# Abelian vs Holomorphic Differentials vs Quadratic Differentials

In dynamics, they talk about Abelian differentials on surfaces, are they the same as holomorphic differentials?

Quadratic differentials are multiple valued and can change sign as you move around a zero.

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An Abelian differential is just a traditional name for a holomorphic or meromorphic differential on a compact Riemann surface.

A quadratic differential is just an element of $S^2(\Omega^1)$, the symmetric square of the sheaf of differentials. I do not really see what you mean when you say «[they] are multiple valued and can change sign as you move around a zero».

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quadratic differentials are not single-valued unless you move to the double cover. the branching occurs where the quadratic differential is 0, right? mainly i wasn't sure what "abelian" differentials were. – cactus314 Jan 17 '11 at 2:18
@john, as I said, I do not know what you mean by all that. – Mariano Suárez-Alvarez Jan 17 '11 at 2:25
John, could you give a reference? Like Mariano, I'm confused as to what you mean. Maybe it's some terminological difference. – arsmath Jan 17 '11 at 18:57
I think that what John means is that if one tries to write a quadratic differential in the form $(df)^2$ for some function $f$, then one may not be able to find such an $f$ globally, but in trying to do so, one constructs a double cover of the Riemann surface, branched at the zeroes of the quadratic differential. – Matt E Jan 17 '11 at 20:00