1 April 2014 Holonomy reductions of Cartan geometries and curved orbit decompositions
A. Čap, A. R. Gover, M. Hammerl
Duke Math. J. 163(5): 1035-1070 (1 April 2014). DOI: 10.1215/00127094-2644793


We develop a holonomy reduction procedure for general Cartan geometries. We show that, given a reduction of holonomy, the underlying manifold naturally decomposes into a disjoint union of initial submanifolds. Each such submanifold corresponds to an orbit of the holonomy group on the modeling homogeneous space and carries a canonical induced Cartan geometry. The result can therefore be understood as a “curved orbit decomposition.” The theory is then applied to the study of several invariant overdetermined differential equations in projective, conformal, and CR geometry. This makes use of an equivalent description of solutions to these equations as parallel sections of a tractor bundle. In projective geometry we study a third-order differential equation that governs the existence of a compatible Einstein metric, and in conformal geometry we discuss almost-Einstein scales. Finally, we discuss analogues of the two latter equations in CR geometry, which leads to invariant equations that govern the existence of a compatible Kähler–Einstein metric.


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A. Čap. A. R. Gover. M. Hammerl. "Holonomy reductions of Cartan geometries and curved orbit decompositions." Duke Math. J. 163 (5) 1035 - 1070, 1 April 2014. https://doi.org/10.1215/00127094-2644793


Published: 1 April 2014
First available in Project Euclid: 26 March 2014

zbMATH: 1298.53042
MathSciNet: MR3189437
Digital Object Identifier: 10.1215/00127094-2644793

Primary: 53C29
Secondary: 32V05 , 53A20 , 53A30 , 53B15 , 58D19 , 58J70

Rights: Copyright © 2014 Duke University Press

Vol.163 • No. 5 • 1 April 2014
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