# Variational formulas for curves of fixed degree

## TL;DR

This paper extends the deformation theory of curves of fixed degree in graded manifolds, providing variational formulas, examples, and conditions for deformability, thus advancing geometric analysis in this setting.

## Contribution

It introduces a variational framework for curves of fixed degree in graded manifolds, generalizing classical results and offering new deformation criteria and applications.

## Key findings

- Derived variational formulas for length functional
- Provided sufficient conditions for curve deformation
- Extended classical deformation theory to graded manifolds

## Abstract

We consider a length functional for $C^1$ 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.