Derivational formulas of a submanifold of a generalized Riemannian space

Svetislav M. Minčić, Ljubica S. Velimirović

In the introduction is given basic information on a generalized Riemannian space, as a differentiable manifold endowed with asymmetric basic tensor, and a subspace is defined (in local coordinates). In \S1., for a tensor whose certain indices are related to the space and the others to the subspace, four kinds of covariant derivative are introduced and, in this manner, also four connections. Derivational formulas for tangents of the submanifold are expressed by means of the unit normals (Theorem 1.1 and Theorem 1.2). It is proved that by applying the third or the fourth kind of covariant derivative one concludes that induced connection is symmetric (Theorem 1.2). \S2. is related to the induced connection of the normal bundle (eq. (2.9)). In this case also are possible four kinds of covariant derivatives on the obtained normal submanifold $X^N_{N-M}$ (eq. (2.10)). In Theorem 2.1. is given the presentation of covariant derivative of the normals, using the first and the second kind of covariant derivatives. Theorem 2.2. is related to the properties of the coefficients of this connection. In Theorem 2.3. is proved that, applying the third and the fourth kind of covariant derivative at $X^N_{N-M}$, we express the covariant derivative of normals by means of tangents, and in this case the induced connection at $X^N_{N-M}$ is unique $(\underset{1}{\bar\Gamma}=\underset{2}{\bar\Gamma})$.