Frenet formulas for a general subspace of a Riemann space

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Introduction. The Frenet formulas for a curve in ordinary space have been extended by Blaschket to a curve in a Riemann space Vm. In ??1-4 of the present paper it is shown that by utilizing a properly defined covariant differentiation similar formulas can be obtained for any subspace V. of a Vm.t For a curve the curvatures are arbitrary functions of the parameter; for a general subspace the corresponding quantities are functions of the coordinates xi which must satisfy certain integrability conditions, the Gauss, Codazzi, Ricci equations. In ?5 a curve in the subspace is considered and Meusnier's Theorem extended, while in ?6 certain relations of V, to its osculating geodesic spaces are discussed. 1. Complete tensors and complete derivatives. Consider a Riemann space Vm with definite fundamental tensor a,,, and let Vn with fundamental tensor gii be a subspace given by (1. 1) ya = ya(xl, , xn) (a = 1, m),