On locally convex hypersurfaces in Hadamard manifolds
A complete simply connected Riemannian manifold of nonpositive sectional curvature is called here a Hadamard manifold. For an immersion \(f:N^n \to M^{n+1}\) of class \(C^2\) into a Riemannian space \(M^{n+1}\) the product of principal curvatures: \(K=k_1,k_2,\dots,k_n\) is called the Gaussian curvature. It is proved that if here \(N^n\) is compact connected, \(n\geq 2\), \(M^{n+1}\) is a Hadamard manifold, \(f\) is regular, and either 1) the Gaussian curvature of \(F=f(N^n)\) is positive at every point, or 2) the second quadratic form of \(F\) (at each point under a suitable choice of the unit normal) is nonpositive definite and of rank \(\geq 1\), or 3) the external Ricci curvature of \(F\) is nonnegative and the external scalar curvature is positive, then \(F\) is a convex hypersurface homeomorphic to the sphere. It is remarked that after this paper had been submitted, the author learned that the result under assumption 2) had in essence been proved by Alexander [see \textit{F. W. Warner}, Trans. Am. Math. Soc., 122, 341-356 (1966; Zbl 0139.15601)] based on a different idea that suggests an approach to solving the problem without constraints on the rank of the second quadratic form.
scientific article; zbMATH DE number 1531350
scientific article