The submanifolds X<sub>m</sub> of the manifold <sup>*</sup>g-MEX<sub>n</sub>. I. The induced connection on X<sub>m</sub> of *g-MEX<sub>n</sub>


Abstract


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

DOI Code: 10.1285/i15900932v18n2p213

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