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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?

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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.

Sam Lisi
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  • Is the first Chern class being 0 a sufficient condition for triviality? – Todd Mar 23 '20 at 12:28
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    @Todd: yes, as long as $S$ is a Riemann surface. If $S$ is higher dimensional, the first Chern class vanishing is no longer sufficient. Ask a new question if you need an explanation for this. I don't have room in this comment box to explain it. (Or just look in Milnor & Stasheff) – Sam Lisi Mar 24 '20 at 20:01