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 Teichmueller 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
.
References
- Kurt Strebel, Quadratic differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 5. Springer-Verlag, Berlin, 1984. xii+184 pp. ISBN 3-540-13035-7
- Y. Imayoshi and M. Taniguchi, M. An introduction to Teichmüller spaces. Translated and revised from the Japanese version by the authors. Springer-Verlag, Tokyo, 1992. xiv+279 pp. ISBN 4-431-70088-9