scholarly journals Critical associated metrics on contact manifolds III

1991 ◽  
Vol 50 (2) ◽  
pp. 189-196 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Vector Field ◽  
Contact Structure ◽  
Contact Manifold ◽  
Characteristic Vector ◽  
Sphere Bundle ◽  
Contact Manifolds ◽  
Parameter Group ◽  
Group Of Isometries ◽  
Characteristic Vector Field ◽  
Regular Contact

AbstractIn the first paper of this series we studied on a compact regular contact manifold the integral of the Ricci curvature in the direction of the characteristic vector field considered as a functional on the set of all associated metrics. We showed that the critical points of this functional are the metrics for which the characteristic vector field generates a 1-parameter group of isometries and conjectured that the result might be true without the regularity of the contact structure. In the present paper we show that this conjecture is false by studying this problem on the tangent sphere bundle of a Riemannian manifold. In particular the standard associated metric is a critical point if and only if the base manifold is of constant curvature +1 or −1; in the latter case the characteristic vector field does not generate a 1-parameter group of isometries.

scholarly journals Critical associated metrics on contact manifolds

1984 ◽  
Vol 37 (1) ◽  
pp. 82-88 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Vector Field ◽  
Critical Points ◽  
Compact Manifold ◽  
Riemannian Metrics ◽  
Contact Manifold ◽  
Characteristic Vector ◽  
Contact Form ◽  
Contact Manifolds ◽  
Characteristic Vector Field ◽  
Regular Contact

AbstractDefining a function on the set of all Riemannian metrics associated to a contact form on a compact manifold by taking the integral of the Ricci curvature in the direction of the characteristic vector field, it is shown that on a compact regular contact manifold the only critical points of this function are the metrics for which the characteristic vector field generates a group of isometrics.

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.

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.

Second Variation of the "Total Scalar Curvature" on Contact Manifolds

1995 ◽  
Vol 38 (1) ◽  
pp. 16-22 ◽  
Author(s):  
D. E. Blair ◽  
D. Perrone
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Local Minimum ◽  
Contact Manifold ◽  
Second Variation ◽  
Contact Manifolds ◽  
Total Scalar Curvature ◽  
The Second Variation ◽  
Sasakian Metric ◽  
Characteristic Vector Field

AbstractLet M2n+1 be a compact contact manifold and 𝓐 the set of associated metrics. Using the scalar curvature R and the *-scalar curvature R*, in [5] we defined the "total scalar curvature", by and showed that the critical points of I(g) on 𝓐 are the K-contact metrics, i.e. metrics for which the characteristic vector field is Killing. In this paper we compute the second variation of I(g) and prove that the index of I(g) and of —I(g) are both positive at each critical point. As an application we show that the classical total scalar curvature A(g) = ∫M R dVg restricted to 𝓐 cannot have a local minimum at any Sasakian metric.

scholarly journals Harmonic characteristic vector fields on contact metric three-manifolds

2003 ◽  
Vol 67 (2) ◽  
pp. 305-315 ◽  
Author(s):  
Domenico Perrone
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Harmonic Map ◽  
Characteristic Vector ◽  
Dense Open Subset ◽  
Sphere Bundle ◽  
Sasaki Metric ◽  
Characteristic Vector Field ◽  
Characteristic Vector Fields ◽  
Unit Tangent Sphere Bundle

In this paper we show that a contact metric three-manifold is a generalised (k, μ)-space on an everywhere dense open subset if and only if its characteristic vector field ξ determines a harmonic map from the manifold into its unit tangent sphere bundle equipped with the Sasaki metric. Moreover, we classify the contact metric three-manifolds whose characteristic vector field ξ is strongly normal (or equivalently, is harmonic and minimal).

On the non-regularity of tangent sphere bundles

1978 ◽  
Vol 82 (1-2) ◽  
pp. 13-17 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Riemannian Manifold ◽  
Large Class ◽  
Constant Curvature ◽  
Positive Constant ◽  
Contact Structure ◽  
Compact Riemannian Manifold ◽  
Contact Structures ◽  
Sphere Bundle ◽  
Contact Manifolds ◽  
Tangent Sphere

SynopsisClassically the tangent sphere bundles have formed a large class of contact manifolds; their contact structures are not in general regular, however. Specifically we prove that the natural contact structure on the tangent sphere bundle of a compact Riemannian manifold of non-positive constant curvature is not regular.

scholarly journals On Generalized ϕ-Recurrent (ϵ,δ)-Trans-Sasakian Manifolds

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
C. S. Bagewadi ◽  
Dakshayani A. Patil
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Characteristic Vector ◽  
Sasakian Manifolds ◽  
Characteristic Vector Field

We study generalized ϕ-recurrent (ϵ,δ)-trans-Sasakian manifolds. A relation between the associated 1-forms A and B and relation between characteristic vector field ξ and the vector fields ρ1, ρ2 for a generalized ϕ-recurrent.

scholarly journals Ricci semisymmetric almost Kenmotsu manifolds with nullity distributions

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 179-186
Author(s):  
Sharief Deshmukh ◽  
Uday De ◽  
Peibiao Zhao
Keyword(s):  
Vector Field ◽  
Characteristic Vector ◽  
Kenmotsu Manifolds ◽  
Almost Kenmotsu Manifolds ◽  
Nullity Distribution ◽  
Characteristic Vector Field

The object of the present paper is to characterize Ricci semisymmetric almost Kenmotsu manifolds with its characteristic vector field ? belonging to the (k,?)'-nullity distribution and (k,?)-nullity distribution respectively. Finally, an illustrative example is given.

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