A section of the square of the canonical bundle, locally ; its coefficient changes by the square of the coordinate derivative. A holomorphic quadratic differential has holomorphic coefficient, and its flat metric of a quadratic differential is .
For , the length element is , locally Euclidean in flat coordinates. Zeros give cone points, not punctures. Its infimal length in a homotopy class need not determine the hyperbolic length of that class.
The area of the flat metric of a quadratic differential is . Multiplying by a scalar multiplies this area by and lengths by . It is preserved by the SL2R action on differentials.
A surface with local Euclidean charts whose changes of coordinate are , outside finitely many cone points. A nonzero holomorphic quadratic differential defines such a surface; a zero of order has cone angle .
A quadratic differential with holomorphic local coefficients. If it is nonzero, its zeros have total order on a compact genus- Riemann surface. Away from its zeros, a nonzero differential has flat coordinates whose changes of coordinate have the form .
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A **quadratic differential** is a mathematical concept that arises primarily in the fields of complex analysis and differential geometry, often used to study the properties of Riemann surfaces and their associated geometric structures. In a more formal description, a quadratic differential on a Riemann surface can be seen as a section of the tensor product of the cotangent bundle with itself, specifically a differential form of type (2,0).