Symmetries in Riemann-Cartan geometries (Q6614603)

A Riemann-Cartan geometry is a pseudo-Riemannian metric together with a compatible connection, perhaps with torsion. The authors consider spherically symmetric Riemann-Cartan geometries of Lorentz signature on \(4\)-dimensional manifolds. Among these, they determine explicitly (a) all those which have a \(7\)-dimensional symmetry group, and (b) all those which are static, and (c) all those which are stationary. Their method is the Cartan-Karlhede algorithm. This algorithm proceeds by describing the bundle of all frames on which the curvature and torsion and their covariant derivatives become normalized. This normalization occurs over open subsets of the Lorentz manifold on which the curvature and torsion and those covariant derivatives satisfy various possible nondegeneracy hypotheses. As one proceeds to higher covariant derivatives, these bundles have successively smaller structure groups, until they reach some minimum. At that point, the symmetries act transitively on the bundle.