Abstract
The only umbilical surfaces of Euclidean space are planes and spheres. We propose to study umbilical hypersurfaces in Riemannian space with the added condition that they constitute a system of parallels. As will be shown, such families only exist in spaces of constant curvature. We will adhere throughout to the spirit of Weatherburn's notation [3]. Greek indices take the values 1, 2, * M, m=fn+1; the range of Latin indices is 1, , n. Consider the equations ya=ya(xx, XI , , xn, s). For each value of s we have a hypersurface and if the x's are fixed, the equations represent a trajectory of the family of hypersurfaces. Since we want a system of parallels [3, pp. 80-82 ], the trajectories used are geodesics orthogonal to the hypersurfaces. Making s the arc length 39ya=Nc is the unit normal and for the intrinsic derivative of ya, we find
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