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On the Ricci and Weingarten maps of a hypersurface

1965
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Proceedings of the American Mathematical Society
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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

doi:10.1090/s0002-9939-1965-0176483-0
fatcat:t66kiiy5kzb2lkbtbygnytlrx4