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.
Each quadratic differential on a domain
in the complex plane may be written as
Such a "local" quadratic differential is holomorphic if and only if
for a general Riemann surface
and a quadratic differential
defines a quadratic differential on a domain in the complex plane.
is an abelian differential on a Riemann surface, then
is a quadratic differential.
A holomorphic quadratic differential
determines a Riemannian metric
on the complement of its zeroes.
is defined on a domain in the complex plane, and
, then the associated Riemannian metric is
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