I'm trying to understand this question and I was hoping someone could help me. One thing in particular is confusing me.
Question: They are starting with a riemannian manifold $(M,g)$ and considering the "metric-induced" almost complex structure on $T^*M$. What exactly is the metric induced almost complex structure? Is there a nice way to visualize/think of it?
Thoughts: Does this come from first defining a metric $g_0$ (induced by $g$) on $T^*M$ so that we would then have the canonical symplectic form $\omega_0$ and and riemannian $g_0$ on $T^*M$, which would determine an almost complex structure via the compatible triples. If this is the case, how exactly is $g_0$ defined using $g$? (a reference for this construction will definitely suffice.)