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On submanifolds immersed in a manifold with quarter symmetric connection. (English) Zbl 0982.53016
Let ${M}^{n+1}$ be a ${C}^{\infty }$-manifoldd with a quarter symmetric metric connection $\stackrel{˙}{\nabla }$ in the sense of R. S. Mishra and S. N. Pandey [Tensor 34, 1-7 (1980; Zbl 0451.53017)]. It is proved that the connection $\nabla$ induced on a hypersurface ${M}^{n}$ (as well as on a submanifold ${M}^{n-1}$ of codimension 2) of such an ${M}^{n+1}$ is also quarter symmetric. The hypersurface ${M}^{n}$ (resp. the submanifold ${M}^{n-1}$) will be totally umbilic with respect to $\stackrel{˙}{\nabla }$ if and only if it is totally umbilic with respert to $\nabla$. The Gauss, Weingarten and Codazzi equations are deduced.
MSC:
 53B25 Local submanifolds 53B05 Linear and affine connections