scholarly journals On Sectional Curvatures of (ε)-Sasakian Manifolds

2007 ◽  
Vol 2007 ◽  
pp. 1-8 ◽  
Author(s):  
Rakesh Kumar ◽  
Rachna Rani ◽  
R. K. Nagaich
Keyword(s):  
Sectional Curvature ◽  
Curvature Tensor ◽  
Riemannian Curvature ◽  
Sasakian Manifolds ◽  
Bisectional Curvature ◽  
Riemannian Curvature Tensor ◽  
Sectional Curvatures ◽  
Totally Real

We obtain some basic results for Riemannian curvature tensor of (ε)-Sasakian manifolds and then establish equivalent relations amongφ-sectional curvature, totally real sectional curvature, and totally real bisectional curvature for (ε)-Sasakian manifolds.

scholarly journals On the geometry of the tangent bundle with gradient Sasaki metric

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Lakehal Belarbi ◽  
Hichem Elhendi
Keyword(s):  
Curvature Tensor ◽  
Tangent Bundle ◽  
Riemannian Curvature ◽  
Curvature Scalar ◽  
Sasaki Metric ◽  
Content Type ◽  
Civita Connection ◽  
Riemannian Curvature Tensor ◽  
Sectional Curvatures ◽  
Levi Civita Connection

PurposeLet (M, g) be a n-dimensional smooth Riemannian manifold. In the present paper, the authors introduce a new class of natural metrics denoted by gf and called gradient Sasaki metric on the tangent bundle TM. The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature, scalar and sectional curvatures.Design/methodology/approachIn this paper the authors introduce a new class of natural metrics called gradient Sasaki metric on tangent bundle.FindingsThe authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.Originality/valueThe authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.

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.

THE HOMOGENEOUS LIFT TO THE (1, 1)-TENSOR BUNDLE OF A RIEMANNIAN METRIC

2013 ◽  
Vol 10 (04) ◽  
pp. 1350006 ◽  
Author(s):  
ESMAEIL PEYGHAN ◽  
HASSAN NASRABADI ◽  
AKBAR TAYEBI
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Curvature Tensor ◽  
Differentiable Manifold ◽  
Riemannian Metric ◽  
Riemannian Curvature ◽  
Riemannian Curvature Tensor ◽  
Geometric Objects ◽  
Levi Civita Connection ◽  
Tensor Bundle

By using a Riemannian metric on a differentiable manifold, the homogeneous lift metric is introduced on the (1, 1)-tensor bundle of the Riemannian manifold. Some geometric objects related to this metric, such as the Levi-Civita connection, Riemannian curvature tensor and sectional curvature are calculated. Also, a para-Nordenian structure on the (1, 1)-tensor bundle with this metric is constructed and interesting properties of this structure are studied.

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.

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.

Riemannian Curvature Tensor in the Cartesian Coordinate Using the Golden Metric Tensor

2016 ◽  
Vol 14 (2) ◽  
pp. 1-8
Author(s):  
D Koffa ◽  
J Omonile ◽  
L Gani ◽  
S Howusu
Keyword(s):  
Curvature Tensor ◽  
Riemannian Curvature ◽  
Metric Tensor ◽  
Riemannian Curvature Tensor ◽  
Cartesian Coordinate

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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