Non-existence of orthogonal coordinates on the complex and quaternionic projective 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 $m,q\ge 2$.

10 pages; new version containing a missing term in formula (12) (without consequence for the rest of the paper) and some further references

References in corpus (2)

• Separability in Riemannian Manifolds
• Are Orthogonal Separable Coordinates Really Classified?