# Metric induced almost complex structure on cotangent bundle

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.)

• I think as metric and symplectic forms are both non degenerate, they give maps TM to T^* M, now you can define J by composing one map with the inverse of other. Mar 27, 2020 at 1:09

• Thank you for the reference. I don't understand what you mean by metric on target though. The metric we start with is on $M$. We have the projection from $T^*M$ to $M$, but this doesn't pull back the metric. Jan 6, 2017 at 22:43
• I’m sorry but I did not find anything about induced metric on the cotangent bundle in your reference, in that page I could only find discussion about open subset of $\mathbb C^n$. Mar 27, 2020 at 1:07