Variational formulas for curves of fixed degree
Abstract
We consider a length functional for curves of fixed degree in graded manifolds equipped with a Riemannian metric. The first variation of this length functional can be computed only if the curve can be deformed in a suitable sense, and this condition is expressed via a differential equation along the curve. In the classical differential geometry setting, the analogous condition was considered by Bryant and Hsu in [Invent. Math., 114(2):435-461, 1993, J. Differential Geom., 36(3):551-589, 1992], who proved that it is equivalent to the surjectivity of a holonomy map. The purpose of this paper is to extend this deformation theory to curves of fixed degree providing several examples and applications. In particular, we give a useful sufficient condition to guarantee the possibility of deforming a curve.
Keywords
Cite
@article{arxiv.1902.04015,
title = {Variational formulas for curves of fixed degree},
author = {Giovanna Citti and Gianmarco Giovannardi and Manuel Ritoré},
journal= {arXiv preprint arXiv:1902.04015},
year = {2021}
}
Comments
In the second version of the manuscript we add Appendix B where we extend the holonomy map to curves with square integrable derivatives