scholarly journals Metric f-Contact Manifolds Satisfying the (κ, μ)-Nullity Condition

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 891
Author(s):  
Alfonso Carriazo ◽  
Luis M. Fernández ◽  
Eugenia Loiudice
Keyword(s):  
Explicit Expression ◽  
Sectional Curvature ◽  
Curvature Tensor ◽  
Tensor Field ◽  
Contact Manifold ◽  
Contact Manifolds

We prove that if the f-sectional curvature at any point of a ( 2 n + s ) -dimensional metric f-contact manifold satisfying the ( κ , μ ) nullity condition with n > 1 is independent of the f-section at the point, then it is constant on the manifold. Moreover, we also prove that a non-normal metric f-contact manifold satisfying the ( κ , μ ) nullity condition is of constant f-sectional curvature if and only if μ = κ + 1 and we give an explicit expression for the curvature tensor field in such a case. Finally, we present some examples.

scholarly journals A New Class of Contact Pseudo Framed Manifolds with Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
K. L. Duggal
Keyword(s):  
Vector Field ◽  
Real Function ◽  
Tensor Field ◽  
Contact Manifold ◽  
Ricci Soliton ◽  
Contact Manifolds ◽  
Normal Contact ◽  
New Class ◽  
Almost Ricci Soliton ◽  
Characteristic Vector Field

In this paper, we introduce a new class of contact pseudo framed (CPF)-manifolds M , g , f , λ , ξ by a real tensor field f of type 1,1 , a real function λ such that f 3 = λ 2 f where ξ is its characteristic vector field. We prove in our main Theorem 2 that M admits a closed 2-form Ω if λ is constant. In 1976, Blair proved that the vector field ξ of a normal contact manifold is Killing. Contrary to this, we have shown in Theorem 2 that, in general, ξ of a normal CPF-manifold is non-Killing. We also have established a link of CPF-hypersurfaces with curvature, affine, conformal collineations symmetries, and almost Ricci soliton manifolds, supported by three applications. Contrary to the odd-dimensional contact manifolds, we construct several examples of even- and odd-dimensional semi-Riemannian and lightlike CPF-manifolds and propose two problems for further consideration.

scholarly journals On W0 and W2 ø-Symmetric Contact Manifold Admitting Quarter-Symmetric Metric Connection

2021 ◽  
Vol 2070 (1) ◽  
pp. 012075
Author(s):  
K. T. Pradeep Kumar ◽  
B.M. Roopa ◽  
K.H. Arun Kumar
Keyword(s):  
Curvature Tensor ◽  
Contact Manifold ◽  
Contact Manifolds ◽  
Metric Connection

Abstract The paper deals locally W0 and W2 curvature tensor of ø-symmetric K-contact manifolds with quarter-symmetric metric connection and some results are obtained.

The Gauss-Bonnet Integrand for a Class of Riemannian Manifolds Admitting Two Orthogonal Complementary Foliations

1983 ◽  
Vol 26 (3) ◽  
pp. 358-364 ◽  
Author(s):  
O. Gil-Medrano ◽  
A. M. Naveira
Keyword(s):  
Sectional Curvature ◽  
Euler Characteristic ◽  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Tensor Field ◽  
General Assumption ◽  
Totally Geodesic ◽  
Geodesic Foliation ◽  
Characteristic Connection ◽  
Totally Geodesic Foliation

AbstractWith the general assumption that the manifold admits two orthogonal complementary foliations, one of which is totally geodesic, we study the components of the curvature tensor field of the characteristic connection.In the case where the manifold is compact, orientable of dimension 6 or 8 and the dimension of the totally geodesic foliation is 4, we relate the sign of the Euler characteristic of the manifold and that of the sectional curvature of the leaves of both foliations.

scholarly journals $W_{2}$-CURVATURE TENSOR ON K-CONTACT MANIFOLDS

2021 ◽  
pp. 995
Author(s):  
Krishnendu De
Keyword(s):  
Curvature Tensor ◽  
Sufficient Conditions ◽  
Contact Manifold ◽  
Sasakian Manifold ◽  
Contact Manifolds

The object of the present paper is to obtain sufficient conditions  for a K-contact manifold to be a Sasakian manifold.

scholarly journals Strongly pseudo-convex CR space forms

2019 ◽  
Vol 6 (1) ◽  
pp. 279-293 ◽  
Author(s):  
Jong Taek Cho
Keyword(s):  
Sectional Curvature ◽  
Hyperbolic Space ◽  
Space Form ◽  
Contact Manifold ◽  
Holomorphic Sectional Curvature ◽  
Sphere Bundle ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Pseudo Convex ◽  
Unit Tangent Sphere Bundle

AbstractFor a contact manifold, we study a strongly pseudo-convex CR space form with constant holomorphic sectional curvature for the Tanaka-Webster connection. We prove that a strongly pseudo-convex CR space form M is weakly locally pseudo-Hermitian symmetric if and only if (i) dim M = 3, (ii) M is a Sasakian space form, or (iii) M is locally isometric to the unit tangent sphere bundle T1(𝔿n+1) of a hyperbolic space 𝔿n+1 of constant curvature −1.

scholarly journals Open books and exact symplectic cobordisms

2018 ◽  
Vol 29 (04) ◽  
pp. 1850026 ◽  
Author(s):  
Mirko Klukas
Keyword(s):  
Disjoint Union ◽  
Contact Manifold ◽  
Higher Dimensions ◽  
Fiber Bundles ◽  
Open Book ◽  
Contact Manifolds ◽  
Open Books ◽  
Symplectic Cobordism ◽  
The Given ◽  
Symplectic Cobordisms

Given two open books with equal pages, we show the existence of an exact symplectic cobordism whose negative end equals the disjoint union of the contact manifolds associated to the given open books, and whose positive end induces the contact manifold associated to the open book with the same page and concatenated monodromy. Using similar methods, we show the existence of strong fillings for contact manifolds associated with doubled open books, a certain class of fiber bundles over the circle obtained by performing the binding sum of two open books with equal pages and inverse monodromies. From this we conclude, following an outline by Wendl, that the complement of the binding of an open book cannot contain any local filling obstruction. Given a contact [Formula: see text]-manifold, according to Eliashberg there is a symplectic cobordism to a fibration over the circle with symplectic fibers. We extend this result to higher dimensions recovering a recent result by Dörner–Geiges–Zehmisch. Our cobordisms can also be thought of as the result of the attachment of a generalized symplectic [Formula: see text]-handle.

scholarly journals NATURAL CONNECTION WITH TOTALLY SKEW-SYMMETRIC TORSION ON ALMOST CONTACT MANIFOLDS WITH B-METRIC

2012 ◽  
Vol 09 (05) ◽  
pp. 1250044 ◽  
Author(s):  
MANCHO MANEV
Keyword(s):  
Curvature Tensor ◽  
Torsion Tensor ◽  
Contact Manifolds ◽  
Natural Connection ◽  
Civita Connection ◽  
Levi Civita Connection

A natural connection with totally skew-symmetric torsion on almost contact manifolds with B-metric is constructed. The class of these manifolds, where the considered connection exists, is determined. Some curvature properties for this connection, when the corresponding curvature tensor has the properties of the curvature tensor for the Levi-Civita connection and the torsion tensor is parallel, are obtained.

On geometry of Kenmotsu manifolds with N-connection

2019 ◽  
pp. 48-60
Author(s):  
A. Bukusheva
Keyword(s):  
Vector Field ◽  
Coordinate System ◽  
Curvature Tensor ◽  
Tensor Field ◽  
Lie Derivative ◽  
Kenmotsu Manifold ◽  
Metric Tensor ◽  
Civita Connection ◽  
Kenmotsu Manifolds ◽  
Levi Civita Connection

A Kenmotsu manifold with a given N-connection is considered. From the integrability of the distribution of a Kenmotsu manifold it follows that the N-connection belongs to the class of the quarter-symmetric connections. Among the N-connections, the class of connections adapted to the structure of the Kenmotsu manifold is specified. In particular, it is proved that an N-connection preserves the structure endomorphism φ of the Kenmotsu manifold if and only if the endomorphisms N and φ commute. A formula expressing the N-connection in terms of the Levi-Civita connection is obtained. The Chrystoffel symbols of the Levi-Civita connection and of the N-connection of the Kenmotsu manifold with respect to the adapted coordinates are computed. The properties of the invariants of the interior geometry of the Kenmotsu manifolds are investigated. The invariants of the interior geometry are the following: the Schouten curvature tensor; the 1-form  defining the distribution D; the Lie derivative 0   L g of the metric tensor g along the vector field ;  the tensor field P with the components given with respect to the adapted coordinate system by the formula Pacd  ncad . The field P is called in the work the Schouten — Wagner tensor. It is proved that the Schouten — Wagner tensor of the interior connection of the Kenmotsu manifold is zero. The conditions that satisfies the endomorphism N defining the metric N-connection are found. At the end of the work, an example of a Kenmotsu manifold with a metric N-connection preserving the structure endomorphism φ is given.

scholarly journals Certain results on generalized (k,\mu)-contact manifolds

2017 ◽  
Vol 37 (3) ◽  
pp. 131
Author(s):  
Pradip Majhi ◽  
Gopal Ghosh
Keyword(s):  
Contact Manifold ◽  
Contact Manifolds

The object of the present paper is to study generalized (k,\mu)-contact manifolds. At first we consider \phi-semisymmetric generalized (k,\mu)-contact manifolds. Beside these we study extended pseudo projective at generalized (k,\mu)-contact manifolds. Also (k,\mu )-contact manifold satisfying \barP^e.S = 0 is also considered. As a consequence we obtain several corollaries.

scholarly journals Space time manifolds and contact structures

1990 ◽  
Vol 13 (3) ◽  
pp. 545-553 ◽  
Author(s):  
K. L. Duggal
Keyword(s):  
Contact Structure ◽  
Contact Manifold ◽  
Contact Structures ◽  
Timelike Vector ◽  
Contact Manifolds ◽  
New Class ◽  
Dimensional Spacetime ◽  
Globally Hyperbolic Spacetimes ◽  
Hyperbolic Spacetimes ◽  
Regular Contact

A new class of contact manifolds (carring a global non-vanishing timelike vector field) is introduced to establish a relation between spacetime manifolds and contact structures. We show that odd dimensional strongly causal (in particular, globally hyperbolic) spacetimes can carry a regular contact structure. As examples, we present a causal spacetime with a non regular contact structure and a physical model [Gödel Universe] of Homogeneous contact manifold. Finally, we construct a model of 4-dimensional spacetime of general relativity as a contact CR-submanifold.

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