Integral formulas in riemannian geometry
01 Mar 1973-Bulletin of The London Mathematical Society (Oxford University Press (OUP))-Vol. 5, Iss: 1, pp 124-125
About: This article is published in Bulletin of The London Mathematical Society.The article was published on 1973-03-01. It has received 331 citations till now. The article focuses on the topics: Riemannian geometry & Fundamental theorem of Riemannian geometry.
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TL;DR: In this paper, the existence of an almost pseudo conformally symmetric Riemannian manifold is shown by a non-trivial concrete example, which is a type of non-conformally flat Riemanian manifold.
Abstract: The object of the present paper is to study a type of non-conformally flat Riemannian manifold called almost pseudo conformally symmetric manifold. The existence of an almost pseudo conformally symmetric manifold is also shown by a non-trivial concrete example.
14 citations
TL;DR: In this paper, it was shown that if the curvatures of plane sections containing the characteristic vector field of a contact metric manifold M are non-vanishing, then a second order parallel tensor on M is a constant multiple of the associated metric tensor.
Abstract: If the sectional curvatures of plane sections containing the characteristic vector field of a contact metric manifold M are non-vanishing, then we prove that a second order parallel tensor on M is a constant multiple of the associated metric tensor. Next, we prove for a contact metric manifold of dimension greater than 3 and whose Ricci operator commutes with the fundamental collineation that, if its Weyl conformal tensor is harmonic, then it is Einstein. We also prove that, if the Lie derivative of the fundamental collineation along the characteristic vector field on a contact metric 3-manifold M satisfies a cyclic condition, then M is either Sasakian or locally isometric to certain canonical Lie-groups with a left invariant metric. Next, we prove that if a three-dimensional Sasakian manifold admits a non-Killing projective vector field, it is of constant curvature 1. Finally, we prove that a conformally recurrent Sasakian manifold is locally isometric to a unit sphere.
14 citations
01 Jan 2010
TL;DR: In this article, a study of conharmonic curvature tensors has been made on the four dimensional spacetime of general relativity and the existence of Killing and confor- mal Killing vectors on such spacetime have been established.
Abstract: The signiflcance of conharmonic curvature tensor is very well known in the difierential geometry of certain F-structures (e.g., complex, almost complex, Kahler, almost Kahler, Hermitian, almost Hermitian structures, etc.). In this paper, a study of conharmonic curvature ten- sor has been made on the four dimensional spacetime of general relativity. The spacetime satisfying Einstein fleld equations and having vanishing conharmonic tensor is considered and the existence of Killing and confor- mal Killing vectors on such spacetime have been established. Perfect ∞uid cosmological models have also been studied.
13 citations
TL;DR: In this article, a pseudo generalized quasi-Einstein manifold is introduced and some properties of such a manifold with several non-trivial examples are studied. But these properties are restricted to Riemannian man-ifolds.
Abstract: The object of the present paper is to introduce a type of non-flat Riemannian man- ifold called pseudo generalized quasi-Einstein manifold and studied some properties of such a manifold with several non-trivial examples.
13 citations
TL;DR: In this article, the causal character of projective vector fields and curvature on a Lorentzian manifold was studied. And the existence of such vector fields was investigated in terms of affine, homothetic and killing vector fields.
Abstract: Some relations between the causal character of projective vector fields and curvature on a Lorentzian manifold M are studied. As a consequence, obstructions to the existence of such vector fields are found. Affine, homothetic and Killing vector fields are considered specifically.
13 citations