Isometric immersion into sphere (Q1067622)

Let M be a submanifold of a Riemannian manifold \(\tilde M\) and B(M) its unit normal bundle. M is called locally convex if at each point \((P,\nu)\in B(M)\), there is a neighbourhood of P on M, which lies on one side of the geodesic hypersurface of \(\bar M\) with \(\nu\) as its unit normal at P. The author proves the following : Let M be a compact, connected, oriented n-dimensional \((n\geq 2)\) and locally convex submanifold of the unit sphere \(S^{n+N}\), which lies on an open hemisphere. Suppose M has a point where all second fundamental forms are definite. Then M belongs to a totally geodesic sphere \(S^{n+1}\subset S^{n+N}\) and is the boundary of a convex body of \(S^{n+1}\). The analogous problem with Euclidean space as ambient space has been considered by \textit{M. P. Do Carmo} and \textit{E. Lima} [Arch. Math. 20, 173-175 (1969; Zbl 0177.500)].