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 .
References
Kurt Strebel, Quadratic differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 5. Springer-Verlag, Berlin, 1984. xii + 184 pp.
ISBN3-540-13035-7.
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.
ISBN4-431-70088-9.
Frederick P. Gardiner, Teichmüller Theory and Quadratic Differentials. Wiley-Interscience, New York, 1987. xvii + 236 pp.
ISBN0-471-84539-6.