Hypersurfaces with harmonic curvature

Abstract

where Rijk denotes the covariant derivative of Ricci tensor Rfy. This conditionis essentially weaker than that for the parallel Recci tensor. In fact Derdzinski [2] gave an example of a 4-dimentional Riemannian manifold with harmonic curvaiure whose Ricci tensor is not parallel. Recently E. Omachi [5] investigated compact hypersurfaces with harmonic curvature in a Euclidean space or a sphere and gave a classification of such hypersurfaces provided that the mean curvature is constant. This paper is concerned with hypersurfaces with harmonic curvature isometrically immersed into a Riemannian manifold of constant curvature. In the firstsection,a concept of Codazzi type for a symmetric (0, 2)-tensor is introduced and a sufficientcondition for a symmetric tensor of Codazzi type to be parallelis given. A similar condition for a symmetric tensor of Codazzi type is also treated by S. Y. Cheng an S. T. Yau [1]. In the second section, the result proved in the first section is applied to hypersurfaces with harmonic curvature immersed in a Riemannian manifold of constant curvature, in which Omachi's result [5] is generalized without the assumption of compactness. Finally we study also the case where the assumotion that the mean curvature is constant is omitted.