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Weil–Petersson metric
Mathematical metric for Riemann surfaces
Mathematical metric for Riemann surfaces
In mathematics, the Weil–Petersson metric is a Kähler metric on the Teichmüller space T**g,n of genus g Riemann surfaces with n marked points. It was introduced by using the Petersson inner product on forms on a Riemann surface (introduced by Hans Petersson).
Definition
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.
References
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