# Orthogonal coordinates in Riemannian geometry

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Paul Gauduchon

created by daniele on 21 Oct 2018

modified on 29 Oct 2018

8 nov 2018
-- 17:00

Aula 5, DiMaI, Firenze

**Abstract.**

On a Riemannian manifold of dimension $n$, a system of coordinates $x_1,...,x_n$ is called orthogonal if the metric has the local expression $g = \sum _{i = 1} ^n a _i ^2 dx _i \otimes dx_i$. Such coordinates clearly exist if the metric is locally conformally flat, in particular if $n = 2$, and it has been shown by Dennis DeTurck and Dean Yang in 1984 that they always exist if $n = 3$. If $n > 3$, the opposite is expected in general, as the problem becomes overdetermined starting from $n = 4$, but it seems that very few has been known so far in this case. In this talk, I'll try to somewhat clarify the situation by providing some non-existence results for some classes of manifolds, in particulat for the complex projective spaces (joint work with Andrei Moroianu).