We present elementary derivations of the Gauss–Bonnet type and Bianchi type identities for Riemann–Cartan geometry. The identities are derived directly in terms of the vierbein field eaμ and the spin‐connection field 𝒱abμ, in the spirit of gauge theory, and are suitable for discussing Riemann–Cartan type gravitational theories.
REFERENCES
1.
2.
D. W. Sciama, in Recent Developments in General Relativity (Pergamon, New York, 1962).
A general review is given by
F. W.
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B. S. DeWitt, The Dynamical Theory of Groups and Fields (Gordon and Breach, New York, 1965);
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4.
A major portion of this section, which we include for self‐containment, can be found in T. W. B. Kibble, in Ref. 2.
5.
For the Dirac matrices, we follow the convention of J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw‐Hill, New York, 1964).
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The Gauss‐Bonnet identity (24) has been used, e.g., in
S. W.
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© 1980 American Institute of Physics.
1980
American Institute of Physics
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