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

in the complex plane is holomorphic, the pull-back defines a quadratic differential on a domain in the complex plane.

Relation to abelian differentials

is an abelian differential is a quadratic differential.

Singular Euclidean structure

determines a Riemannian metric is holomorphic, the curvature.

References

• Kurt Strebel, Quadratic differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 5. Springer-Verlag, Berlin, 1984. xii + 184 pp. .

• 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. .

• Frederick P. Gardiner, Teichmüller Theory and Quadratic Differentials. Wiley-Interscience, New York, 1987. xvii + 236 pp. .