Rigidity of Compact Pseudo-Riemannian Homogeneous Spaces for Solvable Lie Groups

Baues, Oliver; Globke, Wolfgang

doi:10.1093/imrn/rnw320

Cited by 6 publications

(14 citation statements)

References 19 publications

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“…In favorable cases, nil‐invariance of

⟨ \cdot, \cdot ⟩

already implies invariance. For solvable Lie algebras

frakturg

, this is always the case, as was first shown in . These results are briefly summarized in Section 4.…”

Section: Introduction and Main Resultsmentioning

confidence: 62%

“…Pseudo‐Riemannian homogeneous spaces of arbitrary index were studied by Baues and Globke for solvable Lie groups

G

. They found that, for solvable

G

, the finite volume condition implies that the stabilizer

H

is a lattice in

G

and that the metric on

M

is induced by a bi‐invariant metric on

G

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…They found that, for solvable

G

, the finite volume condition implies that the stabilizer

H

is a lattice in

G

and that the metric on

M

is induced by a bi‐invariant metric on

G

. Also, it was observed in that the nil‐invariance condition holds for the isometry Lie algebras of finite volume homogeneous spaces, where it appears as a direct consequence of the Borel density theorem. The main result in amounts to showing the surprising fact that any nil‐invariant symmetric bilinear form on a solvable Lie algebra

frakturg

is, in fact, an invariant form (a concise proof is also provided in Appendix ).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The first two authors studied nil‐invariant symmetric bilinear forms on solvable Lie algebras in . The main result [, Theorem 1.2] is: Theorem Let

frakturg

be a solvable Lie algebra and

⟨ \cdot, \cdot ⟩

a nil‐invariant symmetric bilinear form on

frakturg

. Then

⟨ \cdot, \cdot ⟩

is invariant.…”

Section: Review Of the Solvable Casementioning

confidence: 99%

“…In particular,

frakturg

is abelian if

⟨ \cdot, \cdot ⟩

is definite. This implies [, Proposition 5.4]: Proposition Let

(g, ⟨ \cdot, \cdot ⟩)

be a solvable metric Lie algebra with nil‐invariant symmetric bilinear form

⟨ \cdot, \cdot ⟩

. After a finite sequence of reductions with respect to totally isotropic and central ideals,

(g, ⟨ \cdot, \cdot ⟩)

reduces to an abelian metric Lie algebra.…”

Section: Review Of the Solvable Casementioning

confidence: 99%

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Isometry Lie algebras of indefinite homogeneous spaces of finite volume

Baues

Globke

Zeghib

2019

Proc. London Math. Soc.

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Let frakturg be a real finite‐dimensional Lie algebra equipped with a symmetric bilinear form ⟨·,·⟩. We assume that ⟨·,·⟩ is nil‐invariant. This means that every nilpotent operator in the smallest algebraic Lie subalgebra of endomorphisms containing the adjoint representation of frakturg is an infinitesimal isometry for ⟨·,·⟩. Among these Lie algebras are the isometry Lie algebras of pseudo‐Riemannian manifolds of finite volume. We prove a strong invariance property for nil‐invariant symmetric bilinear forms, which states that the adjoint representations of the solvable radical and all simple subalgebras of non‐compact type of frakturg act by infinitesimal isometries for ⟨·,·⟩. Moreover, we study properties of the kernel of ⟨·,·⟩ and the totally isotropic ideals in frakturg in relation to the index of ⟨·,·⟩. Based on this, we derive a structure theorem and a classification for the isometry algebras of indefinite homogeneous spaces of finite volume with metric index at most 2. Examples show that the theory becomes significantly more complicated for index greater than 2. We apply our results to study simply connected pseudo‐Riemannian homogeneous spaces of finite volume.

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“…In favorable cases, nil‐invariance of

⟨ \cdot, \cdot ⟩

already implies invariance. For solvable Lie algebras

frakturg

, this is always the case, as was first shown in . These results are briefly summarized in Section 4.…”

Section: Introduction and Main Resultsmentioning

confidence: 62%

“…Pseudo‐Riemannian homogeneous spaces of arbitrary index were studied by Baues and Globke for solvable Lie groups

G

. They found that, for solvable

G

, the finite volume condition implies that the stabilizer

H

is a lattice in

G

and that the metric on

M

is induced by a bi‐invariant metric on

G

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…They found that, for solvable

G

, the finite volume condition implies that the stabilizer

H

is a lattice in

G

and that the metric on

M

is induced by a bi‐invariant metric on

G

frakturg

is, in fact, an invariant form (a concise proof is also provided in Appendix ).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The first two authors studied nil‐invariant symmetric bilinear forms on solvable Lie algebras in . The main result [, Theorem 1.2] is: Theorem Let

frakturg

be a solvable Lie algebra and

⟨ \cdot, \cdot ⟩

a nil‐invariant symmetric bilinear form on

frakturg

. Then

⟨ \cdot, \cdot ⟩

is invariant.…”

Section: Review Of the Solvable Casementioning

confidence: 99%

“…In particular,

frakturg

is abelian if

⟨ \cdot, \cdot ⟩

is definite. This implies [, Proposition 5.4]: Proposition Let

(g, ⟨ \cdot, \cdot ⟩)

be a solvable metric Lie algebra with nil‐invariant symmetric bilinear form

⟨ \cdot, \cdot ⟩

. After a finite sequence of reductions with respect to totally isotropic and central ideals,

(g, ⟨ \cdot, \cdot ⟩)

reduces to an abelian metric Lie algebra.…”

Section: Review Of the Solvable Casementioning

confidence: 99%

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Isometry Lie algebras of indefinite homogeneous spaces of finite volume

Baues

Globke

Zeghib

2019

Proc. London Math. Soc.

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Compact pseudo-Riemannian homogeneous Einstein manifolds of low dimension

Globke

Nikolayevsky

2017

Differential Geometry and its Applications

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Abstract. Let M be pseudo-Riemannian homogeneous Einstein manifold of finite volume, and suppose a connected Lie group G acts transitively and isometrically on M . In this situation, the metric on M induces a bilinear form ⟨⋅, ⋅⟩ on the Lie algebra g of G which is nil-invariant, a property closely related to invariance. We study such spaces M in three important cases. First, we assume ⟨⋅, ⋅⟩ is invariant, in which case the Einstein property requires that G is either solvable or semisimple. Next, we investigate the case where G is solvable. Here, M is compact and M = G Γ for a lattice Γ in G. We show that in dimensions less or equal to 7, compact quotients M = G Γ exist only for nilpotent groups G. We conjecture that this is true for any dimension. In fact, this holds if Schanuel's Conjecture on transcendental numbers is true. Finally, we consider semisimple Lie groups G, and find that M splits as a pseudo-Riemannian product of Einstein quotients for the compact and the non-compact factors of G.

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Existence of Cocompact Lattices in Lie Groups With a Bi-invariant Metric of Index 2

Kath

2020

International Mathematics Research Notices

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We study the existence of cocompact lattices in Lie groups with bi-invariant metric of signature (2, n − 2). We assume in addition that the Lie groups under consideration are simply-connected, indecomposable and solvable. Then their centre is one-or two-dimensional. In both cases, a parametrisation of the set of such Lie groups is known. We give a necessary and sufficient condition for the existence of a lattice in terms of these parameters. For groups with one-dimensional centre this problem is related to Salem numbers. MSC2010: 53C50, 22E40, 57S30

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Rigidity of Compact Pseudo-Riemannian Homogeneous Spaces for Solvable Lie Groups

Cited by 6 publications

References 19 publications

Isometry Lie algebras of indefinite homogeneous spaces of finite volume

Isometry Lie algebras of indefinite homogeneous spaces of finite volume

Compact pseudo-Riemannian homogeneous Einstein manifolds of low dimension

Existence of Cocompact Lattices in Lie Groups With a Bi-invariant Metric of Index 2

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