Some theorems on Codazzi tensors and their applications to hypersurfaces in Riemannian manifold of constant curvature
Let (M,g) be a Riemannian manifold and B a Codazzi-tensor field [''Discussion on Codazzi-tensors'', cf. \textit{D. Ferus}, \textit{W. Kühnel}, \textit{U. Simon} and \textit{B. Wegner}, Global differential geometry and global analysis, Lect. Notes Math. 838 (1981; Zbl 0437.00004); pp. 243- 299] with eigenvalues \(k_ 1,...,k_ n\); denote by \(C_ r\) the r-th elementary symmetric function of the eigenvalues. The author calculates \(\Delta C_ r\) [cf. also \textit{R. Walter}, Math. Ann. 270, 125-145 (1985; Zbl 0536.53054)], where \(\Delta\) denotes the Laplacian. As an application he proves several theorems of the following type: Let (M,g) be of positive sectional curvature, B positive definite and Codazzi. Assume there exists a \(C^ 2\)-function \(g: {\mathbb{R}}^ m\to {\mathbb{R}}\) such that \(g(C_ 1,...,C_ n)=const\) and \(C_ 1\) attains its minimum at some interior point of M, then \(B=k\cdot id\) on M. - As corollaries one gets various new and interesting characterizations of totally umbilical hypersurfaces of Riemannian manifolds, extending classical characterizations of ovaloids by curvature-functions.
elementary symmetric function of the eigenvalues
scientific article; zbMATH DE number 4005089