2019/02/11 by Giovanna Citti, Citti, Giovanna, Gianmarco Giovannardi +3
Engineering · Mathematics · #49Q20 (Primary) #53C17 (Secondary) #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.1902.04015
openalex publication_date 2019/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a length functional for C1 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.