Geodesic
Do Barbero-Immirzi connections exist in different dimensions and signatures?
Communications in Mathematics, Tome 20 (2012) no. 1, pp. 3-11.

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We shall show that no reductive splitting of the spin group exists in dimension $3\le m\le 20$ other than in dimension $m=4$. In dimension $4$ there are reductive splittings in any signature. Euclidean and Lorentzian signatures are reviewed in particular and signature $(2,2)$ is investigated explicitly in detail. Reductive splittings allow to define a global $\mbox{SU} (2)$-connection over spacetime which encodes in an weird way the holonomy of the standard spin connection. The standard Barbero-Immirzi (BI) connection used in LQG is then obtained by restriction to a spacelike slice. This mechanism provides a good control on globality and covariance of BI connection showing that in dimension other than $4$ one needs to provide some other mechanism to define the analogous of BI connection and control its globality.
Classification : 53C07
Keywords: Barbero-Immirzi connection; global connections; Loop Quantum Gravity 
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     title = {Do {Barbero-Immirzi} connections exist in different dimensions and signatures?},
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Fatibene, L.; Francaviglia, M.; Garruto, S. Do Barbero-Immirzi connections exist in different dimensions and signatures?. Communications in Mathematics, Tome 20 (2012) no. 1, pp. 3-11. http://geodesic.mathdoc.fr/item/COMIM_2012__20_1_a1/