On Metrizability of Invariant Affine Connections

The metrizability problem for a symmetric affine connection on a manifold, invariant with respect to a group of diffeomorphisms G, is considered. We say that the connection is G-metrizable, if it is expressible as the Levi-Civita connection of a G-invariant metric field. In this paper we analyze the G-metrizability equations for the rotation group G = SO(3), acting canonically on three- and four-dimensional Euclidean spaces. We show that the property of the connection to be SO(3)-invariant allows us to find complete explicit description of all solutions of the SO(3)-metrizability equations.Comment: 17 pages, To appear in IJGMMP vol.9 No.1 (2012

Spinors in Higher Dimensional and Locally Anisotropic Spaces

The theory of spinors is developed for locally anisotropic (la) spaces, in brief la-spaces, which in general are modeled as vector bundles provided with nonlinear and distinguished connections and metric structures (such la-spaces contain as particular cases the Lagrange, Finsler and, for trivial nonlinear connections, Kaluza-Klein spaces). The la-spinor differential geometry is constructed. The distinguished spinor connections are studied and compared with similar ones on la-spaces. We derive the la-spinor expressions of curvatures and torsions and analyze the conditions when the distinguished torsion and nonmetricity tensors can be generated from distinguished spinor connections. The dynamical equations for gravitational and matter field la-interactions are formulated.Comment: 54 pages, Revtex, an extension of the paper published in J. Math. Phys. 37 (1996), 508--52

The metrizability problem for Lorentz-invariant affine connections

The invariant metrizability problem for affine connections on a manifold, formulated by Tanaka and Krupka for connected Lie groups actions, is considered in the particular cases of Lorentz and Poincaré (inhomogeneous Lorentz) groups. Conditions under which an affine connection on the open submanifold R×(R 3 \{(0,0,0)}) of the Euclidean space R 4 coincides with the Levi-Civita connection of some SO(3,1), respectively (R 4 × s SO(3,1)) -invariant metric field are studied. We give complete description of metrizable Lorentz-invariant connections. Explicit solutions (metric fields) of the invariant metrizability equations are found and their properties are discussed. Read More: http://www.worldscientific.com/doi/abs/10.1142/S0219887816501103Web of Science138art. no. 165011