# Non-existence of orthogonal coordinates on the complex and quaternionic projective spaces

## TL;DR

This paper proves that higher-dimensional complex and quaternionic projective spaces with their canonical metrics do not admit local orthogonal coordinate systems, unlike 3-dimensional Riemannian manifolds.

## Contribution

It establishes the non-existence of orthogonal coordinates on complex and quaternionic projective spaces for dimensions two and above.

## Key findings

- Orthogonal coordinates exist locally in 3D Riemannian manifolds.
- Higher-dimensional complex and quaternionic projective spaces lack such coordinates.
- This non-existence is proven for canonical metrics on these spaces.

## Abstract

DeTurck and Yang have shown that in the neighbourhood of every point of a $3$-dimensional Riemannian manifold, there exists a system of orthogonal coordinates (that is, whith respect to which the metric has diagonal form). We show that this property does not generalize to higher dimensions. In particular, the complex projective spaces $\mathbb{CP}^m$ and the quaternionic projective spaces $\mathbb{HP}^q$, endowed with their canonical metrics, do not have local systems of orthogonal coordinates for $m,q\ge 2$.