Generalization of normal curvature of a curve in a Riemannian V_n

Abstract

In the present paper a congruence of curves through points of a hypersurface V_n imbedded in a Riemannian V_n+1 has been considered. In analogy with the normal curvature of a curve C in V_n, the generalized normal curvature of C at any point of it, relative to the curve of the congruence through that point, has been defined as the negative of the resolved part along C, of the derived vector of the unit tangent to the curve of the congruence through the point along C. The concepts of normal curvature of a hypersurface, principal directions, principal curvatures, lines of curvature, conjugate directions, asymptotic directions and asymptotic lines have been generalized and generalizations of several known theorems on the curvature of a hypersurface V_n in V_n+1 have been obtained.