Abstract
A well-known theorem of Wolpert shows that the Weil-Petersson symplectic form on Teichm\"uller 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 Teichm\"uller 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.
Highlights
Given a closed oriented surface S of genus g 2, Teichmüller space T (S) is classically defined as the space of complex structures on S, up to biholomorphisms isotopic to the identity
Given two simple closed geodesics c, c on a fixed closed hyperbolic surface (S, h), let us denote by tc and tc the infinitesimal twists along c and c, namely tc =
As observed by Mess, the isomorphism between Minkowski space R2,1 and the Lie algebra isom(H2) ∼= so(2, 1) induces a correspondence between the tangent space of T (S), in the model HA1d ρ(π1(S), isom(H2)) of the representation variety, and the translation part of the holonomies of manifolds M as above. This enables us to show that the map which associated to a balanced geodesic graph (G, w) the deformation t(G,w) in T[h]T (S) is surjective
Summary
Given a closed oriented surface S of genus g 2, Teichmüller space T (S) is classically defined as the space of complex structures on S, up to biholomorphisms isotopic to the identity. It is naturally identified to what is sometimes called Fricke space F(S), namely the space of hyperbolic metrics on S up to isometries isotopic to the identity. The Weil–Petersson metric is a Kähler metric on T (S), whose definition only involves the point of view of T (S) as the space of complex structures on S. Given two simple closed geodesics c, c on a fixed closed hyperbolic surface (S, h), let us denote by tc and tc the infinitesimal twists along c and c , namely tc =.
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