Structures Symplectiques sur les Espaces de Courbes Projectives et Affines

TL;DR

This paper introduces symplectic and Poisson structures on spaces of nondegenerate curves in affine and projective geometries, linking algebraic structures like Virasoro and Gel'fand-Dikii brackets with differential geometry.

Contribution

It defines symplectic and Poisson structures on curve spaces and connects algebraic and geometric frameworks in a novel way.

Findings

Symplectic structure on nondegenerate curves in affine manifolds.

Poisson structures on spaces of projective curves.

Connection between Virasoro algebra, Gel'fand-Dikii bracket, and differential geometry.

Abstract

A symplectic structure on the space of nondegenerate and nonparametrized curves in a locally affine manifold is defined. We also consider several interesting spaces of nondegenerate projective curves endowed with Poisson structures. This construction connects the Virasoro algebra and the Gel'fand-Dikii bracket with the projective differential geometry.