Sectional curvature and general relativity
Seminar Room 1, Newton Institute
The geometrical idea of sectional curvature in space-times is introduced and interpreted. It is then shown that, with the exception of plane waves and spaces of constant curvature (and always for non-flat vacuum metrics),the sectional curvature function uniquely determines the space-time metric. Thus the suggestion is made that the sectional curvature function is a possible alternative variable for general relativity. Some of the properties of the sectional curvature function are then explored. These include (i) a certain critical point structure of this function and its relationship to the Petrov classification of the Weyl tensor and the Segre classification of the energy-momentum tensor,(ii) wave surfaces and null geodesic congruences,(iii) the concept of a sectional curvature-preserving vector field (iv) a generalisation of the Einstein space condition and a sectional curvature based concept of conformal flatness and (v) an alternative mathematical description of the sectional curvature function using quadric surfaces.