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Parabolic geometries and canonical Cartan connections. (English) Zbl 0996.53023
Author’s abstract: Let \(G\) be a (real or complex) semisimple Lie group, whose Lie algebra \(\mathfrak g\) is endowed with a so called \(|k|\)-grading, i.e. a grading of the form \(\mathfrak g=\mathfrak g_{-k}\oplus\cdots\oplus\mathfrak g_k\), such that no simple factor of \(G\) is of type \(A_1\). Let \(P\) be the subgroup corresponding to the subalgebra \(\mathfrak p=\mathfrak p_0\oplus\cdots\oplus\mathfrak p_k\). The aim of this paper is to clarify the geometrical meaning of Cartan connections corresponding to the pair \((G,P)\) and to study basic properties of these geometric structures.
Let \(G_0\) be the (reductive) subgroup of \(P\) corresponding to the subalgebra \(\mathfrak g_0\). A principal \(P\)-bundle \(E\) over a smooth manifold \(M\) endowed with a (suitably normalized) Cartan connection \(\omega\in \Omega^1(E,\mathfrak g)\) automatically gives rise to a filtration of the tangent bundle \(TM\) of \(M\) and to a reduction to the structure group \(G_0\) of the associated graded vector bundle to the  filtered vector bundle \(TM\).
We prove that in almost all cases the principal \(P\) bundle together with the Cartan connection is already uniquely determined by this underlying structure (which can be  easily understood geometrically), while in the remaining cases one has to make an additional choice (which again can be easily interpreted geometrically) to determine the bundle and the Cartan connection.
The canonical Cartan connections were constructed long time ago in some particular cases. For CR-structures it was done by E. Cartan [“”Notice sur les traveaux scientifique”, Œuvres Complètes, Partie I, Vol. I 72-85 (1979; Zbl 0049.30302)], for hypersurfaces in \(\mathbb C^2\) by N. Tanaka [Hokkaido Math. J. 8, 23-84 (1952; Zbl 0409.17013)], and by S. S. Chern and J. K. Moser [Acta Math. 133, 219-271 (1974; Zbl 0302.32015)] for arbitrary CR-manifolds. As an application of the results obtained, the existence of a canonical Cartan connection for the significantly more general class of partially integrable almost-CR-manifolds is shown.

MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
32V05 CR structures, CR operators, and generalizations
53A40 Other special differential geometries
53A55 Differential invariants (local theory), geometric objects
53B15 Other connections
53C05 Connections (general theory)
58A30 Vector distributions (subbundles of the tangent bundles)
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