When is symplectic pullback bundle trivial

Question

Suppose $x : S \to M$ is a smooth map, where $M$ is a symplectic manifold and $S$ is a Riemann surface. Consider the pullback bundle $x^*TM \to S$. When is this bundle trivial (as symplectic vector bundle)? If $S$ has non-empty boundary then the answer is "always". But for instance if $S$ is a sphere then it is not always trivial. What is the obstruction in general? And how does one check whether a given bundle is trivial?

Answer 1

A symplectic vector bundle has a contractible family of complex structures on it, so a symplectic vector bundle is essentially a complex vector bundle. In particular then, you want to look at the first Chern class of the bundle.