In mathematics, the Weil–Petersson metric is a
Kähler metric Kähler may refer to:
People
*Birgit Kähler (born 1970), German high jumper
* Erich Kähler (1906–2000), German mathematician
* Heinz Kähler (1905–1974), German art historian and archaeologist
* Luise Kähler (1869–1955), German trade union ...
on the
Teichmüller space
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmülle ...
''T''
''g'',''n'' of genus ''g''
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s with ''n'' marked points. It was introduced by using the
Petersson inner product In mathematics the Petersson inner product is an inner product defined on the space
of entire modular forms. It was introduced by the German mathematician Hans Petersson.
Definition
Let \mathbb_k be the space of entire modular forms of weight k ...
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 differential In mathematics, a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle. If the section is holomorphic, then the quadratic differential is said to be holomorphic. The vector space of h ...
s at ''R''. Since the Riemann surface has a natural
hyperbolic metric, at least if it has negative
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
, one can define a
Hermitian inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
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 Kähler may refer to:
People
*Birgit Kähler (born 1970), German high jumper
* Erich Kähler (1906–2000), German mathematician
* Heinz Kähler (1905–1974), German art historian and archaeologist
* Luise Kähler (1869–1955), German trade union ...
. proved that it has negative holomorphic
sectional,
scalar, and
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
s. The Weil–Petersson metric is usually not complete.
Generalizations
The Weil–Petersson metric can be defined in a similar way for some
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
s of higher-dimensional varieties.
See also
*
Ramanujan–Petersson conjecture
References
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External links
*
Weil-Petersson metric on
nLab
The ''n''Lab is a wiki for research-level notes, expositions and collaborative work, including original research, in mathematics, physics, and philosophy, with a focus on methods from type theory, category theory, and homotopy theory. The ''n''Lab ...
{{DEFAULTSORT:Weil-Petersson metric
Riemann surfaces
Moduli theory