Isometry group and geometry of compact stationary Lorentz...

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

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:

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.

g ∈ G : gK ∩ K 6= ∅}

is compact.

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

(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:

Theorem

Non-compactness

Cases:

Theorem

Non-compactness

Cases:

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

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

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

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

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

A is an isometry of (R2,g)

and also of (T2,g)

1 →∞)

Consider A =

(a bc d

A is an isometry of (R2,g)

and also of (T2,g)

1 →∞)

Consider A =

(a bc d

1 →∞)

Consider A =

(a bc d

1 →∞)

Consider A =

(a bc d

1 →∞)

Consider A =

(a bc d

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)

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

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)