scholarly journals On the Riemannian curvature tensor of an almost-product manifold

1986 ◽  
Vol 36 (1) ◽  
pp. 72-86
Author(s):  
Francisco J. Carreras
Keyword(s):  
Curvature Tensor ◽  
Riemannian Curvature ◽  
Product Manifold ◽  
Riemannian Curvature Tensor ◽  
Almost Product Manifold

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

scholarly journals QUASI-CONFORMAL CURVATURE TENSOR OF GENERALIZED SASAKIAN-SPACE-FORMS

2020 ◽  
Vol 35 (1) ◽  
pp. 089
Author(s):  
Braj B. Chaturvedi ◽  
Brijesh K. Gupta
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Space Form ◽  
Riemannian Curvature ◽  
Space Forms ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Sasakian Space Forms ◽  
Riemannian Curvature Tensor

The present paper deals the study of generalised Sasakian-space-forms with the conditions Cq(ξ,X).S = 0, Cq(ξ,X).R = 0 and Cq(ξ,X).Cq = 0, where R, S and Cq denote Riemannian curvature tensor, Ricci tensor and quasi-conformal curvature tensor of the space-form, respectively and at last, we have given some examples to improve our results.

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.

On Strongly Convex Indicatrices in Minkowski Geometry

2002 ◽  
Vol 45 (2) ◽  
pp. 232-246 ◽  
Author(s):  
Min Ji ◽  
Zhongmin Shen
Keyword(s):  
Vector Space ◽  
Curvature Tensor ◽  
Riemannian Metric ◽  
Minkowski Geometry ◽  
Riemannian Curvature ◽  
Strongly Convex ◽  
Riemannian Curvature Tensor ◽  
Cartan Torsion ◽  
The Mean ◽  
The Relationship

AbstractThe geometry of indicatrices is the foundation of Minkowski geometry. A strongly convex indicatrix in a vector space is a strongly convex hypersurface. It admits a Riemannian metric and has a distinguished invariant—(Cartan) torsion. We prove the existence of non-trivial strongly convex indicatrices with vanishing mean torsion and discuss the relationship between the mean torsion and the Riemannian curvature tensor for indicatrices of Randers type.

scholarly journals Curvature Estimates in Asymptotically Flat Lorentzian Manifolds

2005 ◽  
Vol 57 (4) ◽  
pp. 708-723 ◽  
Author(s):  
Felix Finster ◽  
Margarita Kraus
Keyword(s):  
Fundamental Form ◽  
Curvature Tensor ◽  
Lorentzian Manifold ◽  
Second Fundamental Form ◽  
Riemannian Curvature ◽  
Lorentzian Manifolds ◽  
Asymptotically Flat ◽  
Riemannian Curvature Tensor ◽  
Curvature Estimates

AbstractWe consider an asymptotically flat Lorentzian manifold of dimension (1, 3). An inequality is derived which bounds the Riemannian curvature tensor in terms of the ADM energy in the general case with second fundamental form. The inequality quantifies in which sense the Lorentzianmanifold becomes flat in the limit when the ADM energy tends to zero.

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