Dual of a Vector Bundle

Question

I am not quite getting the idea of morphisms between vector bundles. I read and reread the definition but I didn't quite get it.

Can someone provide me with an example of a morphism between a vector bundle and its dual for instance?

Thanks

Answer 1

Let $X$ be a manifold, and let $\omega$ be a symplectic form on $X$. We can use the symplectic form to define a map between the tangent and cotangent bundles. Namely, we have $$TX\rightarrow T^*X$$ $$(x,v)\mapsto \omega(x)(v,\cdot).$$