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Indefinite trans-Sasakian manifold with semi-symmetric metric connection. (English) Zbl 1328.53055
Summary: The objective of the present paper is to study indefinite trans-Sasakian manifold with a semi-symmetric metric connection. We find the relations between curvature tensors, Ricci curvature tensors and scalar curvature of indefinite trans-Sasakian manifolds with semi-symmetric metric connection and with metric connection. Also, we prove some results on quasi-projective and \(\varphi\)-projective manifolds with respect to semi-symmetric metric connection. It is shown that the manifold satisfying \(\bar{R}:\bar{S} = 0\) is an \(\eta\)-Einstein manifold if \(\alpha = 0\) and \(\beta =\) constant. It is also proved that the manifold satisfying \(\bar{P}:\bar{S} = 0\) is an \(\eta\)-Einstein manifold if \(\alpha = 0\) and \(\beta =\) constant. Finally, we obtain the conditions for the manifold with semi-symmetric metric connection to be conformally at and \(\xi\)-conformally flat.
MSC:
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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