Let $(M,g)$ be a Riemannian manifold, $g' = g + f$ be another metric. Is it possible to get local coordinates such that at a point $P \in M$, $g_{ij} = \delta_{ij}$ and $f_{ij} = 0$ for all $i \not = j$.
If (in any coordinates) $\sum_j g_{ij} g'_{jk} = \sum_j g'_{ij} g_{jk}$ linear algebra gives us the result via simultaneous diagonalization, but is it possible without this condition?
In particular i have a Kähler metric $g$ and $g'_{i\bar{j}} = g_{i\bar{j}} + \partial_{i\bar{j}} \varphi$. The result is needed for the proof of estimates for the Calabi conjecture in Thierry Aubin - Some nonlinear problems in Riemannian geometry.
Another way of phrasing this problem is that you're looking for an orthonormal basis (with respect to $g$) that diagonalizes $f$. Since $f$ is real symmetric, this is just the spectral theorem for Hermitian matrices.