Curvature tensors and Ricci solitons with respect to Zamkovoy connection in anti-invariant submanifolds of trans-Sasakian manifold

Abstract

The present paper deals with the study of some properties of anti-invariant submanifolds of trans-Sasakian manifold with respect to a new non-metric affine connection called Zamkovoy connection. The nature of Ricci flat, concircularly flat, $\xi$-projectively flat, $M$-projectively flat, $\xi$-$M$-projectively flat, pseudo projectively flat and $\xi$-pseudo projectively flat anti-invariant submanifolds of trans-Sasakian manifold admitting Zamkovoy connection are discussed. Moreover, Ricci solitons on Ricci flat, concircularly flat, $M$-projectively flat and pseudo projectively flat anti-invariant submanifolds of trans-Sasakian manifold admitting the aforesaid connection are studied. At last, some conclusions are made after observing all the results and an example of an anti-invariant submanifold of a trans-Sasakian manifold is given in which all the results can be verified easily.

Keywords

• anti-invariant submanifold
• trans-sasakian manifold
• zamkovoy connection
• $\eta$-einstein manifold
• ricci curvature tensor
• concircular curvature tensor
• projective curvature tensor
• $m$-projective curvature tensor
• pseudo projective curvature tensor
• ricci soliton\looseness-1