Integrability conditions of derivational equations of a submanifold of a generalized Riemannian space

The present work is a continuation of [5] and [6]. In [5] we have obtained derivational equations of a submanifold XM of a generalized Riemannian space GRN . Since the basic tensor in GRN is 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 1 and the 2 kind of covariant derivative. The present work deals in the cited matter, using the 3 and the 4 kind 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.