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 CP m and the quaternionic projective spaces HP q , endowed with their canonical metrics, do not have local systems of orthogonal coordinates for m, q ≥ 2.

Spacetimes with all Penrose limits diagonalisable*

“…Next for type (2, 1, 1) we can dispose of the vacuum case by the same argument as for type (3,1) here: by ( 7) any PND of the Weyl spinor is gsf but in vacuum only repeated ones should be, so there are no vacuum solutions of this type, but there could be nonvacuum solutions. It will follow from the argument in the algebraically general case that, since the scalar invariant I is nonzero in this case the Weyl spinor is proportional to a Killing spinor with a real function of proportionality.…”

“…Call this process diagonalisation, then a metric in four or more dimensions cannot always be diagonalised. The problem was considered in [17] where some non-diagonalisable 4dimensional Lorentzian metrics were given, and has recently been considered in [3] where some Riemannian metrics in dimension 4 or more are shown to be nondiagonalisable.…”

“…Call this process diagonalisation, then a metric in four or more dimensions cannot always be diagonalised. The problem was considered in [23] where some nondiagonalisable 4-dimensional (4D) Lorentzian metrics were given, and has recently been considered in [4] where some Riemannian metrics in dimension 4 or more are shown to be non-diagonalisable. In general it is still a difficult problem to decide whether a given 4-metric Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.…”