Cylindricity of isometric immersions between hyperbolic spaces

Abstract:

The motivation for this paper was to prove the following analogue of the Euclidean cylinder theorem: any umbilic-free isometric immersion $\eta :{H^{n - 1}} \to {H^n}$ between hyperbolic spaces takes the form of a hyperbolic $(n - 2)$-cylinder over a uniquely determined parallelizing curve in ${\bar H^n}$. Our approach is through the more general study of isometric immersions generated by one-parameter families of hyperbolic k-planes without focal points. A by-product of this study is a natural extension to curves in ${\bar H^n}$ of the notion of a parallel family of k-planes along a curve in ${H^n}$; the extension is based on spherical symmetry of variation fields. Existence and uniqueness properties of this extended notion of parallelism are considered.