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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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up vote 2 down vote accepted

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. –  john mangual 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
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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
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