  In 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.
