# Results related to generalizations of Hilbert&#x27;s non-immersibility theorem for the hyperbolic plane - Mathematics > Differential Geometry

Results related to generalizations of Hilbert&#x27;s non-immersibility theorem for the hyperbolic plane - Mathematics > Differential Geometry - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Abstract: We discuss generalizations of the well-known theorem of Hilbert that there isno complete isometric immersion of the hyperbolic plane into Euclidean 3-space.We show that this problem is expressed very naturally as the question of theexistence of certain homotheties of reflective submanifolds of a symmetricspace. As such, we conclude that the only other non-compact cases to whichthis theorem could generalize are the problem of isometric immersions with flatnormal 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 theseproblems, and for the compact cases we prove that the theorem does in factgeneralize: local embeddings exist, but complete immersions do not.

Autor: David Brander

Fuente: https://arxiv.org/