On constant curvature submanifolds of space forms

TL;DR

This paper establishes a converse to classical results, characterizing constant curvature submanifolds in space forms with minimal normal bundle rank, revealing their holonomic nature and transformation properties.

Contribution

It proves that submanifolds with constant curvature and minimal normal bundle rank are characterized by their substantial codimension being n-1, extending classical results by Cartan and Moore.

Findings

Submanifolds have substantial codimension p=n-1.

They are holonomic and admit Bäcklund and Ribaucour transformations.

The results provide a converse to classical theorems by Cartan and Moore.

Abstract

We prove a converse to well-known results by E. Cartan and J. D. Moore. Let $f\colon M^n_c\to\Q^{n+p}_{\tilde c}$ be an isometric immersion of a Riemannian manifold with constant sectional curvature $c$ into a space form of curvature $\tilde c$, and free of weak-umbilic points if $c>\tilde{c}$. 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 B\"{a}cklund and Ribaucour transformations.