We consider corrections to Einstein gravity arising from the inclusion of Lorentz Chern-Simons terms, as well as Gauss-Bonnet terms, in a theory containing dilatons and Kalb-Ramond axions as in string theory. We show that all four-dimensional space-times with a maximally symmetric two-dimensional subspace have vanishing Lorentz Chern-Simons terms, which do not contribute to gravitational equations of motion or Bianchi identities. In addition, space-times conformally related to such have exact Lorentz Chern-Simons terms. This greatly simplifies the equations of motion for space-times of this form, such as Robertson-Walker and Schwarzschild. On the contrary, we show that for both the Kerr metric, and the Bianchi IX metric, Lorentz Chern-Simons terms cannot be neglected and lead to a non-trivial modification of the gravitational equations of motion, as well as the Bianchi identity for the Kalb-Ramond field strength, dH = tr R ∧ R ≠ 0.
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We wish to thank L. Jensen and Dr. Page for helpful comments on the manuscript. We would also like to thank D. Me Manus and J. Twamley for assistance with the computer algebra. The work of B.A. C. was supported in part by the Natural Sciences and Engineering Research Council of Canada. The work of M.D., N.K. and K.A.O. was supported in part by DOE grant DE-AC02-83ER-40105. In addition the work of K.A.O. was also supported in part by a Presidential Young Investigator Award.