Riemann compatible tensors

Carlo Alberto Mantica¹ ; Luca Guido Molinari²

¹ I.I.S. Lagrange Via L. Modignani 65 20161, Milano, Italy
² Physics Department Università degli Studi di Milano and I.N.F.N. sezione di Milano Via Celoria 16 20133 Milano, Italy

Colloquium Mathematicum, Tome 128 (2012) no. 2, pp. 197-210

Résumé

Derdziński and Shen's theorem on the restrictions on the Riemann tensor imposed by existence of a Codazzi tensor holds more generally when a Riemann compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity for a new “Codazzi deviation tensor”, with a geometric significance. The above general properties are studied, with their implications on Pontryagin forms. Examples are given of manifolds with Riemann compatible tensors, in particular those generated by geodesic mappings. Compatibility is extended to generalized curvature tensors, with an application to Weyl's tensor and general relativity.

Détail

DOI : 10.4064/cm128-2-5

Mots-clés : derdzi ski shens theorem restrictions riemann tensor imposed existence codazzi tensor holds generally riemann compatible tensor exists several properties shown remain valid broader setting riemann compatibility equivalent bianchi identity codazzi deviation tensor geometric significance above general properties studied their implications pontryagin forms examples given manifolds riemann compatible tensors particular those generated geodesic mappings compatibility extended generalized curvature tensors application weyls tensor general relativity