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Sectional curvature and general relativity

Presented by: 
G Hall [Aberdeen]
Friday 7th October 2005 - 16:00 to 17:00
INI Seminar Room 1

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.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons