How can we show that if we have some symmetric $(2,0)$-tensor on $\mathbb{R}^{2n}:$ $$g=\sum_{\mu,\lambda}^{2n}g_{\mu,\lambda}dx^{\mu}\otimes dx^{\lambda}$$ and $g$ satisfies $g(TX,TY)=g(X,Y)$, where $X$ and $Y$ represent any tangent vectors in the tangent space $T\mathbb{R}^{2n}$ and $T$ being a $(1,1)$-tensor on $\mathbb{R}^{2n}$ such that: $$T\bigg(\frac{\partial}{\partial x_i}\bigg)=\frac{\partial}{\partial x_{i+n}}, T\bigg(\frac{\partial}{\partial x_{i+n}}\bigg)=-\frac{\partial}{\partial x_i}$$
Then if $\alpha$ is a closed two-form, prove that for any $i,j,k=1,2,...,n$: $$\frac{\partial g_{i,j}}{\partial x_j}-\frac{\partial g_{i,k}}{\partial x_k}=\frac{\partial g_{k,j+n}}{\partial x_{i+n}}$$
HINT:Define $\alpha$ to be a $(2,0)-$tensor such that $\alpha (X,Y)=g(X,TY)$ for any tangent vectors $X,Y$ and notice that we can show $\alpha$ is a $2-$form such that $\alpha(X,Y)=-\alpha(Y,X)$. Finally also express $\alpha$ as a sum of wedge products of $dx^i$'s and $dx^{j+n}$'s.
I have no idea how to begin this problem, can someone give me some hints?
It is interesting to see that your professor gave you a final prep problem that is almost similar to a take-home midterm problem that I gave to my students in the "Calculus on Manifolds" course last year -- even the symbols used are almost the same.
I am afraid that your question above is not properly stated. The 2-form $\alpha$ should be defined in the question (before ``if $\alpha$ is closed'') rather than in the hint (that's super strange to formulate the question this way). I hope it is not your professor's fault.
For the complete question, you may visit my website https://piazza.com/ust.hk/spring2017/math4033/resources
You can find the above question in the "Midterm" area. It was Q1 of my last year's midterm. I have also posted the solution there.
I have also observed that you asked a number of questions very similar to homework, exams, and lecture notes of my course (both last year and this year). Some crucial conditions were, however, not well stated in your posts. I hope it's not the fault of your professor. For complete version of those questions, you may visit my last year's website (link above) or this year's one:
https://canvas.ust.hk/courses/16957