On the Ricci and Weingarten maps of a hypersurface

N. Hicks
1965 Proceedings of the American Mathematical Society  
The purpose of this note is to prove a classical type relation between the Ricci map R* and the Weingarten map i of a hypersurface in a flat Riemannian manifold. Indeed, if H is the mean curvature of the hypersurface, then L2-HL+R* = 0. This can be viewed, equivalently, as a relation between the Ricci tensor and the second and third fundamental forms. Some obvious corollaries follow. Let M he an «-dimensional CM Riemannian manifold, let X and F be vectors in Mm, the tangent space at a point m
more » ... pace at a point m in M, and let R(X, Y) be the skew-symmetric linear transformation valued curvature tensor determined by X and F (see Helgason [2]). We say
doi:10.1090/s0002-9939-1965-0176483-0 fatcat:t66kiiy5kzb2lkbtbygnytlrx4