Curvature and the Petrov Canonical Forms

Source abstract

The Petrov classification for the curvature tensor of an Einstein space M4 is related to the critical-point theory of the sectional-curvature function σ, regarded as a function on the manifold of nondegenerate tangent 2-planes at each point of the space. It is shown that the Petrov type is determined by the number of critical points. Furthermore, all the invariants in the canonical form can be computed from a knowledge of the critical value and the Hessian quadratic form of σ at any single critical point.