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.

Identities of the Riemannian Curvature Tensor of Almost C(ʎ)-Manifolds

2020 ◽  
pp. 49-54
Author(s):  
Aligadzhi R. Rustanov ◽  
Elena A. Polkina ◽  
Svetlana V. Kharitonova
Keyword(s):  
Curvature Tensor ◽  
Real Line ◽  
Kähler Manifold ◽  
Riemannian Curvature ◽  
Cosymplectic Manifold ◽  
The Real ◽  
Local Classification ◽  
Kahler Manifold ◽  
Riemannian Curvature Tensor

The geometry of the Riemannian curvature tensor of an almost C(λ)-manifold is studied. We have obtained several identities of the Riemannian curvature tensor of almost C(λ)-manifolds. Four additional identities are distinguished from these identities, on the basis of which four classes of almost C(λ)-manifolds are determined. A local classification of each of the distinguished classes of almost C(λ)-manifolds is obtained. It is proved that the set of almost C(λ)-manifolds of class R_1 coincides with the set of almost C(λ)-manifolds of class R_2, and it is also proved that the set of almost C(λ)-manifolds of class R_3 coincides with the set of almost C(λ)- manifolds of class R_4. We have found that an almost C(λ)-manifold, dimension greater than 3, is a manifold of class R_4 if and only if it is a cosymplectic manifold, i.e. when it is locally equivalent to the product of the Kähler manifold and the real line.

scholarly journals A (CHR)3-flat trans-Sasakian manifold

2019 ◽  
Vol 12 (2) ◽  
Author(s):  
Koji Matsumoto
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Curvature Tensor ◽  
Ricci Tensor ◽  
Tensor Field ◽  
Einstein Manifold ◽  
Sasakian Manifold ◽  
Hermitian Manifolds ◽  
Riemannian Curvature Tensor ◽  
Contact Riemannian Manifold

In [4] M. Prvanovic considered several curvaturelike tensors defined for Hermitian manifolds. Developing her ideas in [3], we defined in an almost contact Riemannian manifold another new curvaturelike tensor field, which is called a contact holomorphic Riemannian curvature tensor or briefly (CHR)3-curvature tensor. Then, we mainly researched (CHR)3-curvature tensor in a Sasakian manifold. Also we proved, that a conformally (CHR)3-flat Sasakian manifold does not exist. In the present paper, we consider this tensor field in a trans-Sasakian manifold. We calculate the (CHR)3-curvature tensor in a trans-Sasakian manifold. Also, the (CHR)3-Ricci tensor ρ3  and the (CHR)3-scalar curvature τ3  in a trans-Sasakian manifold have been obtained. Moreover, we define the notion of the (CHR)3-flatness in an almost contact Riemannian manifold. Then, we consider this notion in a trans-Sasakian manifold and determine the curvature tensor, the Ricci tensor and the scalar curvature. We proved that a (CHR)3-flat trans-Sasakian manifold is a generalized   ɳ-Einstein manifold. Finally, we obtain the expression of the curvature tensor with respect to the Riemannian metric g of a trans-Sasakian manifold, if the latter is (CHR)3-flat.

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.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document