On constant curvature submanifolds of space forms

Abstract

We prove a converse to well-known results by E. Cartan and J. D. Moore. Let f:Mcn→Qc˜n+p be an isometric immersion of a Riemannian manifold with constant sectional curvature c into a space form of curvature c˜, and free of weak-umbilic points if c>c˜. We show that the substantial codimension of f is p=n−1 if, as shown by Cartan and Moore, the first normal bundle possesses the lowest possible rank n−1. These submanifolds are of a class that has been extensively studied due to their many properties. For instance, they are holonomic and admit Bäcklund and Ribaucour transformations.