Riemannian metrics induced by two immersions

Abstract:

We consider the situation where a Riemannian manifold ${M^n}$ can be isometrically immersed into spaces ${N^{n + 1}}(c)$ and ${N^{n + q}}(\tilde c)$ with constant curvatures $c < \tilde c$, $q \leqslant n - 3$, and show that this implies the existence, at each point $p \in M$, of an umbilic subspace ${U_p} \subset {T_p}M$, for both immersions, with ${U_p} \geqslant n - q$. In particular, if ${M^n}$ can be isometrically immersed as a hypersurface into two spaces of distinct constant curvatures, ${M^n}$ is conformally flat.