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Holonomy reductions of Cartan geometries and curved orbit decompositions. (English) Zbl 1298.53042

Let \(G\) be a Lie group and let \(P\subset G\) be a closed subgroup, with Lie algebras \(\mathfrak{g}\) and \(\mathfrak{p}\), respectively. A Cartan geometry of type \((G,P)\) on a manifold \(M\) is a \(P\)-principal bundle \(\mathcal{G}\to M\) endowed with a Cartan connection \(\omega\in\Omega^1(\mathcal{G},\mathfrak{g})\). The homogeneous model of Cartan geometries of type \((G,P)\) is the bundle \(G\to G/P\) with the left Maurer-Cartan form \(\omega^{MC}\in\Omega^1(G,\mathfrak g)\) as the Cartan connection.
The curvature \(K\in\Omega^2(\mathcal G,\mathfrak g)\) of \(\omega\) is defined by \(K(\xi,\eta)=d\omega(\xi,\eta)+[\omega(\xi),\omega(\eta)]\), which is exactly the failure of \(\omega\) to satisfy the Maurer-Cartan equation. A Cartan geometry is called torsion free if its curvature form \(K\) has values in \(\mathfrak p\subset\mathfrak g\).
A Cartan connection \(\omega\in\Omega^1(\mathcal{G},\mathfrak{g})\) is easily seen to extend canonically to a \(G\)-principal connection \(\hat\omega\in\Omega^1(\hat{\mathcal{G}},\mathfrak{g})\) on the \(G\)-principal bundle \(\hat{\mathcal G}:=\mathcal G\times_PG\). The extension is characterized by the fact that \(i^*\hat\omega=\omega\), where \(i:\mathcal G\to\hat{\mathcal G}\) is the inclusion.
Let \((\mathcal G,\omega)\) be a Cartan geometry of type \((G,P)\), and let \(\mathcal O\) be a homogeneous space of the group \(G\). Then a holonomy reduction of \(G\)-type \(\mathcal O\) of the geometry \((\mathcal G,\omega)\) is a parallel section of the bundle \(\mathcal G\times_P\mathcal O\). Let \((\mathcal G\to M,\omega)\) be a Cartan geometry of type \((G,P)\) together with a holonomy reduction of type \(\mathcal O\) described by \(s:\hat{\mathcal G}\to\mathcal O\). Then for a point \(x\in M\) the \(P\)-type of \(x\) with respect to \(s\) is the \(P\)-orbit \(s({\mathcal G}_x)\subset\mathcal O\). If \(P\backslash\mathcal O\) denotes the set of \(P\)-orbits in \(\mathcal O\), then it is easy to see that the base manifold \(M\) decomposes into a disjoint union of submanifolds \(M_i\), \(i\in P\backslash\mathcal O\), according to \(P\)-type as \(M=\cup_iM_i\); each component \(M_i\) is called a curved orbit.
Given a manifold equipped with a Cartan connection, the authors show that a holonomy reduction of this connection determines a decomposition of the underlying manifold into a disjoint union of initial submanifolds. Each such submanifold inherits a canonical geometry from the original data. More precisely, the main theorem of the paper is stated as follows:
Let \((\mathcal G,\omega)\) be a Cartan geometry of type \((G,P)\) which is endowed with a holonomy reduction of type \(\mathcal O\). Consider a \(P\)-orbit \(i\in P\backslash\mathcal O\) such that the curved orbit \(M_i\) is nonempty, and consider the corresponding groups \(P_i\subset H_i\) (where \((H_i,P_i)\) denotes an abstract representative of the associated isomorphism class of groups with a distinguished subgroup). Then we have the following:
(i) Choose a representative \(\alpha\in\mathcal O\) for the orbit \(i\), let \(G_\alpha\in G\) be its stabilizer, and consider the holonomy reduction of the homogeneous model \(G/P\) determined by \(\alpha\). Then, for each \(x\in M_i\), there exist neighborhoods \(N\) of \(x\) in \(M\), and \(N'\) of \(eP\) in \(G/P\), and a diffeomorphism \(\varphi:N\to N'\) with \(\varphi(x)=eP\) and \(\varphi(M_i\cap N)=(G_\alpha\cdot{}eP)\cap N'\). In particular, \(M_i\) is an initial submanifold of \(M\).
(ii) \(M_i\) carries a canonical Cartan geometry \((\mathcal G_i\to M_i,\omega_i)\) of type \((H_i,P_i)\). Choosing a representative \(\alpha\) for \(i\in P\backslash\mathcal O\) as in (i) and identifying \((H_i,P_i)\) with \((G_\alpha,P_\alpha)\), we obtain an embedding of principal bundles \(\mathbf{j}_\alpha:{\mathcal G}_i\to{\mathcal G}|_{M_i}\) such that \(\mathbf{j}_\alpha^*\omega=\omega_i\). Thus \((\mathcal G_i,\omega_i)\) can be realized as a \(P_\alpha\)-subbundle in \({\mathcal G}|_{M_i}\) on which \(\omega\) restricts to a Cartan connection of type \((G_\alpha,P_\alpha)\).
(iii) For the embedding \(\mathbf{j}_\alpha\) from (ii), the curvatures \(K\) of \(\omega\) and \(K_i\) of \(\omega_i\) are related as \(K_i=\mathbf{j}_\alpha^*K\). In particular, if \(\omega\) is torsion free, so is \(\omega_i\).
In the last section, the authors study several BGG (Bernstein-Gelfand-Gelfand) equations in projective, conformal and CR geometry.

MSC:

53C29 Issues of holonomy in differential geometry
53B15 Other connections
58J70 Invariance and symmetry properties for PDEs on manifolds
32V05 CR structures, CR operators, and generalizations
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