Isometry group and geometry of compact stationary Lorentz ...piccione/Downloads/slidesOSU_september2014.pdfIsometry group and geometry of compact stationary Lorentz manifolds Ohio

Isometry group and geometry of compactstationary Lorentz manifolds

Ohio State University

Joint work with A. Zeghib, ENS-Lyon, FranceErgodic Theory and Dynamical Systems 34/5 , 2014

Paolo Piccione

Departamento de MatemáticaInstituto de Matemática e Estatística

Universidade de São Paulo

September 11th, 2014

Paolo Piccione Geometry of compact stationary manifolds

Moduli space of a compact manifold

M compact manifold

Diff(M) group of diffeomorphisms of M

Riem(M) set of Riemannian metrics on M

action: Diff(M)× Riem(M) −→ Riem(M) (pull-back)

Moduli space of M: X = Riem(M)/Diff(M) space of orbits

Diff(M) acts properly on Riem(M) =⇒ X is Hausdorff.

a function F : X −→ R is a Riemannian invariant (diameter,volume, total curvature, injectivity radius, etc.)



M compact manifold









M compact manifold









M compact manifold









M compact manifold









M compact manifold









M compact manifold









M compact manifold








Action on Lorentzian metrics

Super question: When does Diff(M) act properly on Lor(M)?

Recall: G acts properly on X if for all compact K ⊂ X , the setof return times:

GK ={

g ∈ G : gK ∩ K 6= ∅}

is compact.

Gromov: the difficulty in studying global properties ofLorentzian manifolds lies in the fact that Lor(M)/Diff(M) is notHausdorff.





GK ={

g ∈ G : gK ∩ K 6= ∅}

is compact.






GK ={

g ∈ G : gK ∩ K 6= ∅}

is compact.



Isometry group

If the Diff(M)-action onLor(M) is proper =⇒

for all g ∈ Lor(M),Stabilizer(g) is compact.

Stabilizer(g) = Iso(M,g)

Questions:

When is the isometry group of a compact Lorentzianmanifold compact/non compact?

In the non compact case: classify Lorentz manifolds forwhich G = Iso(M,g) acts non properly.


Isometry group




Questions:




Isometry group




Questions:




Isometry group




Questions:




Global geometry: Riemannian vs. Lorentzian

Compact Riemannianmanifolds:

are complete andgeodesically completeare geodesicallyconnectedhave compact isometrygroup

Compact Lorentz manifolds:

may be geodesicallyincompletemay fail to begeodesically connectedhave possibly noncompact isometry group


Non-compactness

Assume G = Iso(M,g) non compact. Then:

Cases:

(a) G0 identity connected component of G non compact

or

(b) Γ = G/G0 the discrete part infinite

Theorem

If G0 has a timelike orbit (i.e., (M,g) has a somewhere timelikeKilling field), then (a) and (b) are mutually exclusive.


Non-compactness


Cases:


or


Theorem



Non-compactness


Cases:


or


Theorem



Non-compactness


Cases:


or


Theorem



Isometry group and frame bundles

Mn smooth manifold, L(M) frame bundle, GL(n)-principalbundle

g semi-Riemannian metric on M of index k

Iso(M,g) isometry group

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

p ∈ Lg(M), orbit: Op ={ψ◦ : ψ ∈ Iso(M,g)

}is a closed

subset of Lg(M)

Iso(M,g) 3 ψ 7−→ ψ ◦ p ∈ Op is a homeomorphism


Two direct consequences

Proposition

If (M,g) is a compact Riemannian manifold, then Iso(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 isometriesgenerated by K is pre-compact.

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

Compact stationary Lorentzian manifolds admit at least two nontrivial closed geodesics.



Proposition







Proposition






Lack of compactness of Iso(M,g)

Unlike Riemannian isometries, Lorentzian isometries:need not be equicontinuousmay generate chaotic dynamics on the manifold

Example

Dynamics of Lorentzian isometries can be of Anosov type,evocative of the fact that in General Relativity one can havecontractions in time and expansion in space.

q Lorentzian quadratic form in Rn,Iso(Rn+1,q) = O(q) ∼= O(n,1) non compact.The orthogonal frame bundle Fr(M,g) has non compactfibers. Iso(M,g) is identified topologically with any of itsorbits in Fr(M,g).




Example






Example


q Lorentzian quadratic form in Rn,Iso(Rn+1,q) = O(q) ∼= O(n,1) non compact.

The orthogonal frame bundle Fr(M,g) has non compactfibers. Iso(M,g) is identified topologically with any of itsorbits in Fr(M,g).




Example




An easy examples of non compactness (D’Ambra)

Consider A =

(a bc d

)∈ SL(2,Z) satisfying:

A diagonalizable on R (for instance, b = c):Eigenvalues: λ1, λ2 ∈ R, λ1 > 1, λ2 = λ−1

1

e1,e2 eigenvectors

Define a Lorentz metric g on R2 by settingg(e1,e1) = g(e2,e2) = 0, g(e1,e2) = 1.

g induces a (flat) metric on the torus T2 = R2/Z2

A is an isometry of (R2,g) and also of (T2,g)

Iso(T2,g) is not compact,since Ak →∞ as k → +∞ (λk

1 →∞)



Consider A =

(a bc d


A diagonalizable on R (for instance, b = c):

Eigenvalues: λ1, λ2 ∈ R, λ1 > 1, λ2 = λ−11

e1,e2 eigenvectorsDefine a Lorentz metric g on R2 by settingg(e1,e1) = g(e2,e2) = 0, g(e1,e2) = 1.




1 →∞)



Consider A =

(a bc d



1

e1,e2 eigenvectors

Define a Lorentz metric g on R2 by settingg(e1,e1) = g(e2,e2) = 0, g(e1,e2) = 1.




1 →∞)



Consider A =

(a bc d



1



A is an isometry of (R2,g)

and also of (T2,g)


1 →∞)



Consider A =

(a bc d



1



A is an isometry of (R2,g)

and also of (T2,g)


1 →∞)



Consider A =

(a bc d



1





1 →∞)



Consider A =

(a bc d



1





1 →∞)



Consider A =

(a bc d



1





1 →∞)



Consider A =

(a bc d



1





1 →∞)


Classification of connected groups

(M,g) compact Lorentz manifold.

Theorem (D’Ambra, Inventiones 1988)

If (M,g) is analytic and simply connected, then Iso(M,g) iscompact.

Theorem (Adams, Stuck, Zeghib, 1997)

The identity component Iso0(M,g) is direct product:

A× K × HA is abelianK is compactH is locally isomorphic to:

SL(2,R)an oscillator groupa Heisenberg group.




















Geometry of M via isometry group

Theorem (Zeghib)

If Iso0(M,g) contains a group locally isomorphic to SL(2,R),then M̃ is a warped product of ˜SL(2,R) and a Riemannianmanifold.

An analogous result when Iso0(M,g) contains a group locallyisomorphic to an oscillator group.

Oscillator groups: characterized as the only simply connectedsolvable non abelian Lie groups that admit bi-invariant Lorentzmetrics (Medina, Revoy, 1985). G = S1 n Heis

Action of S1 on the Lie algebra heis:Positivity conditions on the eigenvalues =⇒ existence ofbi-invariant Lorentz metricsarithmetic conditions =⇒ existence of lattices.



Theorem (Zeghib)

If Iso0(M,g) contains a group locally isomorphic to SL(2,R),then M̃ is a warped product of ˜SL(2,R) and a Riemannianmanifold.An analogous result when Iso0(M,g) contains a group locallyisomorphic to an oscillator group.





Theorem (Zeghib)






Theorem (Zeghib)





On a (false) conjecture

(M,g) compact Lorentz manifoldK Killing field of (M,g), p ∈ M, g(Kp,Kp) < 0the 1-parameter group of isometries generated by K ispre-compact in Iso0(M,g)

in this situation, Iso(M,g) has a non empty open cone ofvectors that generate a precompact 1-parameter subgroupof Iso(M,g)

Conjecture: Given a connected G Lie group, g = Lie(G), if ghas a non empty open cone of vectors v such that t 7→ exp(t · v)is precompact in G, then G is compact.

Counterexample: G = SL(2,R)














An algebraic criterion for pre-compactness

Theorem

Let G be a connected Lie group, K ⊂ G a maximal compactsubgroup and k ⊂ g their Lie algebras. Let m be anAdK -invariant complement of k in g.

Then, g has a non empty open cone of vectors that generateprecompact 1-parameter subgroups of G if and only if thereexists v ∈ k such that the restriction adv : m→ m is anisomorphism.


An algebraic criterion for pre-compactness

Theorem

Let G be a connected Lie group, K ⊂ G a maximal compactsubgroup and k ⊂ g their Lie algebras. Let m be anAdK -invariant complement of k in g.Then, g has a non empty open cone of vectors that generateprecompact 1-parameter subgroups of G if and only if thereexists v ∈ k such that the restriction adv : m→ m is anisomorphism.


Proof of the algebraic criterion

Proof

C ={v ∈ g : exp(tv) is precompact

}C ⊂ k′, k′ = Lie(K ′), K ′ ⊂ G maximal compactall maximal compact subgroups are conjugated =⇒C = AdG(k)

F : G × k→ g, F (g, v) = Adg(v)

C = Im(F ) has non empty interior iff F has maximal rank atsome point (Sard)by equivariance, iff it has maximal rank at some point (e, v)

dF(e,v)(g, k) = [g, v] + k = [m, v] + k.



Proof


}

C ⊂ k′, k′ = Lie(K ′), K ′ ⊂ G maximal compactall maximal compact subgroups are conjugated =⇒C = AdG(k)

F : G × k→ g, F (g, v) = Adg(v)


dF(e,v)(g, k) = [g, v] + k = [m, v] + k.



Proof


}C ⊂ k′, k′ = Lie(K ′), K ′ ⊂ G maximal compact

all maximal compact subgroups are conjugated =⇒C = AdG(k)

F : G × k→ g, F (g, v) = Adg(v)


dF(e,v)(g, k) = [g, v] + k = [m, v] + k.



Proof



F : G × k→ g, F (g, v) = Adg(v)


dF(e,v)(g, k) = [g, v] + k = [m, v] + k.



Proof



F : G × k→ g, F (g, v) = Adg(v)


dF(e,v)(g, k) = [g, v] + k = [m, v] + k.



Proof



F : G × k→ g, F (g, v) = Adg(v)

C = Im(F ) has non empty interior iff F has maximal rank atsome point (Sard)

by equivariance, iff it has maximal rank at some point (e, v)

dF(e,v)(g, k) = [g, v] + k = [m, v] + k.



Proof



F : G × k→ g, F (g, v) = Adg(v)


dF(e,v)(g, k) = [g, v] + k = [m, v] + k.



Proof



F : G × k→ g, F (g, v) = Adg(v)


dF(e,v)(g, k) = [g, v] + k = [m, v] + k.


Iso0(M,g): stationary case

Corollary 1

Let (M,g) be a compact Lorentz manifold that has a Killingvector field which is timelike somewhere. Then, Iso0(M,g) iscompact unless it contains a group locally isomorphic toSL(2,R) or to an oscillator group.

Proof

assume Iso(M,g) = h + a + c, h=Heisenberg, a=abelian,c=compact semi-simplesince c is compact, then we can assume g = h + a

can assume A simply connectedh + a nilpotent =⇒ for no v ∈ h + a the map adv is injective.



Corollary 1


Proof





Corollary 1


Proof

assume Iso(M,g) = h + a + c, h=Heisenberg, a=abelian,c=compact semi-simple

since c is compact, then we can assume g = h + a




Corollary 1


Proof





Corollary 1


Proof


can assume A simply connected

h + a nilpotent =⇒ for no v ∈ h + a the map adv is injective.



Corollary 1


Proof




Non compact isometry group

Corollary 2

If (M,g) admits a somewhere timelike Killing vector field, thenthe two conditions are mutually exclusive:(a) Iso0(M,g) is not compact;(b) Iso(M,g) has infinitely many connected components.

Proof. Use Corollary 1 and Zeghib’s classification:If Iso0(M,g) contains a group locally isomorphic to SL(2,R) orto an oscillator group then:

Iso(M,g) has only a finite number of connectedcomponents;M is not simply connected.



Corollary 2


Proof. Use Corollary 1 and Zeghib’s classification:

If Iso0(M,g) contains a group locally isomorphic to SL(2,R) orto an oscillator group then:




Corollary 2


Proof. Use Corollary 1 and Zeghib’s classification:If Iso0(M,g) contains a group locally isomorphic to SL(2,R) orto an oscillator group then:



The problem

Classify compact Lorentz manifolds (M,g) with:

G0 having a timelike orbit

Γ = G/G0 infinite


Compact Lorentz manifolds with large isometry group

Definition

ρ : Γ→ GL(E) representation.

Then, ρ is said to be:

of Riemannian type if it preserves some positive definiteinner product on E ;of post-Riemannian type if it preserves some positivesemi-definite inner product on E with kernel of dimension 1.

Obs.: ρ : Γ→ GL(E) of Riemannian type⇐⇒ ρ(Γ) precompact.

Proposition

(M,g) compact Lorentz manifold. If the conjugacy action ofΓ = Iso(M,g)/Iso0(M,g) on Iso0(M,g) is not ofpost-Riemannian type, then Iso0(M,g) has a timelike orbit in M,and Iso(M,g) has infinitely many connected components.



Definition

ρ : Γ→ GL(E) representation.Then, ρ is said to be:of Riemannian type if it preserves some positive definiteinner product on E ;

of post-Riemannian type if it preserves some positivesemi-definite inner product on E with kernel of dimension 1.


Proposition




Definition

ρ : Γ→ GL(E) representation.Then, ρ is said to be:of Riemannian type if it preserves some positive definiteinner product on E ;of post-Riemannian type if it preserves some positivesemi-definite inner product on E with kernel of dimension 1.


Proposition




Definition



Proposition




Definition



Proposition



Paradigmatic example

q Lorentz form in Rn

it induces a flat Lorentz metric on Tn = Rn/Zn

Linear isometry group of Tn: O(q,Z) = GL(n,Z) ∩ O(q)

Full isometry group: O(q,Z) nTn

For generic q, O(q,Z) is trivial.If q is rational, by Harich–Chandra–Borel theorem O(q,Z)is big in O(q).When q not rational, many intermediate situations occur.Complicated dynamics of hyperbolic elements A ∈ O(q,Z):they may have Salem numbers in their spectrum.

Theorem (P.P., A. Zeghib)

Compact Lorentzian manifolds with large isometry groups areessentially built up by tori.











































For generic q, O(q,Z) is trivial.

If q is rational, by Harich–Chandra–Borel theorem O(q,Z)is big in O(q).When q not rational, many intermediate situations occur.Complicated dynamics of hyperbolic elements A ∈ O(q,Z):they may have Salem numbers in their spectrum.









For generic q, O(q,Z) is trivial.If q is rational, by Harich–Chandra–Borel theorem O(q,Z)is big in O(q).

When q not rational, many intermediate situations occur.Complicated dynamics of hyperbolic elements A ∈ O(q,Z):they may have Salem numbers in their spectrum.









For generic q, O(q,Z) is trivial.If q is rational, by Harich–Chandra–Borel theorem O(q,Z)is big in O(q).When q not rational, many intermediate situations occur.

Complicated dynamics of hyperbolic elements A ∈ O(q,Z):they may have Salem numbers in their spectrum.






















The structure result

Theorem

Let (M,g) be a compact Lorentz manifold that has asomewhere timelike Killing vector field, and whose isometrygroup Iso(M,g) has infinitely many connected components.Then:

Iso0(M,g) contains a torus Td endowed with a Lorentzform q, such that Γ is a subgroup of O(q,Z);up to finite cover, M is:

either a direct product Td × N, with N compact Riemannianmanifoldor an amalgamated metric product Td ×S1 L, where L is alightlike manifold with an isometric S1-action.


Amalgamated metric products

Amalgamated product X ×S1 Y :

X and Y manifolds carrying a smooth action of S1.Z = (X × Y )/S1 diagonal action.Assume X Lorentzian, Y Riemannian (or lightlike), andaction of S1 isometricIdentify T(x0,y0)Z with Tx0X × {S1 − orbit through y0}⊥

Endow T(x0,y0)Z with the induced metric (Lorentzian).

Long exact homotopy sequence of the fibrationX × Y → (X × Y )/S1:

Z ∼= π1(S1)→ π1(X )× π1(Y )→ π1(Z )→ π0(S1) ∼= {1}

Proposition

If π1(X )× π1(Y ) is not cyclic, then (X × Y )/S1 is not simplyconnected.



Amalgamated product X ×S1 Y :X and Y manifolds carrying a smooth action of S1.

Z = (X × Y )/S1 diagonal action.Assume X Lorentzian, Y Riemannian (or lightlike), andaction of S1 isometricIdentify T(x0,y0)Z with Tx0X × {S1 − orbit through y0}⊥



Z ∼= π1(S1)→ π1(X )× π1(Y )→ π1(Z )→ π0(S1) ∼= {1}

Proposition




Amalgamated product X ×S1 Y :X and Y manifolds carrying a smooth action of S1.Z = (X × Y )/S1 diagonal action.

Assume X Lorentzian, Y Riemannian (or lightlike), andaction of S1 isometricIdentify T(x0,y0)Z with Tx0X × {S1 − orbit through y0}⊥



Z ∼= π1(S1)→ π1(X )× π1(Y )→ π1(Z )→ π0(S1) ∼= {1}

Proposition




Amalgamated product X ×S1 Y :X and Y manifolds carrying a smooth action of S1.Z = (X × Y )/S1 diagonal action.Assume X Lorentzian, Y Riemannian (or lightlike), andaction of S1 isometric

Identify T(x0,y0)Z with Tx0X × {S1 − orbit through y0}⊥



Z ∼= π1(S1)→ π1(X )× π1(Y )→ π1(Z )→ π0(S1) ∼= {1}

Proposition




Amalgamated product X ×S1 Y :X and Y manifolds carrying a smooth action of S1.Z = (X × Y )/S1 diagonal action.Assume X Lorentzian, Y Riemannian (or lightlike), andaction of S1 isometricIdentify T(x0,y0)Z with Tx0X × {S1 − orbit through y0}⊥



Z ∼= π1(S1)→ π1(X )× π1(Y )→ π1(Z )→ π0(S1) ∼= {1}

Proposition







Z ∼= π1(S1)→ π1(X )× π1(Y )→ π1(Z )→ π0(S1) ∼= {1}

Proposition





Endow T(x0,y0)Z with the induced metric (Lorentzian).Long exact homotopy sequence of the fibrationX × Y → (X × Y )/S1:

Z ∼= π1(S1)→ π1(X )× π1(Y )→ π1(Z )→ π0(S1) ∼= {1}

Proposition





Endow T(x0,y0)Z with the induced metric (Lorentzian).Long exact homotopy sequence of the fibrationX × Y → (X × Y )/S1:

Z ∼= π1(S1)→ π1(X )× π1(Y )→ π1(Z )→ π0(S1) ∼= {1}

Proposition



Two interesting consequences

Theorem

Assume Iso(M,g) non compact. If there is a somewheretimelike Killing vector field, then there is an everywhere timelikeKilling vector field.

Theorem

If (M,g) admits a somewhere timelike Killing vector field and Mis simply connected, then Iso(M,g) is compact.

Proof.When Iso0(M,g) contains a group locally isomorphic toSL(2,R) or to an oscillator group use Zeghib’s classification.When Iso(M,g) has infinitely many connected components, usethe structure result.



Theorem


Theorem





Theorem


Theorem




Main ingredients of the proof

(a) a fixed point theorem in linear Lorentzian dynamics(b) a Gauss map

Using (a) and (b), one shows that when G0 has a timeline orbit:

Γ-action is post-Riemannian⇐⇒ Γ is infinite


A fixed point theorem in linear Lorentz dynamics

Theorem

Γ a group, ρ : Γ→ GL(E) linear representation

F space of quadratic form on E , ρF associatedrepresentation

Assume:1 ρ(Γ) not pre-compact2 some Lorentz form has bounded ρF -orbit.

Then, (some finite index subgroup of) Γ preserves a Lorentzform.



Theorem

Γ a group, ρ : Γ→ GL(E) linear representationF space of quadratic form on E , ρF associatedrepresentation





Theorem






Theorem





The Gauss map

Killing fields

Iso(M,g) 3 v∼=7−→ K v ∈ Kill(M,g).

K v infinitesimal generator of t 7→ exp(tv)

Φ ∈ Iso(M,g) =⇒ Φ∗(K v ) = K AdΦ(v)

Gauss map:G : M −→ Sym

(Iso(M,g)

)Gp(v ,w) = g

(K v (p),K w (p)

)Proposition

If the action of Γ on Iso0(M,g) is not of post-Riemannian type,then Iso0(M,g) has somewhere timelike orbits.Proof: Use k(v ,w) =

∫M Gp(v ,w) dp.


The Gauss map

Killing fields





(Iso(M,g)

)Gp(v ,w) = g

(K v (p),K w (p)

)Proposition


∫M Gp(v ,w) dp.


The Gauss map

Killing fields





(Iso(M,g)

)Gp(v ,w) = g

(K v (p),K w (p)

)

Proposition


∫M Gp(v ,w) dp.


The Gauss map

Killing fields





(Iso(M,g)

)Gp(v ,w) = g

(K v (p),K w (p)

)Proposition


∫M Gp(v ,w) dp.


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Isometry group and geometry of compact stationary Lorentz ...piccione/Downloads/slidesOSU_september2014.pdfIsometry group and geometry of compact stationary Lorentz manifolds Ohio