Let $(M,g)$ be a riemannian manfiold.
The cotangent bundle $T^*M$ of a manifold $M$ admits a canonical symplectic form. See here.
Let $g$ be the canonical induced metric on $T^*M$. See here
Every symplectic manifold with a given riemannian metric admits an almost complex structure in a canonical way. See here.
Can we say anything about this almost complex structure? For example, does it take on a nice form because we are using a symplectic structure and metric which in a sense keep track of the fact that $T^*M$ is locally a product?
(Sorry this is a little vague. I'm note exactly sure what to expect, but I hope what I am saying makes a little bit of sense.)