NEW INTEGRABILITY CONDITIONS OF DERIVATIONAL EQUATIONS OF A SUBMANIFOLD IN A GENERALIZED RIEMANNIAN SPACE

The present work is a continuation of [5] and [6]. In [5] we have obtained derivational equations of a submanifold XMof a generalized Riemannian space GRN. Since the basic tensor in GRNis asymmetric and in this way the connection is also asymmetric, in a submanifold the connection is generally asymmetric too. By reason of this, we define 4 kinds of covariant derivative and obtain 4 kinds of derivational equations. In [6] we have obtained integrability conditions and Gauss-Codazzi equations using the 1stand the 2stkind of covariant derivative. The present work deals in the cited matter, using the 3rdand the 4thkind of covariant derivative. One obtains three new integrability conditions for derivational equations of tangents and three such conditions for normals of the submanifold, as the corresponding Gauss-Codazzi equations too.