Non-holonomic Kenmotsu manifolds equipped with generalized Tanaka — Webster connection

Bukusheva, A.V.

doi:10.5922/0321-4796-2020-52-5

Cited by 3 publications

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References 5 publications

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“…The concept of a non-holonomic Kenmotsu manifold was introduced in [19]. In [20], a non-holonomic Kenmotsu manifold is equipped with an N-connection ∇ N with torsion of a special type. A normal almost contact metric manifold M is called a non-holonomic Kenmotsu manifold if L ξ !…”

Section: Theorem 1 a Triple M G ∇ Nmentioning

confidence: 99%

Geometry of sub-Riemannian manifolds equipped with a quasi-semi-Weyl structure

Galaev¹,

Kokin²

2023

Topology - Recent Advances and Applications [Working Title]

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The quasi-semi-Weyl and quasi-statistical structures are based on a connection with torsion. In this paper, as a connection with torsion, we consider the so-called extended connection, which is defined with the help of an intrinsic connection, i.e., a connection in the distribution of a sub-Riemannian manifold, and with the help of an endomorphism that preserves the specified distribution and is called a structural endomorphism. It is proved that the extended connection is a connection of the quasi-semi-Weyl structure of a sub-Riemannian manifold of contact type only if the distribution of the sub-Riemannian manifold is involutive. In order to be able to consider sub-Riemannian manifolds with a not necessarily involutive distribution, the concepts of a sub-Riemannian quasi-semi-Weyl structure and a sub-Riemannian quasi-statistical structure are introduced in the paper, which are a modifications to the case of sub-Riemannian manifolds of contact type of quasi-semi-Weyl and quasi- statistical structure. The form of a structural endomorphism for the connection of a sub-Riemannian quasi-statistical structure is found. As an example, we consider nonholonomic Kenmotsu manifolds, which are sub-Riemannian manifolds of contact type endowed with an additional structure. It is proved that the restriction of the structural endomorphism to the distributions of such manifolds differs from the identity transformation only by a factor. It turns out that the intrinsic connection underlying the extended connection of a sub-Riemannian quasi-semi-Weyl structure on a sub-Riemannian manifold of contact type is compatible with the metric constraint tensor on the distribution of this manifold.

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Section: Theorem 1 a Triple M G ∇ Nmentioning

confidence: 99%

Geometry of sub-Riemannian manifolds equipped with a quasi-semi-Weyl structure

Galaev¹,

Kokin²

2023

Topology - Recent Advances and Applications [Working Title]

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On the geometry of generalized nonholonomic Kenmotsu manifolds

Bukusheva

2022

Differ. Geom. Mnogoobr. Figur

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The concept of a generalized nonholonomic Kenmotsu manifold is introduced. In contrast to the previously defined nonholonomic Kenmotsu manifold, the manifold studied in the article is an almost normal almost contact metric manifold of odd rank. The manifold is equipped with a metric connection with torsion, which is called the canonical connection in this work. The main properties of the canonical connection are studied. The canonical connection is an analogue of the generalized Tanaka-Webster connection. In this paper, we prove that the canonical connection is the only metric connection with torsion of a special structure that preserves the structural 1-form and the Reeb vector field. We study the intrinsic geometry of a generalized nonholonomic Kenmotsu manifold equipped with a canonical connection. It is proved that if a generalized nonholonomic Kenmotsu manifold is an Einstein manifold with respect to a canonical connection, then it is Ricci-flat with respect to this connection. An example of a generalized nonholonomic Kenmotsu manifold that is not a nonholonomic Kenmotsu manifold is given.

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On the geometry of sub-Riemannian manifolds equipped with a canonical quarter-symmetric connection

Galaev

2023

Differ. Geom. Mnogoobr. Figur

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In this article, a sub-Riemannian manifold of contact type is understood as a Riemannian manifold equipped with a regular distribution of codimension-one and by a unit structure vector field orthogonal to this distribution. This vector field is called a structural. On a sub-Riemannian manifold of contact type, a quarter-symmetric connection is defined, which is associated with an endomorphism that preserves the distribution of the sub-Riemannian manifold. It is proved that if the connection under study is metric, then the endomorphism associated to it is uniquely defined. The structure of the associated endomorphism is found. In the case when the structure vector field is a field of infinitesimal isometries, the quarter-symmetric connection is called the canonical N-connection. An expression is found for the curvature tensor of the canonical N-connection in terms of the Riemann curvature tensor. The properties of the Schouten curvature tensor are investigated, which provide, in particular, the necessary symmetries of the curvature tensor of an N-connection for its sectional curvature to be well-defined. A relation between the sectional curvature of the canonical N-connection and the sectional curvature of the Levi-Civita connection is found. Necessary and sufficient conditions are found under which the sectional curvature of the N-connection and the sectional curvature of the Levi-Civita connection coincide.

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Non-holonomic Kenmotsu manifolds equipped with generalized Tanaka — Webster connection

Cited by 3 publications

References 5 publications

Geometry of sub-Riemannian manifolds equipped with a quasi-semi-Weyl structure

Geometry of sub-Riemannian manifolds equipped with a quasi-semi-Weyl structure

On the geometry of generalized nonholonomic Kenmotsu manifolds

On the geometry of sub-Riemannian manifolds equipped with a canonical quarter-symmetric connection

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Non-holonomic Kenmotsu manifolds equipped with generalized Tanaka — Webster connection

Cited by 3 publications

References 5 publications

Geometry of sub-Riemannian manifolds equipped with a quasi-semi-Weyl structure

Geometry of sub-Riemannian manifolds equipped with a quasi-semi-Weyl structure

On the geometry of generalized nonholonomic Kenmotsu manifolds

On the geometry of sub-Riemannian manifolds equip­ped with a canonical quarter-symmetric connection

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On the geometry of sub-Riemannian manifolds equipped with a canonical quarter-symmetric connection