Transitive group actions and Ricci curvature properties.

Miatello, Isabel Dotti

doi:10.1307/mmj/1029003823

Cited by 10 publications

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“…[10]. We note that Corollary 1.3 generalizes [9,Corollary 3]. This gives some hope that the Alekseevskii conjecture is indeed true as one would expect an Einstein space to have more symmetries than other metrics and there are metrics on G/K whose symmetry groups are as large as G × K .…”

Section: Theorem 11 (Bochner) Let M Be a Compact Riemannian Manifoldmentioning

confidence: 60%

“…If such a space admits a transitive unimodular group of isometries, then it is known to admit a transitive semi-simple group of isometries [9]. However, little else is known about these spaces in general, save some very interesting cases worked out by Nikonorov [27] We follow Bochner's approach to glean even more about the geometry of homogeneous spaces with negative Ricci curvature.…”

Section: Theorem 11 (Bochner) Let M Be a Compact Riemannian Manifoldmentioning

confidence: 99%

“…In the general homogeneous case, one can deduce an alternate proof to the corollary above using the techniques in [9]. Additionally, Jorge Lauret has shown us a different proof which uses the relationship between Ric and the moment map; Yurii Nikonorov has pointed out to us that the corollary also follows from [7,Theorem 4].…”

Section: Corollary 18 Let G/k Be a Homogeneous Space Where G Acts Almentioning

confidence: 99%

“…In [23] it is shown that SL(n, R) admits metrics of negative Ricci curvature for n ≥ 3; although, in those examples the eigenvalues of Ric appear to be so widely spread that Einstein metrics might not exist. The case of transitive, unimodular groups of isometries was further investigated in [9] where it was shown that for such a space with negative Ricci curvature there must exist a transitive semi-simple group of isometries. For more in this direction, see [27] and Proposition 1.2 below.…”

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A step towards the Alekseevskii conjecture

Jablonski

Petersen²

2016

Math. Ann.

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We refine existing structure results for non-compact, homogeneous, Einstein manifolds and provide a reduction in the classification problem of such spaces. Using this work, we verify the (Generalized) Alekseevskii Conjecture for a large class of homogeneous spaces.A longstanding open question in the study of Riemannian homogeneous spaces is the classification of non-compact, Einstein spaces. In the 1970s, it was conjectured by D. Alekseevskii that any (non-compact) homogeneous Einstein space of negative scalar curvature is diffeomorphic to R n . Equivalently, this conjecture can be phrased as follows:Classical Alekseevskii Conjecture: Given a homogeneous Einstein space G/K with negative scalar curvature, K must be a maximal compact subgroup of G.

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Section: Theorem 11 (Bochner) Let M Be a Compact Riemannian Manifoldmentioning

confidence: 60%

Section: Theorem 11 (Bochner) Let M Be a Compact Riemannian Manifoldmentioning

confidence: 99%

Section: Corollary 18 Let G/k Be a Homogeneous Space Where G Acts Almentioning

confidence: 99%

mentioning

confidence: 99%

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A step towards the Alekseevskii conjecture

Jablonski

Petersen²

2016

Math. Ann.

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“…Let us fix (·, ·) on m and consider arbitrary Ad(H )-invariant inner product ·, · on m. After simultaneous diagonalization of the forms ·, · and (·, ·) by using Schur's lemma we obtain ·, · = x 1 (·, ·)| m 1 ⊥x 2 (·, ·)| m 2 ⊥ · · · ⊥x s (·, ·)| The following theorem for homogeneous manifolds with unimodular groups of motions will be useful for our goals. PROPOSITION 3 (Dotti-Miatello [17] (s, Q), where s is a Lie algebra, and Q is some inner product on s.…”

Section: Preliminariesmentioning

confidence: 99%

Noncompact Homogeneous Einstein 5-Manifolds

Nikonorov¹

2005

Geom Dedicata

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The Alekseevskii conjecture in low dimensions

Arroyo

Lafuente

2016

Math. Ann.

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The long-standing Alekseevskii conjecture states that a connected homogeneous Einstein space G/K of negative scalar curvature must be diffeomorphic to R n . This was known to be true only in dimensions up to 5, and in dimension 6 for non-semisimple G. In this work we prove that this is also the case in dimensions up to 10 when G is not semisimple. For arbitrary G, besides 5 possible exceptions, we show that the conjecture holds up to dimension 8.

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Transitive group actions and Ricci curvature properties.

Cited by 10 publications

References 0 publications

A step towards the Alekseevskii conjecture

A step towards the Alekseevskii conjecture

Noncompact Homogeneous Einstein 5-Manifolds

The Alekseevskii conjecture in low dimensions

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