# Generalized Einstein tensor for a Weyl manifold and its applications

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## Date

2013

## Authors

Özdeğer, Abdülkadir

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## Publisher

Springer Heidelberg

## Open Access Color

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## Abstract

It is well known that the Einstein tensor G for a Riemannian manifold defined by R (alpha) (beta) = g (beta gamma) R (gamma I +/-) where R (gamma I +/-) and R are respectively the Ricci tensor and the scalar curvature of the manifold plays an important part in Einstein's theory of gravitation as well as in proving some theorems in Riemannian geometry. In this work we first obtain the generalized Einstein tensor for a Weyl manifold. Then after studying some properties of generalized Einstein tensor we prove that the conformal invariance of the generalized Einstein tensor implies the conformal invariance of the curvature tensor of the Weyl manifold and conversely. Moreover we show that such Weyl manifolds admit a one-parameter family of hypersurfaces the orthogonal trajectories of which are geodesics. Finally a necessary and sufficient condition in order that the generalized circles of a Weyl manifold be preserved by a conformal mapping is stated in terms of generalized Einstein tensors at corresponding points.

## Description

## Keywords

Weyl manifold, Einstein-Weyl manifold, Einstein tensor, Generalized Einstein tensor, Generalized circle

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## Fields of Science

## Citation

2

## WoS Q

Q3

## Scopus Q

Q2

## Source

## Volume

29

## Issue

2

## Start Page

373

## End Page

382