Identity regarding problem 2.4 of Walds General Relativity
I was going through some of the problems in Wald's General Relativity and in problem 4 chapter 2 I found something that confuses me. So, basically we are asked to show that in any coordinate chart $(x^1,\ldots,x^n)$ the following equation holds \begin{align} \Bigr(\partial_\nu Y^{\gamma }\mu - \partial\mu Y^{\gamma * }\nu\Bigr)dx^\mu\otimes dx^\nu &= \Bigr(C{\alpha \beta}^\gamma Y^{\alpha }\mu Y^{\beta }\nu\Bigr)dx^\mu\otimes dx^\nu, \tag{$1$} \end{align} where ${Y\alpha}_{\alpha=1}^n$ is a frame in a $n$ dimensional manifold, ${Y^{\gamma }}{\gamma=1}^n$ its dual frame and $C_{\alpha\beta}^\gamma=\bigr(Y^{\gamma }\bigr)\bigr[Y_{\alpha},Y_{\beta}\bigr]$ the structure constant for the commutator. This can easily be shown to be the case by evaluating both sides of eq. 1 at $Y_\rho\otimes T_\sigma$. In local coordinates the LHS of (1) takes the form \begin{align} \Bigr(\partial_\nu Y^{\gamma }\mu - \partial\mu Y^{\gamma * }\nu\Bigr)dx^\mu\otimes dx^\nu\Bigr(Y\rho^\lambda \partial_\lambda\otimes Y_\sigma^\eta\partial_\eta\Bigr)&=\Bigr(\partial_\nu Y^{\gamma }\mu - \partial\mu Y^{\gamma * }\nu\Bigr)Y\rho^\mu Y_\sigma^\nu\ &=-Y_\mu^{\gamma}Y_\sigma^\nu\partial_\mu Y_\rho^\nu+ Y^{\gamma }\nu Y^{\mu}\rho\partial_\mu Y^\nu_\sigma\ &=-Y^{\gamma}\mu \bigr( Y\sigma^\nu \partial_\nu Y_\rho^\mu - Y_\rho^\nu \partial_\nu Y_\sigma^\mu \bigr)\ &=Y^{\gamma}\bigr[Y_\rho,Y_\sigma\bigr]\ &=C_{\rho \sigma}^\gamma. \end{align} Similarly, by evaluating the RHS we arrive at the same result. 324
Since ${Y_\rho\otimes Y_\sigma}{\rho,\sigma=1}^n$ is a basis for the space of tensors of rank 2, this successfully proves the desired result. However, and this is what confuses me, we may have as well evualuated both sides of eq. 1 with another basis, namely ${\partial/\partial x^\rho\otimes\partial/\partial x^\sigma}{\rho,\sigma=1}^n$. In which case the LHS becomes $ \partial_\sigma Y_\rho^{\gamma }-\partial_\rho Y_\sigma^{\gamma }$; where as the RHS can be written as \begin{align} C_{\alpha \beta}^\gamma Y^{\alpha }\rho Y^{\beta }\sigma&=\Bigr(\partial\nu Y^{\gamma }\mu - \partial_\mu Y^{\gamma * }\nu\Bigr)Y\alpha^\mu Y_\beta^\nu Y^{\alpha }_\rho Y^{\beta }\sigma. \end{align} So this seems to imply the relation \begin{align} Y\alpha^\mu Y^{\alpha *}\rho &= \delta^\mu\rho, \end{align} which seems reasonable, but I do not quite understand it. Is this relation true for any frame ${Y_\alpha}$? If so, how does one prove it? 818