Umbilical submanifolds of $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$

Abstract

In this article we complete the classification of the umbilical submanifolds of a Riemannian product of space forms, addressing the case of a conformally flat product $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$, which has not been covered in previous works on the subject. We show that there exists precisely a $p$-parameter family of congruence classes of umbilical submanifolds of $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$ with substantial codimension~$p$, which we prove to be at most $\mbox{min}\,\{k+1, n-k+2\}$. We study more carefully the cases of codimensions one and two and exhibit, respectively, a one-parameter family and a two-parameter family (together with three extra one-parameter families) that contain precisely one representative of each congruence class of such submanifolds. In particular, this yields another proof of the classification of all (congruence classes of) umbilical submanifolds of $\mathbb{S}^n\times \mathbb{R}$, and provides a similar classification for the case of $\mathbb{H}^n\times \mathbb{R}$. We determine all possible topological types, actually, diffeomorphism types, of a complete umbilical submanifold of $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$. We also show that umbilical submanifolds of the product model of $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$ can be regarded as rotational submanifolds in a suitable sense, and explicitly describe their profile curves when $k=n$. As a consequence of our investigations, we prove that every conformal diffeomorphism of $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$ onto itself is an isometry.