scholarly journals The Left-Invariant Contact Metric Structure on the Sol Manifold

2020 ◽  
Vol 162 (1) ◽  
pp. 77-90
Author(s):  
V.I. Pan’zhenskii ◽  
◽  
B A.O. Rastrepina ◽  
Keyword(s):  
Contact Metric Structure

On nonexistence of Kenmotsu structure on аст-hypersurfaces of cosymplectic type of a Kählerian manifold

2019 ◽  
pp. 23-28
Author(s):  
G. Banaru
Keyword(s):  
Hermitian Manifold ◽  
Structural Equations ◽  
Type I ◽  
Almost Hermitian Manifold ◽  
Almost Contact Metric Structure ◽  
Almost Contact Metric ◽  
Contact Metric Structure ◽  
Metric Structures ◽  
Kählerian Manifold ◽  
Almost Contact Metric Structures

Almost contact metric (аст-)structures induced on oriented hypersurfaces of a Kählerian manifold are considered in the case when these аст- structures are of cosymplectic type, i. e. the contact form of these structures is closed. As it is known, the Kenmotsu structure is the most important non-trivial example of an almost contact metric structure of cosymplectic type. The Cartan structural equations of the almost contact metric structure of cosymplectic type on a hypersurface of a Kählerian manifold are obtained. It is proved that an almost contact metric structure of cosymplectic type on a hypersurface of a Kählerian manifold of dimension at least six cannot be a Kenmotsu structure. Moreover, it follows that oriented hypersurfaces of a Kählerian manifold of dimension at least six do not admit non-trivial almost contact metric structures of cosymplectic type that belong to any well studied class of аст-structures. The present results generalize some results on almost contact metric structures on hypersurfaces of an almost Hermitian manifold obtained earlier by V. F. Kirichenko, L. V. Stepanova, A. Abu-Saleem, M. B. Banaru and others.

Lifting semi-invariant submanifolds to distribu­tion of almost contact metric manifolds

2020 ◽  
pp. 39-48
Author(s):  
A. V. Bukusheva
Keyword(s):  
Parallel Transport ◽  
Linear Connection ◽  
Contact Metric Manifold ◽  
Almost Contact Metric Structure ◽  
Almost Contact Metric Manifold ◽  
Contact Metric Manifolds ◽  
Almost Contact Metric ◽  
Invariant Submanifolds ◽  
Contact Metric Structure ◽  
Almost Contact Metric Manifolds

Let M be an almost contact metric manifold of dimension n = 2m + 1. The distribution D of the manifold M admits a natural structure of a smooth manifold of dimension n = 4m + 1. On the manifold M, is defined a linear connection that preserves the distribution D; this connection is determined by the interior connection that allows parallel transport of admissible vectors along admissible curves. The assigment of the linear connection is equivalent to the assignment of a Riemannian metric of the Sasaki type on the distribution D. Certain tensor field of type (1,1) on D defines a so-called prolonged almost contact metric structure. Each section of the distribution D defines a morphism of smooth manifolds. It is proved that if a semi-invariant sub­manifold of the manifold M and is a covariantly constant vec­tor field with respect to the N-connection , then is a semi-invariant submanifold of the manifold D with respect to the prolonged almost contact metric structure.

scholarly journals A semi-symmetric metric connection on an integrated contact metric structure manifold

2016 ◽  
Vol 2 (12) ◽  
pp. 194
Author(s):  
Shalini Singh
Keyword(s):  
Riemannian Manifold ◽  
Linear Connection ◽  
Differentiable Manifold ◽  
Geometrical Properties ◽  
Metric Connection ◽  
Contact Metric Structure

In 1924, A. Friedmann and J. A. Schoten [1] introduced the idea of a semi-symmetric linear connection in a differentiable manifold. Hayden [2] has introduced the idea of metric connection with torsion in a Riemannian manifold. The properties of semi-symmetric metric connection in a Riemannian manifold have been studied by Yano [3] and others [4], [5]. The purpose of the present paper is to study some properties of semi-symmetric metric connection on an integrated contact metric structure manifold [6], several useful algebraic and geometrical properties have been studied.

scholarly journals Quaternionic Heisenberg groups as naturally reductive homogeneous spaces

2015 ◽  
Vol 12 (08) ◽  
pp. 1560007 ◽  
Author(s):  
Ilka Agricola ◽  
Ana Cristina Ferreira ◽  
Reinier Storm
Keyword(s):  
Homogeneous Spaces ◽  
Heisenberg Groups ◽  
Canonical Connection ◽  
Vertical Distributions ◽  
Metric Connection ◽  
Contact Metric Structure ◽  
Horizontal And Vertical Distributions ◽  
Generalized Killing Spinors ◽  
Reductive Homogeneous Spaces ◽  
Naturally Reductive

In this paper, we describe the geometry of the quaternionic Heisenberg groups from a Riemannian viewpoint. We show, in all dimensions, that they carry an almost 3-contact metric structure which allows us to define the metric connection that equips these groups with the structure of a naturally reductive homogeneous space. It turns out that this connection, which we shall call the canonical connection because of its analogy to the 3-Sasaki case, preserves the horizontal and vertical distributions and even the quaternionic contact (qc) structure of the quaternionic Heisenberg groups. We focus on the 7-dimensional case and prove that the canonical connection can also be obtained by means of a cocalibrated G2 structure. We then study the spinorial properties of this group and present the noteworthy fact that it is the only known example of a manifold which carries generalized Killing spinors with three different eigenvalues.

A 3-DIMENSIONAL SASAKIAN METRIC AS A YAMABE SOLITON

2012 ◽  
Vol 09 (04) ◽  
pp. 1220003 ◽  
Author(s):  
RAMESH SHARMA
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Constant Curvature ◽  
Constant Scalar Curvature ◽  
Complete Manifold ◽  
3 Dimensional ◽  
Infinitesimal Automorphism ◽  
Contact Metric Structure ◽  
Sasakian Metric ◽  
Yamabe Soliton

If a 3-dimensional Sasakian metric on a complete manifold (M, g) is a Yamabe soliton, then we show that g has constant scalar curvature, and the flow vector field V is Killing. We further show that, either M has constant curvature 1, or V is an infinitesimal automorphism of the contact metric structure on M.

scholarly journals HOMOGENEOUS AND H-CONTACT UNIT TANGENT SPHERE BUNDLES

2010 ◽  
Vol 88 (3) ◽  
pp. 323-337 ◽  
Author(s):  
G. CALVARUSO ◽  
D. PERRONE
Keyword(s):  
Riemannian Manifolds ◽  
Einstein Manifolds ◽  
Tangent Sphere ◽  
Contact Metric Structure ◽  
Metric Structures ◽  
Standard Contact ◽  
Unit Tangent Sphere Bundles ◽  
Contact Unit ◽  
Two Parameter ◽  
Kaluza Klein

AbstractWe prove that all g-natural contact metric structures on a two-point homogeneous space are homogeneous contact. The converse is also proved for metrics of Kaluza–Klein type. We also show that if (M,g) is an Einstein manifold and $\tilde G$ is a Riemannian g-natural metric on T1M of Kaluza–Klein type, then $(T_1 M,\tilde \eta ,\tilde G)$ is H-contact if and only if (M,g) is 2-stein, so proving that the main result of Chun et al. [‘H-contact unit tangent sphere bundles of Einstein manifolds’, Q. J. Math., to appear. DOI: 10.1093/qmath/hap025] is invariant under a two-parameter deformation of the standard contact metric structure on T1M. Moreover, we completely characterize Riemannian manifolds admitting two distinct H-contact g-natural contact metric structures, with associated metric of Kaluza–Klein type.

scholarly journals A Theorem on Analytic Mappings of Complex Manifolds

1966 ◽  
Vol 27 (2) ◽  
pp. 543-557 ◽  
Author(s):  
Minoru Kurita
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Complex Space ◽  
Main Idea ◽  
Complex Manifolds ◽  
Normal Contact ◽  
Contact Metric Structure ◽  
Complex Projective ◽  
Analytic Mappings

We prove in this paper a theorem on analytic mappings of the complex space Cn into the complex projective space Pn. The theorem is closely related to that of S. S. Chern in [1], and the main idea of the proof is the same with the latter, though the calculations are rather different. The background of our calculation is the normal contact metric structure which was found by S. Sasaki and Y. Hatakeyama [4].

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