Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic plane

TL;DR

This paper explores generalizations of Hilbert's theorem on the non-immersibility of the hyperbolic plane into Euclidean space, extending to certain symmetric spaces and confirming non-existence of complete immersions in compact cases.

Contribution

It characterizes when Hilbert's theorem can be extended to other symmetric spaces and proves non-existence of complete immersions in compact cases.

Findings

Non-existence of complete isometric immersions of hyperbolic space into Euclidean space.

Generalization of the theorem to certain symmetric space cases.

Existence of local embeddings in compact cases, but no complete immersions.

Abstract

We discuss generalizations of the well-known theorem of Hilbert that there is no complete isometric immersion of the hyperbolic plane into Euclidean 3-space. We show that this problem is expressed very naturally as the question of the existence of certain homotheties of reflective submanifolds of a symmetric space. As such, we conclude that the only other (non-compact) cases to which this theorem could generalize are the problem of isometric immersions with flat normal bundle of the hyperbolic space $H^n$ into a Euclidean space $E^{n+k}$, $n \geq 2$, and the problem of Lagrangian isometric immersions of $H^n$ into $\cc^n$, $n \geq 2$. Moreover, there are natural compact counterparts to these problems, and for the compact cases we prove that the theorem does in fact generalize: local embeddings exist, but complete immersions do not.