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.

Variational formulas for curves of fixed degree / Giovanna Citti, Gianmarco Giovannardi, Manuel Ritoré. - In: ADVANCES IN DIFFERENTIAL EQUATIONS. - ISSN 1079-9389. - ELETTRONICO. - 27:(2022), pp. 5-6.333-5-6.384.

Variational formulas for curves of fixed degree

Gianmarco Giovannardi;
2022

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.
27
333
384
Giovanna Citti, Gianmarco Giovannardi, Manuel Ritoré
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2158/1285610
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