I am trying to understand what a holomorphic quadratic differential is, i have read a local definition on two books: Jürgen-Jost-"Compact Riemman Surfaces" and Kurt Strebel-"Quadratic-Differentials". The definition that they use is local:
Definition: Let ( M , g) be a Riemann surface with a conformal metric and $z$ a local conformal coordinate, we say that $\varphi dz^2$ is a holomorphic quadratic differential if $\varphi$ is holomorphic.
That is the definition in Jürgen-Jost book, in Kurt Strebel "Quadratic-Differentials" They only add that a transformation rule for other coordinates is needed.
I would like to understand a global definition in terms of sections, in wikipedia i found this:
"a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle"
I guess holomorphic cotangent bundle means the bundle of holomorphic 1-forms. I don't understand the "symmetric square part". I would like to have a global definition and a local coordinate representation. Of course I don't trust at all about a wikipedia link. So I would like to ask, is this definition right? Is there a good reference i should read ?