Some Properties of Generalized Einstein Tensor for a Pseudo-Ricci Symmetric Manifold

Abstract

<jats:p>The object of the paper is to study some properties of the generalized Einstein tensor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>G</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:mfenced></mml:math> which is recurrent and birecurrent on pseudo-Ricci symmetric manifolds <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:msub><mml:mrow><mml:mfenced open="(" close=")"><mml:mrow><mml:mtext>PRS</mml:mtext></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>. Considering the generalized Einstein tensor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>G</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:mfenced></mml:math> as birecurrent but not recurrent, we state some theorems on the necessary and sufficient conditions for the birecurrency tensor of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>G</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:mfenced></mml:math> to be symmetric.</jats:p>

On Generalized Recurrent and Generalized Concircularly Recurrent Weyl Manifolds

“…Locally symmetric manifold has been generalized to recurrent manifolds [2], generalized recurrent manifolds [3]- [4], concircularly recurrent manifolds [5], generalized concircularly recurrent manifolds [6], Ricci recurrent manifolds [7], generalized Ricci recurrent manifolds [8] etc. Recently, there are some studies on generalizations of symmetric manifolds [9][10][11].…”