On Some Deformations of Riemann Surfaces. I

Abstract

We define a family of infinitesimal deformations of compact Riemann surfaces of genus $g > 2$ which generalizes the Fenchel-Nielsen deformations. Those new deformations are associated to smooth vector fields on the circle. We compute a representation of the deformations in terms of Poincaré series and determine the corresponding Eichler cohomology classes.