I am a beginner in complex geometry. I am reading the proof of the Calabi conjecture for the first Chern class is negative. In the beginning, we will let the metric $\tilde{g}_{ij}=g_{ij}+\partial_i\bar{\partial}_j\varphi$ and $g_{ij}$ is fixed. Clearly, we can take normal coordinate to $\tilde{g}$ and then buy standard linear algebra and apply the unitary transform to diagonalize $g$. In other words, we create a good frame for the calculation.
My question is, why is such construction legal and independent of local coordinates and frames?