Riemann compatible tensors

Volume 128 / 2012

Carlo Alberto Mantica, Luca Guido Molinari Colloquium Mathematicum 128 (2012), 197-210 MSC: Primary 53B20, Secondary 53B21. DOI: 10.4064/cm128-2-5


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.


  • Carlo Alberto ManticaI.I.S. Lagrange
    Via L. Modignani 65
    20161, Milano, Italy
  • Luca Guido MolinariPhysics Department
    Università degli Studi di Milano
    I.N.F.N. sezione di Milano
    Via Celoria 16
    20133 Milano, Italy

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