Umbilical spacelike submanifolds of arbitrary co-dimension
Differential Geometry
2017-02-16 v2 General Relativity and Quantum Cosmology
Abstract
Given a semi-Riemannian manifold, we give necessary and sufficient conditions for a Riemannian submanifold of arbitrary co-dimension to be umbilical along normal directions. We do that by using the so-called \emph{total shear tensor}, i.e., the trace-free part of the second fundamental form. We define the \emph{shear space} and the \emph{umbilical space} as the spaces generated by the total shear tensor and by the umbilical vector fields, respectively. We show that the sum of their dimensions must equal the co-dimension.
Keywords
Cite
@article{arxiv.1701.06045,
title = {Umbilical spacelike submanifolds of arbitrary co-dimension},
author = {Nastassja Cipriani and José M. M. Senovilla},
journal= {arXiv preprint arXiv:1701.06045},
year = {2017}
}
Comments
7 pages, no figures, contribution to the Proceedings of the VIII International Meeting on Lorentzian Geometry (GELOMA, M\'alaga, September 20-23, 2016)