Quadratic differential

Local form

Each quadratic differential on a domain U in the complex plane may be written as f ( z ) d z ⊗ d z where z is the complex variable and f is a complex valued function on U . Such a `local' quadratic differential is holomorphic if and only if f is holomorphic. Given a chart μ for a general Riemann surface R and a quadratic differential q on R , the pull-back ( μ − 1 ) ∗ ( q ) defines a quadratic differential on a domain in the complex plane.

Relation to abelian differentials

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

Singular Euclidean structure

A holomorphic quadratic differential q determines a Riemannian metric | q | on the complement of its zeroes. If q is defined on a domain in the complex plane and q = f ( z ) d z ⊗ d z , then the associated Riemannian metric is | f ( z ) | ( d x 2 + d y 2 ) where z = x + i y . Since f 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 z such that f ( z ) = 0 .