Non existence of homogeneous contact metric manifolds of non positive curvature

Lotta, Antonio

doi:10.2748/tmj/1294170347

Cited by 8 publications

(5 citation statements)

References 13 publications

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“…Turning to the homogeneous case, i.e., contact manifolds admitting a transitive Lie group of diffeomorphisms preserving the contact form, a contact metric manifold is said to be homogeneous if it admits a transitive Lie group of diffeomorphisms preserving the structure tensors (φ, Z η , η, g). This case was studied by A. Lotta [12], and he proved the following two theorems.…”

Section: Theoremmentioning

confidence: 96%

On the Sign of the Curvature of a Contact Metric Manifold

Blair

2019

Mathematics

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In this expository article, we discuss the author’s conjecture that an associated metric for a given contact form on a contact manifold of dimension ≥5 must have some positive curvature. In dimension 3, the standard contact structure on the 3-torus admits a flat associated metric; we also discuss a local example, due to Krouglov, where there exists a neighborhood of negative curvature on a particular 3-dimensional contact metric manifold. In the last section, we review some results on contact metric manifolds with negative sectional curvature for sections containing the Reeb vector field.

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Section: Theoremmentioning

confidence: 96%

On the Sign of the Curvature of a Contact Metric Manifold

Blair

2019

Mathematics

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“…A contact metric manifold is said to be homogenous if it admits a transitive Lie group of di eomorphisms which preserves the structure tensors (ϕ, Zη , η, g). We now have the following results of A. Lotta [79]. At y = − , z = this is positive, so R with the standard Darboux form and this metric has some positive curvature.…”

Section: Theorem 2 a Compact Negatively Curved Riemannian Manifold Hmentioning

confidence: 82%

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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“…• D. E. Blair (see [2], p. 120) conjectured the non-existence of contact Riemannian manifolds having non positive sectional curvature, with the exception of the flat 3-dimensional case. In this direction, A. Lotta [74] got the following (as a consequence of a more general theorem and by using the classification given in Theorem 21).…”

Section: Some Results In Arbitrary Dimensionmentioning

confidence: 99%

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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Non existence of homogeneous contact metric manifolds of non positive curvature

Cited by 8 publications

References 13 publications

On the Sign of the Curvature of a Contact Metric Manifold

On the Sign of the Curvature of a Contact Metric Manifold

A Survey of Riemannian Contact Geometry

Contact Semi-Riemannian Structures in CR Geometry: Some Aspects

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