2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/167815In a family of compact, canonically polarized, complex manifolds the first variation of the lengths of closed geodesics is computed. As an application, we show the coincidence of the Fenchel-Nielsen and Weil-Petersson symplectic forms on the Teichmueller spaces of compact Riemann surfaces in a purely geometric way. The method can also be applied to situations like moduli spaces of weighted punctured Riemann surfaces, where the methods of Kleinian groups are not available.Differential GeometryComplex Variables32G15; 30F60text