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

DeTurck and Yang have shown that in the neighborhood of every point of a 3-dimensional Riemannian manifold, there exists a system of orthogonal coordinates (that is, with 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 CPm and the quaternionic projective spaces HPq, endowed with their canonical metrics, do not have local systems of orthogonal coordinates for m,q≥2.