A well-known theorem of Wolpert shows that the Weil–Petersson symplectic form on Teichmüller space, computed on two infinitesimal twists along simple closed geodesics on a fixed hyperbolic surface, equals the sum of the cosines of the intersection angles. We define an infinitesimal deformation starting from a more general object, namely a balanced geodesic graph, by which any tangent vector to Teichmüller space can be represented. We then prove a generalization of Wolpert’s formula for these deformations. In the case of simple closed curves, we recover the theorem of Wolpert.
[Gol80] Discontinuous Groups and the Euler Class (1980) (Ph. D. Thesis) | MR 2630832
[Thu86] Earthquakes in two-dimensional hyperbolic geometry, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984) (London Mathematical Society Lecture Note Series) Volume 112, Cambridge University Press, 1986, pp. 91-112 | MR 903860 | Zbl 0628.57009
[Uhl83] Closed minimal surfaces in hyperbolic -manifolds, Seminar on minimal submanifolds (Annals of Mathematics Studies) Volume 103, Princeton University Press, 1983, pp. 147-168 | MR 795233 | Zbl 0529.53007