Journal of the
Korean Mathematical Society JKMS

Recently, Chung and et al.\,\,([11], 1991c) introduced a new concept of a manifold, denoted by $^*g$-SEX$_n$ , in Einstein's $n$-dimensional ${}^*g$-unified field theory. The manifold $^*g$-SEX$_n$ is a generalized $n$-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor $^*g^{\lambda \nu}$ through the SE-connection which is both Einstein and semi-symmetric. In this paper, they proved a necessary and sufficient condition for the unique existence of SE-connection and presented a beautiful and surveyable tensorial representation of the SE-connection in terms of the tensor $^*g^{\lambda \nu}$. \hfill\break\indent This paper is the first part of the following series of two papers: \hfill\break\indent\qquad\qquad I. The SE-curvature tensor of $^*g$-SEX$_n$ \hfill\break\indent\qquad\quad\enspace\thinspace II. The contracted SE-curvature tensors of $^*g$-SEX$_n$ \hfill\break In the present paper we investigate the properties of SE-curvature tensor of $^*g$-SEX$_n$, with main emphasis on the derivation of several useful generalized identities involving it. In our subsequent paper, we are concerned with contracted curvature tensors of $^*g$-SEX$_n$ and several generalized identities involving them. In particular, we prove the first variation of the generalized Bianchi's identity in $^*g$-SEX$_n$, which has a great deal of useful physical applications.

Keywords: SE-connection, the manifold $^*g$-SEX$_n$, the SE-curvature tensors

MSC numbers: 83E50, 83C05, 58A05