The submanifolds $X_m $ of the manifold $^*g-MEX_n$. I. The induced connection on $X_m$ of $*g-MEX_n$

Abstract

An Einstein's connection which takes the form (2.33) is called an $^*g-ME$-connection. Recently, Chung and et al ([15], 1993)introduced a new manifolds, called an n-dimensional $^*g-ME$-manifold (debnoted by $^*g-MEX_n$).The manifold $^*g-MEX_n$ is a generalized n-dimensional Riemannian manifold $X_n$ on which the differential geometric structure is imposed by the unified field tensor $^*g^{λ \upsilon} m$ satisfying certain conditions through the $^*g-ME$-connection. In the following series of two papers, we investigate the submanifolds $X_m$ of $^*g-MEX_n$: I. The induced connection on $X_m$ of $^*g-MEX_n$ II. The generalized fundamental equations on $X_m$ of $^*g-MEX_n$ In this paper, Part I of the series, we present a brief introduction of n-dimensional $^*g$-unified field theory, the C-nonholonomic frame of reference in $X_n$ at points of $X-m$, and the manifold $^*g-MEX_n$. and then, we introduce the generalized coefficients of the second fundamental form of $X_m$ and prove a necessary and sufficient condition for the induced connection on $X_m$ of $^*-MEX_n$ to be a $^*g-Me$-connection. Our subsequent paper, Part II of the series, deals with the generalized fundamental equations on $X_m$ of $^*-MEX_n$, such as the generalized Gauss formulae, the generalized Weingarten equations, and the Gauss-Codazzi equations.