Weil–Petersson metric

If a point of Teichmüller space is represented by a Riemann surface R, then the cotangent space at that point can be identified with the space of quadratic differentials at R. Since the Riemann surface has a natural hyperbolic metric, at least if it has negative Euler characteristic, one can define a Hermitian inner product on the space of quadratic differentials by integrating over the Riemann surface. This induces a Hermitian inner product on the tangent space to each point of Teichmüller space, and hence a Riemannian metric. ==Properties==

stated, and proved, that the Weil–Petersson metric is a Kähler metric. proved that it has negative holomorphic sectional, scalar, and Ricci curvatures. The Weil–Petersson metric is usually not complete. ==Generalizations==

The Weil–Petersson metric can be defined in a similar way for some moduli spaces of higher-dimensional varieties. == See also ==