Einstein manifolds and contact geometry

Boyer, Charles P.; Galicki, Krzysztof

doi:10.1090/s0002-9939-01-05943-3

Cited by 82 publications

(44 citation statements)

References 27 publications

(28 reference statements)

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“…As (M, g) is complete and the Einstein constant is 2m, by Myers' theorem, (M, g) is compact. Applying the following result of Boyer and Galicki [4]: "A compact Einstein K-contact manifold is Sasakian", we conclude that g is Sasakian, completing the proof.…”

Section: Proofs Of the Resultssupporting

confidence: 54%

Some results on almost Ricci solitons and geodesic vector fields

Sharma

2017

Beitr Algebra Geom

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Section: Proofs Of the Resultssupporting

confidence: 54%

Some results on almost Ricci solitons and geodesic vector fields

Sharma

2017

Beitr Algebra Geom

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“…Cho then proves that a contact Ricci soliton is shrinking and is Einstein K-contact. Now Boyer and Galicki [25] proved that a compact Einstein K-contact manifold is Sasakian; therefore we have as a corollary that a compact contact Ricci solition is Sasakian Einstein.…”

Section: Proposition 1 Ifmentioning

confidence: 63%

A Survey of Riemannian Contact Geometry

Blair

2019

Complex Manifolds

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This survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference “RIEMain in Contact”, 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers’ kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense “tighten up” the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on contact manifolds, whether in terms of “tight” vs. “overtwisted” or whether an associated metric should have some positive curvature. The first section will lay down the basic definitions and examples of the subject of contact metric manifolds. The second section will be a continuation of the first discussing tangent sphere bundles, contact structures on 3-dimensional Lie groups and a brief treatment of submanifolds. Section III will be devoted to the curvature of contact metric manifolds. Section IV will discuss complex contact manifolds and some older style topology. Section V treats curvature functionals and Ricci solitons. A sixth section has been added giving a discussion of the question of whether a Riemannian metric g can be an associated metric for more than one contact structure; at the conference this was an addendum to the third lecture.

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“…[21], Theorem A ). If r L = 2n, that is r = −2n, and M is compact, from Theorem 7.2 of [21], we get that the contact Riemannian structure (η, g) which corresponds to the η-Einstein Lorentzian K-contact structure (g L , η) is η-Einstein Sasakian. So, summing up we get (see also [22], Section 5)…”

Section: A Contact Semi-riemannian Manifold Is Called η-Einstein If Tmentioning

confidence: 95%

“…Recall the following result of C. Boyer and K. Galicki (see [21]): A compact K-contact Einstein manifold is Sasakian Einstein. Therefore, from Corollary 1 we get the following By Corollary 1, only trivial contact Ricci solitons occur in Riemannian settings.…”

Section: Corollary 1 a Contact Riemannian Manifold Is A Contact Riccmentioning

confidence: 98%

Contact Semi-Riemannian Structures in CR Geometry: Some Aspects

Perrone

2019

Axioms

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There is one-to-one correspondence between contact semi-Riemannian structures ( η , ξ , φ , g ) and non-degenerate almost CR structures ( H , ϑ , J ) . In general, a non-degenerate almost CR structure is not a CR structure, that is, in general the integrability condition for H 1 , 0 : = X - i J X , X ∈ H is not satisfied. In this paper we give a survey on some known results, with the addition of some new results, on the geometry of contact semi-Riemannian manifolds, also in the context of the geometry of Levi non-degenerate almost CR manifolds of hypersurface type, emphasizing similarities and differences with respect to the Riemannian case.

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Einstein manifolds and contact geometry

Abstract: Abstract. We show that every K-contact Einstein manifold is SasakianEinstein and discuss several corollaries of this result.

Cited by 82 publications

References 27 publications

Some results on almost Ricci solitons and geodesic vector fields

Some results on almost Ricci solitons and geodesic vector fields

A Survey of Riemannian Contact Geometry

Contact Semi-Riemannian Structures in CR Geometry: Some Aspects

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