In mathematics, the Weil–Petersson metric is a Kähler metric on the Teichmüller space Tg,n of genus g Riemann surfaces with n marked points.
It was introduced by André Weil (1958, 1979) using the Petersson inner product on forms on a Riemann surface (introduced by Hans Petersson).
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.
Ahlfors (1961b) proved that it has negative holomorphic sectional, scalar, and Ricci curvatures.
The Weil–Petersson metric can be defined in a similar way for some moduli spaces of higher-dimensional varieties.