Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic plane - Mathematics > Differential GeometryReport as inadecuate




Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic plane - Mathematics > Differential Geometry - Download this document for free, or read online. Document in PDF available to download.

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.



Author: David Brander

Source: https://arxiv.org/



DOWNLOAD PDF




Related documents