometry group and frame bundles

Mⁿsmooth manifold,L(M)frame bundle,GL(n)-principal bundle

g semi-Riemannian metric onMof indexk Iso(M,g)isometry group

Lg(M)g-orthogonal frame bundle,O(n,k)-principal subbundle

p∈L_g(M), orbit: O_p=

ψ◦:ψ∈Iso(M,g) is aclosed subsetofLg(M)

Iso(M,g)3ψ7−→ψ◦p∈ O_pis a homeomorphism

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Two direct consequences

Proposition

If(M,g)is a compact Riemannian manifold, thenIso(M,g) is compact.

If(M,g)is compact Lorentzian,K ∈Kill(M,g),p∈M, g(Kp,Kp)<0, then the 1-parameter group of isometries generated byK is pre-compact.

Corollary (J. L. Flores, M. A. Javaloyes, P.P.)

Compact stationary Lorentzian manifolds admit at least two non trivial closed geodesics.

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Two direct consequences

Proposition

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Two direct consequences

Proposition

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Riemannian isometries 1

Theorem 1 (Gleason 1952, Yamabe 1953)

IfGis alocally compacttopological group, thenGadmits a (necessarily unique) Lie group structure compatible with the topology, if and only ifG doesn’t havesmall subgroups.

Def.:Ghassmall subgroupsif every neighborhood of 1 contains a nontrivial subgroup ofG.

Theorem 2

IfGis alocally compacttopological group,M is a smooth manifold that carries a continuouseffectiveaction by

diffeomorphismsG×M →MofG, thenGadmits a Lie group structure compatible with the topology, and the action is smooth.

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Riemannian isometries 1

Theorem 2

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Riemannian isometries 1

Theorem 2

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Riemannian isometries 2

(M,g)Riemannian manifold

,dg metric onM induced byg isometries of(M,g) ! isometries of(M,d_g) The isometry group of a locally compact metric space is a locally compact topological group, endowed with the compact-open topology.

It follows from Theorem 2 thatIso(M,g)is a Lie group with the compact-open topology.

This theory does not work for Lorentzian isometries/conformal diffeomorphisms

General question:For which groupsG⊂GL(n,R)the set of automorphisms of aG-manifoldis a Lie transformation group?

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Riemannian isometries 2

(M,g)Riemannian manifold,dg metric onM induced byg

isometries of(M,g) ! isometries of(M,d_g) The isometry group of a locally compact metric space is a locally compact topological group, endowed with the compact-open topology.

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Riemannian isometries 2

(M,g)Riemannian manifold,dg metric onM induced byg isometries of(M,g) ! isometries of(M,d_g)

The isometry group of a locally compact metric space is a locally compact topological group, endowed with the compact-open topology.

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Riemannian isometries 2

(M,g)Riemannian manifold,dg metric onM induced byg isometries of(M,g) ! isometries of(M,d_g) The isometry group of a locally compact metric space is a locally compact topological group, endowed with the compact-open topology.

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Riemannian isometries 2

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Riemannian isometries 2

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Riemannian isometries 2

General question:For which groupsG⊂GL(n,R)the set of

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Definitions and examples

Notation

M: differentiable (smooth) manifold of dimensionn<∞.

G: Lie subgroup (not necessarily closed) ofGL(n).

L(M): the frame bundle ofM.

Obs.:Gmay be non-connected, but we require that it is second-countable.

Definition

AG-structureB_G onMis a reductionπ:B_G →M of the structural group ofL(M)toG.

Characterization

Givenp ∈B_G andg∈GL(n), we havep◦g ∈B_G ⇐⇒g ∈G.

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Definitions and examples

Notation

Definition

AG-structureB_G onM is a reductionπ:B_G →M of the structural group ofL(M)toG.

Characterization

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Definitions and examples

Notation

Definition

AG-structureB_G onM is a reductionπ:B_G →M of the structural group ofL(M)toG.

Characterization

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Definitions and examples

1-structures or parallelisms

WhenG={1}, aG-structure is a choice of frame at each tangent space,TxM; i.e., aparallelismonM.

O(n, ν)-structures

WhenG=O(n, ν), aG-structure is equivalent to a semi-Riemannian metric onM of indexν.

Sp(n)-structures

WhenG=Sp(n), aG-structure is equivalent to analmost symplecticstructure onM. This means that the symplectic form Ωis not necessarily closed. One may write the integrability conditiondΩ =0 in the language ofG-structures, as we’ll see.

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Definitions and examples

O(n, ν)-structures

Sp(n)-structures

WhenG=Sp(n), aG-structure is equivalent to analmost symplecticstructure onM. This means that the symplectic form Ωis not necessarily closed. One may write the integrability conditiondΩ =0 in the language ofG-structures, as we’ll see.

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Definitions and examples

O(n, ν)-structures

Sp(n)-structures

WhenG=Sp(n), aG-structure is equivalent to analmost symplecticstructure onM. This means that the symplectic form Ωis not necessarily closed. One may write the integrability

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Definitions and examples

Example:O(n, ν)-structures

To define a semi-Riemannian metric from a given O(n, ν)-structure:

givenu,v ∈T_xM, put

hu,vi_x :=hp⁻¹(u),p⁻¹(v)i_Rⁿ_ν, p∈π⁻¹(x). (1) Conversely, to define aO(n, ν)-structure from a given

semi-Riemannian metric: givenx ∈M, put

π⁻¹(x) :={p ∈Iso(Rⁿ;TxM) : pis a linear isometry}. (2)

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Definitions and examples

To define a semi-Riemannian metric from a given O(n, ν)-structure: givenu,v ∈T_xM, put

hu,vi_x :=hp⁻¹(u),p⁻¹(v)i_Rⁿ_ν, p∈π⁻¹(x). (1)

Conversely, to define aO(n, ν)-structure from a given semi-Riemannian metric:

givenx ∈M, put

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Definitions and examples

semi-Riemannian metric:

givenx ∈M, put

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Definitions and examples

semi-Riemannian metric:

givenx ∈M, put

π⁻¹(x) :={p∈Iso(Rⁿ;TxM) : pis a linear isometry}. (2)

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Definitions and examples

Isomorphisms

LetB¹_G,B_G² beG-structures onM₁,M₂, respectively.

Each diffeomorphismf :M₁→M₂induces a mapf_?such that the following diagram commutes.

We say thatf is anisomorphismofB_G¹ ontoB_G² iff?(B¹_G) =B_G². In this case,f is anautomorphismwhenM₁=M₂and

B_G¹ =B²_G.

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Definitions and examples

Isomorphisms

We say thatf is anisomorphismofB_G¹ ontoB_G² iff?(B¹_G) =B_G². In this case,f is anautomorphismwhenM₁=M₂and

B_G¹ =B²_G.

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Definitions and examples

Isomorphisms

We say thatf is anisomorphismofB_G¹ ontoB_G² iff_?(B¹_G) =B_G². In this case,f is anautomorphismwhenM₁=M₂and

B_G¹ =B_G².

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First-order structure function

Preliminaries

Letπ:B_G →M be aG-structure. There’s a canonical vector-valued 1-form defined onB_G:

θp:Tp(B_G)→Rⁿ

X 7→p⁻¹◦dπ_p(X) (3)

Differentiatingθ, we get

dθ_p:T_p(B_G)∧T_p(B_G)→Rⁿ (4) ChooseH such thatH⊕ V_p=Tp(B_G). Then, we can define

c_H :Rⁿ∧Rⁿ→Rⁿ

u∧v 7→dθp(X ∧Y), (5) whereX,Y ∈H,θ_p(X) =u,θ_p(Y) =v.

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First-order structure function

Preliminaries

θp:Tp(B_G)→Rⁿ

X 7→p⁻¹◦dπ_p(X) (3) Differentiatingθ, we get

dθ_p:T_p(B_G)∧T_p(B_G)→Rⁿ (4)

ChooseH such thatH⊕ V_p=Tp(B_G). Then, we can define c_H :Rⁿ∧Rⁿ→Rⁿ

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First-order structure function

Preliminaries

θp:Tp(B_G)→Rⁿ

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First-order structure function

Preliminaries

θp:Tp(B_G)→Rⁿ

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First-order structure function

Dependence on horizontal complements

LetH₁,H₂be such thatH₁⊕ V_p'H₂⊕ V_p'T_p(B_G).

Define

S_H₂_,H₁ :Rⁿ→ V_p 'g

u7→X₂−X₁, (6) whereX_i ∈H_i,i =1,2, andθp(X₂) =θp(X₁) =u.

Then, the following equation holds

c_H₂(u∧v)−c_H₁(u∧v) =A(S_H₂_,H₁)(u∧v), (7) where

A:hom(Rⁿ;g)→hom(Rⁿ∧Rⁿ;Rⁿ) is the antisymmetrization operator.

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First-order structure function

Define

A:hom(Rⁿ;g)→hom(Rⁿ∧Rⁿ;Rⁿ) is the antisymmetrization operator.

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First-order structure function

Define

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First-order structure function

Definition

The first-order structure function ofB_G is c:B_G → hom(Rⁿ∧Rⁿ;Rⁿ)

A(hom(Rⁿ;g)) p7→[c_H]

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Remarks

c is an invariant ofB_G.

For someG-structures,c is identically zero, giving no information.

In some casesc measures (obstruction to) integrability. For instance: almost symplectic structures, distributions, almost complex structures (Newlander-Nirenberg theorem)

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First-order structure function

Definition

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Remarks

For someG-structures,c is identically zero, giving no information.

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First-order structure function

Definition

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Remarks

For someG-structures,cis identically zero, giving no information.

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First-order structure function

Definition

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Remarks

For someG-structures,cis identically zero, giving no information.

In some casesc measures (obstruction to) integrability.

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First prolongation of a G-structure

Definition

ChooseH ⊆hom(Rⁿ∧Rⁿ;Rⁿ)such that

H ⊕ A(hom(Rⁿ;g))'hom(Rⁿ∧Rⁿ;Rⁿ)

This choice induces a family of special frames onB_G z :Rⁿ×g→T_p(B_G)

such that:

z(A) =A, forA∈g;

z(Rⁿ× {0}) =H, for whichc_H ∈ H.

These frames are actually aG⁽¹⁾-structureon B_G, called the first prolongation of B_G and denoted byB_G⁽¹⁾.

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First prolongation of a G-structure

Definition

H ⊕ A(hom(Rⁿ;g))'hom(Rⁿ∧Rⁿ;Rⁿ) This choice induces a family of special frames onB_G

z :Rⁿ×g→T_p(B_G) such that:

z(A) =A, forA∈g;

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First prolongation of a G-structure

Definition

z(A) =A, forA∈g;

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First prolongation of a G-structure

Definition

z(A) =A, forA∈g;

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First prolongation of a G-structure

Definition

z(A) =A, forA∈g;

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Prolongations of g

Definition

To understand the groupG⁽¹⁾, define the first prolongation of g⊆Rⁿ⊗(Rⁿ)^∗ by

g⁽¹⁾:=

g⊗(Rⁿ)^∗

∩

Rⁿ⊗S² (Rⁿ)^∗

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We define inductively theith prolongation ofgby g⁽ⁱ⁾:=

g⁽ⁱ⁻¹⁾⊗(Rⁿ)^∗

∩

Rⁿ⊗Sⁱ (Rⁿ)^∗

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Remark

g⁽ⁱ⁾ may be realized as the space of multilinear symmetric functionsT : (Rⁿ)ⁱ⁺¹→Rⁿsuch that

∀v₁, . . . ,v_i ∈Rⁿ, v 7→T(v₁, . . . ,v_i,v)∈g. (11)

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Prolongations of g

Definition

g⁽¹⁾:=

g⊗(Rⁿ)^∗

∩

(9) We define inductively theith prolongation ofgby

g⁽ⁱ⁾:=

g⁽ⁱ⁻¹⁾⊗(Rⁿ)^∗

∩

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Remark

∀v₁, . . . ,v_i ∈Rⁿ, v 7→T(v₁, . . . ,v_i,v)∈g. (11)

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Prolongations of g

Definition

g⁽¹⁾:=

g⊗(Rⁿ)^∗

∩

(9) We define inductively theith prolongation ofgby

g⁽ⁱ⁾:=

g⁽ⁱ⁻¹⁾⊗(Rⁿ)^∗

∩

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Remark

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Prolongations of B

_G

Definition

The groupG⁽¹⁾⊆GL n+dim(g)

consists of all the matrices of the form

I_Rⁿ 0 T I_g

, whereT ∈g⁽¹⁾.

We define, inductively,G⁽ⁱ⁾:= G⁽ⁱ⁻¹⁾(1)

,∀i ≥1. Remark

In general, the Lie algebra ofG⁽ⁱ⁾is a representation ofg⁽ⁱ⁾. Ifg⁽ⁱ⁾={0}for somei, thenG⁽ⁱ⁾={1}.

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Prolongations of B

_G

Definition

I_Rⁿ 0 T I_g

,

whereT ∈g⁽¹⁾.

,∀i ≥1.

Remark

In general, the Lie algebra ofG⁽ⁱ⁾is a representation ofg⁽ⁱ⁾. Ifg⁽ⁱ⁾={0}for somei, thenG⁽ⁱ⁾={1}.

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Prolongations of B

_G

Definition

I_Rⁿ 0 T I_g

,

whereT ∈g⁽¹⁾.

,∀i ≥1.

Remark

In general, the Lie algebra ofG⁽ⁱ⁾is a representation ofg⁽ⁱ⁾. Ifg⁽ⁱ⁾={0}for somei, thenG⁽ⁱ⁾ ={1}.

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The semi-Riemannian isometry group

Example

The groupO(n, ν)is of finite type and order 1.

To see this, let u,v,w ∈Rⁿbe any vectors, and letA∈o(n, ν)⁽¹⁾. Then,

hA(u)v,wi_Rⁿ

ν =hA(v)u,wi_Rⁿ

ν =−hv,A(u)wi_Rⁿ

ν =

−hv,A(w)ui_Rⁿ

ν =hA(w)v,ui_Rⁿ

ν =hA(v)w,ui_Rⁿ

ν =

−hw,A(v)ui_Rⁿ_ν =−hA(v)u,wi_Rⁿ_ν, =⇒ A=0.

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Some other groups

Sp(n), the symplectic group, is ofinfinite type.

CO(n, ν), the semi-Riemannian conformal group, is of finite type and order 2, forn>2.

In general,∀k ∈N∃n∈NandG⊆GL(n)s.t.Gis of finite type and has orderk.

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The semi-Riemannian isometry group

Example

The groupO(n, ν)is of finite type and order 1. To see this, let u,v,w ∈Rⁿbe any vectors, and letA∈o(n, ν)⁽¹⁾. Then,

hA(u)v,wi_Rⁿ

ν =hA(v)u,wi_Rⁿ

ν =

−hv,A(w)ui_Rⁿ

ν =hA(w)v,ui_Rⁿ

ν =hA(v)w,ui_Rⁿ

ν =

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Some other groups

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The semi-Riemannian isometry group

Example

hA(u)v,wi_Rⁿ

ν =hA(v)u,wi_Rⁿ

ν =

−hv,A(w)ui_Rⁿ

ν =hA(w)v,ui_Rⁿ

ν =hA(v)w,ui_Rⁿ

ν =

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Some other groups

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The semi-Riemannian isometry group

Example

hA(u)v,wi_Rⁿ

ν =hA(v)u,wi_Rⁿ

ν =

−hv,A(w)ui_Rⁿ

ν =hA(w)v,ui_Rⁿ

ν =hA(v)w,ui_Rⁿ

ν =

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Some other groups

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The semi-Riemannian isometry group

Example

hA(u)v,wi_Rⁿ

ν =hA(v)u,wi_Rⁿ

ν =

−hv,A(w)ui_Rⁿ

ν =hA(w)v,ui_Rⁿ

ν =hA(v)w,ui_Rⁿ

ν =

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Some other groups

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The semi-Riemannian isometry group

Example

hA(u)v,wi_Rⁿ

ν =hA(v)u,wi_Rⁿ

ν =

−hv,A(w)ui_Rⁿ

ν =hA(w)v,ui_Rⁿ

ν =hA(v)w,ui_Rⁿ

ν =

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Some other groups

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Prolongations of B

_G

Proposition 1

There is a sequence of principle fiber bundles M ←−^π B_G ←−^π¹ B_G⁽¹⁾ ←−^π² B⁽²⁾_G ←−^π³ . . .

such that the structure group ofB⁽ⁱ⁾_G isG⁽ⁱ⁾.B_G⁽ⁱ⁾ is defined as

B_G⁽ⁱ⁻¹⁾(1)

.

Definition

Ifg⁽ⁱ⁾={0}for somei∈N, we say thatG(and anyG-structure, for suchG) is offinite type. The smallest suchi is called the orderof theG-structure.

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Prolongations of B

_G

Proposition 1

There is a sequence of principle fiber bundles M ←−^π B_G ←−^π¹ B_G⁽¹⁾ ←−^π² B⁽²⁾_G ←−^π³ . . .

such that the structure group ofB⁽ⁱ⁾_G isG⁽ⁱ⁾.B_G⁽ⁱ⁾ is defined as

B_G⁽ⁱ⁻¹⁾(1)

. Definition

Ifg⁽ⁱ⁾={0}for somei∈N, we say thatG(and anyG-structure, for suchG) is offinite type. The smallest suchi is called the orderof theG-structure.