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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.)

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Ashley
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    I think there is a large set of compatible almost complex structures, but the set of compatible almost complex structures is contractible in any suitable topology. The advantage is the positivity of $\omega(\ ,J\ )$. This causes the Cauchy Riemann equations for mapped in surfaces (curves) to be elliptic, so analyzing the properties of their dilution sets is generically very tractable. – Charlie Frohman Jan 07 '17 at 03:00

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