scholarly journals Tensor invariants of generalized Kenmotsu manifolds

2021 ◽  
Vol 244 ◽  
pp. 09006
Author(s):  
Ali Abdul Al Majeed Shihab ◽  
Aligadzhi Rustanov
Keyword(s):  
Curvature Tensor ◽  
Riemannian Geometry ◽  
Local Structure ◽  
Second Order ◽  
Geometric Meaning ◽  
Riemannian Curvature ◽  
Geometric Invariants ◽  
Kenmotsu Manifolds ◽  
Riemannian Curvature Tensor ◽  
Symmetry Properties

In this paper, we study the properties of generalized Kenmotsu manifolds, consider the second-order differential geometric invariants of the Riemannian curvature tensor of generalized Kenmotsu manifolds (by the symmetry properties of the Riemannian geometry tensor). The concept of a tensor spectrum is introduced. Nine invariants are singled out and the geometric meaning of these invariants turning to zero are investigated. The identities characterizing the selected classes are singled out. Also, 9 classes of generalized Kenmotsu manifolds are distinguished, the local structure of 8 classes from the selected ones is obtained.

Some curvature properties of paracontact metric manifolds

2018 ◽  
Vol 9 (3) ◽  
pp. 159-165
Author(s):  
Krishanu Mandal ◽  
Uday Chand De
Keyword(s):  
Sectional Curvature ◽  
Curvature Tensor ◽  
Second Order ◽  
Plane Section ◽  
Riemannian Curvature ◽  
Curvature Condition ◽  
Ricci Operator ◽  
Riemannian Curvature Tensor

AbstractThe purpose of this paper is to study Ricci semisymmetric paracontact metric manifolds satisfying{\nabla_{\xi}h=0}and such that the sectional curvature of the plane section containing ξ equals a non-zero constantc. Also, we study paracontact metric manifolds satisfying the curvature condition{Q\cdot R=0}, whereQandRare the Ricci operator and the Riemannian curvature tensor, respectively, and second order symmetric parallel tensors in paracontact metric manifolds under the same conditions. Several consequences of these results are discussed.

scholarly journals ON THE $\mathcal{M}$--PROJECTIVE CURVATURE TENSOR OF A $(k, \mu)$-CONTACT METRIC MANIFOLD

2017 ◽  
pp. 117
Author(s):  
D. G. Prakasha ◽  
Kakasab Mirji
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Sasakian Manifold ◽  
Riemannian Curvature ◽  
Contact Metric Manifold ◽  
Contact Metric Manifolds ◽  
Riemannian Curvature Tensor ◽  
Projective Curvature ◽  
Projective Curvature Tensor

The paper deals with the study of $\mathcal{M}$-projective curvature tensor on $(k, \mu)$-contact metric manifolds. We classify non-Sasakian $(k, \mu)$-contact metric manifold satisfying the conditions $R(\xi, X)\cdot \mathcal{M} = 0$ and $\mathcal{M}(\xi, X)\cdot S =0$, where $R$ and $S$ are the Riemannian curvature tensor and the Ricci tensor, respectively. Finally, we prove that a $(k, \mu)$-contact metric manifold with vanishing extended $\mathcal{M}$-projective curvature tensor $\mathcal{M}^{e}$ is a Sasakian manifold.

On Some Aspects of Geometry of Almost C(λ)-manifolds

2020 ◽  
pp. 19-24
Author(s):  
A.R. Rustanov ◽  
E.A. Polkina ◽  
S.V. Kharitonova
Keyword(s):  
Constant Curvature ◽  
Local Structure ◽  
Ricci Tensor ◽  
Einstein Manifold ◽  
Sufficient Conditions ◽  
Holomorphic Sectional Curvature ◽  
Riemannian Curvature ◽  
Riemannian Curvature Tensor ◽  
Analytic Expressions ◽  
Necessary And Sufficient

In this paper almost C(λ)-manifolds are considered. The local structure of Ricci-flat almost C(λ)-manifolds is obtained. On the space of the adjoint G-structure, necessary and sufficient conditions are obtained under which the al-most C(λ)-manifolds are manifolds of constant curvature and the structure of the Riemannian curvature tensor of an almost C(λ)-manifold of constant curvature is obtained. Relations are obtained that characterize the Einstein almost C(λ)-manifolds. It is proved that a complete almost C(λ)-Einstein manifold is either holomorphically isometrically covered by the product of a real line by a Ricciflat Kähler manifold, or is compact and has a finite fundamental group. For almost C(λ)-manifolds that are -Einstein, analytic expressions for the functions  and  characterizing these manifolds are obtained. It is shown that an almost C(λ)-manifold has an Ф-invariant Ricci tensor. We study also almost C(λ)-manifolds of pointwise constant Ф-holomorphic sectional curvature.

scholarly journals A CLASSIFICATION OF SPHERICAL SYMMETRIC CR MANIFOLDS

2009 ◽  
Vol 80 (2) ◽  
pp. 251-274 ◽  
Author(s):  
G. DILEO ◽  
A. LOTTA
Keyword(s):  
Curvature Tensor ◽  
Riemannian Curvature ◽  
Cr Manifolds ◽  
Hermitian Manifolds ◽  
Riemannian Curvature Tensor ◽  
Webster Metric ◽  
Simply Connected

AbstractIn this paper we get different characterizations of the spherical strictly pseudoconvex CR manifolds admitting a CR-symmetric Webster metric by means of the Tanaka–Webster connection and of the Riemannian curvature tensor. As a consequence we obtain the classification of the simply connected, spherical symmetric pseudo-Hermitian manifolds.

Convergence of Riemannian 4-manifolds with L 2 L^{2} -curvature bounds

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Norman Zergänge
Keyword(s):  
Curvature Tensor ◽  
Curvature Flow ◽  
Riemannian Curvature ◽  
Hausdorff Topology ◽  
Curvature Bounds ◽  
Prove Theorem ◽  
Riemannian Curvature Tensor ◽  
Convergence Results ◽  
Uniform Diameter ◽  
Gromov Hausdorff Topology

Abstract In this work we prove convergence results of sequences of Riemannian 4-manifolds with almost vanishing {L^{2}} -norm of a curvature tensor and a non-collapsing bound on the volume of small balls. In Theorem 1.1 we consider a sequence of closed Riemannian 4-manifolds, whose {L^{2}} -norm of the Riemannian curvature tensor tends to zero. Under the assumption of a uniform non-collapsing bound and a uniform diameter bound, we prove that there exists a subsequence that converges with respect to the Gromov–Hausdorff topology to a flat manifold. In Theorem 1.2 we consider a sequence of closed Riemannian 4-manifolds, whose {L^{2}} -norm of the Riemannian curvature tensor is uniformly bounded from above, and whose {L^{2}} -norm of the traceless Ricci-tensor tends to zero. Here, under the assumption of a uniform non-collapsing bound, which is very close to the Euclidean situation, and a uniform diameter bound, we show that there exists a subsequence which converges in the Gromov–Hausdorff sense to an Einstein manifold. In order to prove Theorem 1.1 and Theorem 1.2, we use a smoothing technique, which is called {L^{2}} -curvature flow. This method was introduced by Jeffrey Streets. In particular, we use his “tubular averaging technique” in order to prove distance estimates of the {L^{2}} -curvature flow, which only depend on significant geometric bounds. This is the content of Theorem 1.3.

scholarly journals On the $k$-th nullity space of the Riemannian curvature tensor

1975 ◽  
Vol 27 (1) ◽  
pp. 25-30
Author(s):  
Shun-ichi Tachibana ◽  
Masami Sekizawa
Keyword(s):  
Curvature Tensor ◽  
Riemannian Curvature ◽  
Riemannian Curvature Tensor
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