Geometry of the submanifolds of \(SEX_ n\). I: The C-nonholonomic frame of reference
[For part II, see the review below Zbl 0698.53009.] In 1950 Einstein proposed [see \textit{A. Einstein}, The meaning of relativity (Princeton 1950; Zbl 0037.420)] a new unified field theory that would include both gravitation and electromagnetism. Although the intent of this theory is physical, its exposition is mainly geometrical. It may be characterized as a set of geometrical postulates for the space- time \(X_ 4\). However, the geometrical consequences of these postulates were not developed very far by Einstein. Characterizing Einstein's 4- dimensional unified field theory as a set of geometrical postulates for \(X_ 4\), \textit{V. Hlavatý} [Geometry of Eintein's unified field theory (Noordhoop 1957; Zbl 0078.433)] gave its mathematical foundation for the first time. Since then the geometrical consequences of these postulates have been developed very far, mainly by Hlavatý. A number of mathematicians and theoretical physicists have contributed to the development of this theory. Generalizing \(X_ 4\) to an n-dimensional generalized Riemannian manifold \(X_ n\), \textit{R. C. Wrede} [Tensor, New. Ser. 8, 95-122 (1958; Zbl 0080.219)] studied Principles A and B of Einstein's unified field theory for the first time. But this solution of Einstein's equations is not surveyable. Hlavatý's solution for the first class in \(X_ 4\) is not surveyable either. Later, \textit{K. T. Chung} [Nuovo Cimento, X. Ser. 27, 1297-1324 (1963; Zbl 0115.436)] gave a very handy and clear solution in the 4-dimensional g-unified field theory using the method of substitution. Recently, \textit{K. T. Chung} and \textit{C. H. Cho} [Nuovo Cimento 100 B, 537-550 (1987)] introduced the concept of the n-dimensional SE manifold \(SEX_ n\), imposing a semisymmetric condition on \(X_ n\), and found a unique representation of Einstein's connection in a beautiful and clear form. Later, \textit{K. T. Chung} and \textit{I. Y. Lee} [Int. J. Theor. Phys. 27, No.9, 1083-1104 (1988; Zbl 0652.53057); and \textit{K. T. Chung} and \textit{I. H. Hwang} [ibid., 1105-1136 (1988; Zbl 0652.53058)] also investigated curvature theory and field equations in \(SEX_ n.\) The purpose of the present paper is to introduce the new concept of a C- nonholonomic frame of reference in a general \(X_ n\) at points of its submanifold and to deal with its consequences in \(X_ n\) and \(SEX_ n\).
scientific article; zbMATH DE number 4144855