Evaluating composition of tensor fields $g(X,J(Y))$

Question

Let $M$ be a smooth manifold, $g$ a Riemannian metric and $J$ an almost complex structure on $M$. Since $g$ is a $(0,2)$ tensor field, we get $$g(X,Y) \in C^\infty(M)$$ for all $X,Y \in \mathfrak{X}(M)$. Furthermore, since $J$ is a $(1,1)$ tensor field we have that $$J(X) \in \mathfrak{X}(M)$$ Hence $$g(X,J(Y))$$ is well-defined. It can be shown that this is a $(0,2)$ field again. Now my question is, how does this tensor field look? I mean, if $u,v \in T_pM$, I would say that we have $$g_p(u,J_p(v))$$ Is this correct? How would this be shown?

Answer 1

I think I got it: Let $p \in M$ and $u,v \in T_pM$. By the extension theorem for vector fields there exist $X,Y \in \mathfrak{X}(M)$, such that $X_p = u$ and $Y_p = v$. Thus $$\begin{align*} \omega_p(u,v) &= \omega_p(X_p,Y_p)\\ &= \omega(X,Y)(p)\\ &= g(X,JY)(p)\\ &= g_p(X_p,(JY)_p)\\ &= g_p(u,J_pY_p)\\ &= g_p(u,J_pv).\end{align*}$$