Quadratic differential

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In mathematics, a quadratic differential is a form on a Riemann surface that locally looks like the square of an abelian differential. It has (at least) two other interpretations that are useful in the study of Riemann surfaces:

• a flat structure on the Riemann surface, with a discrete set of singularities (the zeroes of the quadratic differential); and
• an element of the cotangent space to the Teichmüller space of the Riemann surface, as is shown in deformation theory

Officially it is a section of the symmetric square of the holomorphic cotangent bundle.