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 holomorphic quadratic differentials on a Riemann surface has a natural interpretation as the cotangent space to the Riemann moduli space, or Teichmüller space.

Local form

Each quadratic differential on a domain in the complex plane may be written as, where is the complex variable, and is a complex-valued function on.
Such a "local" quadratic differential is holomorphic if and only if is holomorphic. Given a chart for a general Riemann surface and a quadratic differential on, the pull-back defines a quadratic differential on a domain in the complex plane.

Relation to abelian differentials

If is an abelian differential on a Riemann surface, then is a quadratic differential.

Singular Euclidean structure

A holomorphic quadratic differential determines a Riemannian metric on the complement of its zeroes. If is defined on a domain in the complex plane, and, then the associated Riemannian metric is, where. Since is holomorphic, the curvature of this metric is zero. Thus, a holomorphic quadratic differential defines a flat metric on the complement of the set of such that.