Let

${M}^{n+1}$ be a

${C}^{\infty}$-manifoldd with a quarter symmetric metric connection

$\dot{\nabla}$ 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

$\dot{\nabla}$ if and only if it is totally umbilic with respert to

$\nabla $. The Gauss, Weingarten and Codazzi equations are deduced.