On submanifolds immersed in a manifold with quarter symmetric connection. (English) Zbl 0982.53016

Ali, S.; Nivas, R.

Riv. Mat. Univ. Parma (6) 3, 11-23 (2000).

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.

Reviewer: Ülo Lumiste (Tartu)

MSC:

53B25	Local submanifolds
53B05	Linear and affine connections

Keywords:

submanifold of codimension 2; quarter symmetric metric connection; hypersurface

Citations:

Zbl 0451.53017