The submanifolds of the manifold . I. The induced connection on of

Abstract

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