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.)
Daniel Huybrechts: "Complex Geometry: An Introduction," Springer Universitex, 2005, presents, on page 48, a very nice, clear picture of what is going on in your question. I find his explanation clear, and hope you too find it so.
With regard to your thoughts, Riemannian means isometric. That is, the metric of the target is pulled back to the source, so in the induced metric on the source, the two manifolds are isometric or Riemannian.
I hope this helps.