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Let $(M,g)$ be a Riemannian manifold, and $x\in M$ be a fixed point.

Q Can we find a conformal transformation such that near $x$ we can write $e^{2u}g$ as $(dx^1)^2+\cdots+(dx^n)^2$?

Since the question is local, we can replace $M$ with $\mathbb R^n$.

(PS: Any reference is welcome.)

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    $\begingroup$ Not every metric on $ \mathbb{R}^n $ is conformal to the standard one. $\endgroup$
    – Julian Seipel
    Commented Apr 6, 2022 at 9:59
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    $\begingroup$ You want to compute the Weyl tensor of the conformal structure. This is a nice example where we can apply Cartan's method of equivalence, for example. $\endgroup$
    – Ben McKay
    Commented Apr 6, 2022 at 10:02

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To close this question: the problem of finding whether a pseudo-Riemannian metric is conformally flat is, not surprisingly, independent of signature and solved a very long time ago. In dimensions 4 or more, the metric is conformally flat just when the Weyl tensor vanishes, as discussed at length in the Wikipedia article on the Weyl tensor. In dimension three, the Weyl tensor vanishes and there is a higher order invariant, called the Cotton tensor whose vanishing characterizes conformal flatness in dimension three. In dimensions zero, one and two all pseudo-Riemannian metrics are locally conformally flat.

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