Some theorems on Codazzi tensors (Q1057509)

Let (M,g) be a closed n-dimensional Riemannian manifold with positive sectional curvature, A be a positive definite Codazzi tensor on M. Denote by \(\lambda_ 1,\lambda_ 2,...,\lambda_ n\) the eigenvalues of A with respect to g, and let \(S_{\gamma}=(1/\left( \begin{matrix} \gamma \\ n\end{matrix} \right))\sum \lambda_{i_ 1}\lambda_{i_ 2}...\lambda_{i_ r}\). In this paper we prove that 1) if \(S_{\gamma}=const\) for some \(\gamma\in \{2,3,n\}\), then \(\lambda_ 1=\lambda_ 2=...=\lambda_ n\); 2) if \(F(S_ 1,S_{\gamma})=0\) for some \(\gamma\in \{2,3\}\), where \(\partial_ 1F(S_ 1,S_{\gamma})\geq 0\), \(\partial_ 2F(S_ 1,S_{\gamma})>0\), then \(\lambda_ 1=\lambda_ 2=...=\lambda_ n\); 3) if \(f(\lambda_ 1)+f(\lambda_ 2)+...+f(\lambda_ n)=0\), where \(f(t)=\sum^{\infty}_{1}a_ kt^ k\) is a power series with \(a_ k\geq 0\), and \(\max \{\lambda_ 1,...,\lambda_ n\}<R\), the convergence radius of f(t), then \(\lambda_ 1=\lambda_ 2=...=\lambda_ n\). As consequences of our theorems, some characteristics of a totally umbilical hypersurface in a Riemannian manifold of constant curvature are given.