This paper studies isometric minimal immersions \(f\) of a complete orientable Riemannian \(n\)-manifold \(M\) into the round sphere \(S^{n+k}\). Theorem A: If \(k=1\), then the supremum of Ric\((M)\) is \(\geq n-2\). Moreover, if the supremum equals \(n-2\), then if \(n\) is odd, the universal cover of \(M\) is homomorphic to \(S^n\), and if \(n\) is even, \(f(M)\) is the product of two round spheres of dimension \(n/2\). Theorem B: If \(M\) is compact of Ric\((M)>n(n-3)/(n-1)\) and if \(f\) is an embedding, then \(M\) is homeomorphic to a spherical space form in case \(n=3\), and \(M\) is homeomorphic to \(S^n\) if \(n\neq 3\).