Riemannian Spaces of Class Greater Than Unity

Abstract

The geometry of a Riemannian space V, as a subvariety of a Riemannian space V,,+, was studied first by Voss.' In presenting this theory by means of tensor calculus Ricci2 made use of n linearly independent mutually orthogonal unit vectors of V+, normal to V, . This method was followed in my presentation of the theory for spaces whose fundamental quadratic form is definite or indefinite as the case may be, when V+, is a general space and in particular when it is flat.3 Burstin and Mayer, together and individually, have developed a theory for the case when the fundamental form is definite in accordance with which the n normal vectors are chosen in a particular manner and arranged in groups, each group being treated as a unit, resulting in a generalization to 4 varieties of higher order of the Frenet formulas for a curve. In the present paper a study is made of a Vn of class p(> 1), when the fundamental form is definite or indefinite, and the vectors normal to Vn are chosen in groups somewhat after the manner of Burstin and Mayer, but the groups are not treated as units.