About Pseudo-Riemannian Lichnerowicz Conjecture

Frances, Charles

doi:10.1007/s00031-015-9317-x

Cited by 22 publications

(17 citation statements)

References 10 publications

(5 reference statements)

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“…The study of pseudo-Riemannian manifolds with an essential group of conformal transforms is a topical and actively developing direction of the modern global differential geometry. This is confirmed by the works by Alekseevskii [1], Podoksenov [2], Frances [3]- [4], Frances and Zeghib [5], Frances and Melnick [6] and by other authors as well as by the surveys in monograph [7].…”

Section: Introductionsupporting

confidence: 72%

“…Without loss of generality, by Lemma 2 we can assume that = . Hence, according Lemma 4, in terms of the geodesic coordinates, in some neighbourhood̃︀ of zero, the representative¯of the element satisfies equation (4).…”

Section: Proof Of Theoremmentioning

confidence: 92%

“…Since ∈ { } × ( − 1 , − 1 ), we necessarily have = for each vector in R × {0 − }. We take an arbitrary infinitesimal real non-zero number and substitute = into (4). Taking…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…We stress that the pseudo-Riemannian manifold Ein , is conformally flat. As 4, in [4], there was proved the existence of an infinite set of conformal pseudo-Riemannian metric of various signatures on S 1 × S −1 with essential groups of conformal transform, which are not conformally flat.…”

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confidence: 99%

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Influence of stratification on the groups of conformal transformations of pseudo-Riemannian orbifolds

Zhukova¹

2018

Ufimskii Matematicheskii Zhurnal

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We study the groups of conformal transformations of-dimensional pseudo-Riemannian orbifolds (,) as 3. We extend the Alekseevskii method for studying conformal transformation groups of Riemannian manifolds to pseudo-Riemannian orbifolds. We show that a conformal pseudo-Riemannian geometry is induced on each stratum of such orbifold. Due to this, for ∈ {0, 1} ∪ {3,. .. , − 1}, we obtain exact estimates for the dimensions of the conformal transformation groups of-dimensional pseudo-Riemannian orbifolds admitting-dimensional stratum with essential groups of conformal transforms. A key fact in obtaining these estimates is that each connected transformation group of an orbifold preserves every connected component of each its stratum. The influence of stratification of-dimensional pseudo-Riemann orbifold to the similarity transformation group of this orbifold is also studied for 2. We prove that the obtained estimates for the dimension of the complete essential groups of conformal transformations and the similarity transformation groups of-dimensional pseudo-Riemann orbifolds are sharp; this is done by adducing corresponding examples of locally flat pseudo-Riemannian orbifolds.

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Section: Introductionsupporting

confidence: 72%

Section: Proof Of Theoremmentioning

confidence: 92%

“…Since ∈ { } × ( − 1 , − 1 ), we necessarily have = for each vector in R × {0 − }. We take an arbitrary infinitesimal real non-zero number and substitute = into (4). Taking…”

Section: Proof Of Theoremmentioning

confidence: 99%

mentioning

confidence: 99%

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Influence of stratification on the groups of conformal transformations of pseudo-Riemannian orbifolds

Zhukova¹

2018

Ufimskii Matematicheskii Zhurnal

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“…In particular, φ t cannot even preserve a Borel measure which is nonzero on open sets, let alone a contact form (cf. [5]).…”

Section: Introductionmentioning

confidence: 99%

On the Lichnerowicz conjecture for CR manifolds with mixed signature

Case

Curry

Matveev

2018

Comptes Rendus. Mathématique

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We construct examples of nondegenerate CR manifolds with Levi form of signature (p, q), 2 ≤ p ≤ q, which are compact, not locally CR flat, and admit essential CR vector fields. We also construct an example of a noncompact nondegenerate CR manifold with signature (1, n − 1) which is not locally CR flat and admits an essential CR vector fields. These provide counterexamples to the analogue of the Lichnerowicz conjecture for CR manifolds with mixed signature.

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Conformal Actions of Real-Rank 1 Simple Lie Groups on Pseudo-Riemannian Manifolds

Pecastaing

2019

Transformation Groups

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We investigate conformal actions of cocompact lattices in higher-rank simple Lie groups on compact pseudo-Riemannian manifolds. Our main result gives a general bound on the real-rank of the lattice, which was already known for the action of the full Lie group ([Zim87b]). When the real-rank is maximal, we prove that the manifold is conformally flat. This indicates that a global conclusion similar to that of [BN02] and [FZ05] in the case of a Lie group action might be obtained. We also give better estimates for actions of cocompact lattices in exceptional groups. Our work is strongly inspired by the recent breakthrough of Brown, Fisher and Hurtado on Zimmer's conjecture [BFH16].

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About Pseudo-Riemannian Lichnerowicz Conjecture

Abstract: Abstract. We construct the first known examples of compact pseudoRiemannian manifolds having an essential group of conformal transformations, and which are not conformally flat. Our examples cover all types (p, q), with 2 ≤ p ≤ q.

Cited by 22 publications

References 10 publications

Influence of stratification on the groups of conformal transformations of pseudo-Riemannian orbifolds

Influence of stratification on the groups of conformal transformations of pseudo-Riemannian orbifolds

On the Lichnerowicz conjecture for CR manifolds with mixed signature

Conformal Actions of Real-Rank 1 Simple Lie Groups on Pseudo-Riemannian Manifolds

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