The Equivalence Problem of Curves in a Riemannian Manifold
Differential Geometry
2012-07-20 v1
Abstract
The equivalence problem of curves with values in a Riemannian manifold, is solved. The domain of validity of Frenet's theorem is shown to be the spaces of constant curvature. For a general Riemannian manifold new invariants must thus be added. There are two important generic classes of curves; namely, Frenet curves and a new class, called curves "in normal position". They coincide in dimensions only. A sharp bound for asymptotic stability of differential invariants is obtained, the complete systems of invariants are characterized, and a procedure of generation is presented. Different classes of examples (specially in low dimensions) are analyzed in detail.
Keywords
Cite
@article{arxiv.1207.4720,
title = {The Equivalence Problem of Curves in a Riemannian Manifold},
author = {M. Castrillon Lopez and V. Fernandez Mateos and J. Munoz Masque},
journal= {arXiv preprint arXiv:1207.4720},
year = {2012}
}