## 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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### References:

 [1] W. Ambrose and I. M. Singer, A theorem on holonomy , Trans. Amer. Math. Soc. 75 (1953), 428-443. · Zbl 0052.18002 [2] V. Apostolov, D. M. J. Calderbank, and P. Gauduchon, Hamiltonian 2-forms in Kähler geometry, I: General theory , J. Differential Geom. 73 (2006), 359-412. · Zbl 1101.53041 [3] S. Armstrong, Definite signature conformal holonomy: A complete classification , J. Geom. Phys. 57 (2007), 2024-2048. · Zbl 1407.53050 [4] S. Armstrong, Projective holonomy, I: Principles and properties , Ann. Global Anal. Geom. 33 (2008), 47-69. · Zbl 1147.53037 [5] S. Armstrong, Projective holonomy, II: Cones and complete classifications , Ann. Global Anal. Geom. 33 (2008), 137-160. · Zbl 1147.53038 [6] S. Armstrong, Einstein connections and involutions via parabolic geometries , J. Geom. Phys. 60 (2010), 1424-1440. · Zbl 1194.53022 [7] S. Armstrong and F. Leitner, Decomposable conformal holonomy in Riemannian signature , Math. Nachr. 285 (2012), 150-163. · Zbl 1246.53064 [8] H. Baum, Lorentzian twistor spinors and CR-geometry , Differential Geom. Appl. 11 (1999), 69-96. · Zbl 0930.53033 [9] H. Baum, T. Friedrich, R. Grunewald, and I. Kath, Twistor and Killing Spinors on Riemannian Manifolds , Seminarberichte 108 , Humboldt Universität Sektion Mathematik, Berlin, 1990. · Zbl 0705.53004 [10] M. Berger, Sur les groupes d’holonomie des variétés riemanniennes non symétriques , C. R. Acad. Sci. Paris 237 (1953), 1306-1308. · Zbl 0052.17504 [11] R. L. Bryant, Metrics with exceptional holonomy , Ann. of Math. (2) 126 (1987), 525-576. · Zbl 0637.53042 [12] R. L. Bryant, Recent advances in the theory of holonomy , Astérisque 266 (2000), 351-374, Séminaire Bourbaki 1998/1999, no. 861. · Zbl 1014.53029 [13] R. L. Bryant, “Conformal geometry and 3-plane fields on 6-manifolds” in Developments of Cartan Geometry and Related Mathematical Problems , RIMS Symposium Proceedings (Kyoto University) 1502 (2006), 1-15. [14] D. M. J. Calderbank and T. Diemer, Differential invariants and curved Bernstein-Gelfand-Gelfand sequences , J. Reine Angew. Math. 537 (2001), 67-103. · Zbl 0985.58002 [15] A. Čap, Correspondence spaces and twistor spaces for parabolic geometries , J. Reine Angew. Math. 582 (2005), 143-172. · Zbl 1075.53022 [16] A. Čap, Infinitesimal automorphisms and deformations of parabolic geometries , J. Eur. Math. Soc. (JEMS) 10 (2008), 415-437. · Zbl 1161.32020 [17] A. Čap and A. R. Gover, CR-tractors and the Fefferman space , Indiana Univ. Math. J. 57 (2008), 2519-2570. · Zbl 1162.32019 [18] A. Čap and A. R. Gover, A holonomy characterisation of Fefferman spaces , Ann. Global Anal. Geom. 38 (2010), 399-412. · Zbl 1298.53041 [19] A. Čap, A. R. Gover, and M. Hammerl, Projective BGG equations, algebraic sets, and compactifications of Einstein geometries , J. Lond. Math. Soc. (2) 86 (2012), 433-454. · Zbl 1258.53016 [20] A. Čap and J. Slovák, Parabolic Geometries, I: Background and General Theory . Math. Surveys Monogr., Amer. Math. Soc., Providence, 2009. [21] A. Čap, J. Slovák, and V. Souček, Bernstein-Gelfand-Gelfand sequences , Ann. of Math. (2) 154 (2001), 97-113. · Zbl 1159.58309 [22] É. Cartan, Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre , Ann. Sci. École Norm. Sup. (3) 27 (1910), 109-192. · JFM 41.0417.01 [23] E. Cartan, Sur une classe remarquable d’espaces de Riemann , Bull. Soc. Math. France 54 (1926), 214-264. · JFM 52.0425.01 [24] M. Eastwood, “Notes on projective differential geometry” in Symmetries and Overdetermined Systems of Partial Differential Equations , IMA Vol. Math. Appl. 144 , Springer, New York, 2008, 41-60. · Zbl 1186.53020 [25] C. Fefferman, Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains , Ann. of Math. (2) 103 (1976), 395-416. · Zbl 0322.32012 [26] C. Fefferman and C. R. Graham, “Conformal invariants” in The Mathematical Heritage of Élie Cartan (Lyon, 1984) , Astérisque 1985 , Numero Hors Serie, Soc. Math. France, Paris, 95-116. · Zbl 0602.53007 [27] C. Fefferman and C. R. Graham, The Ambient Metric , Ann. of Math. Stud. 178 , Princeton Univ. Press, Princeton, 2012. · Zbl 1243.53004 [28] T. Friedrich, “On the conformal relation between twistors and Killing spinors” in Proceedings of the Winter School on Geometry and Physics (Srní, 1989) , Rend. Circ. Mat. Palermo (2) Suppl. 22 , Circ. Mat. Palermo, Palermo, 1990, 59-75. · Zbl 0703.53012 [29] A. R. Gover, “Almost conformally Einstein manifolds and obstructions” in Differential Geometry and Its Applications , Matfyzpress, Prague, 2005, 247-260. · Zbl 1121.53032 [30] A. R. Gover, Conformal Dirichlet-Neumann Maps and Poincaré-Einstein Manifolds , SIGMA Symmetry Integrability Geom. Methods Appl. 3 , Natl. Acad. Sci. Ukr., Inst. Math., Dep. Appl. Res., Kiev, 2007. [31] A. R. Gover, Almost Einstein and Poincaré-Einstein manifolds in Riemannian signature , J. Geom. Phys. 60 (2010), 182-204. · Zbl 1194.53038 [32] A. R. Gover and F. Leitner, A class of compact Poincare-Einstein manifolds: Properties and construction , Commun. Contemp. Math. 12 (2010), 629-659. · Zbl 1201.53054 [33] C. R. Graham, On Sparling’s characterization of Fefferman metrics , Amer. J. Math. 109 (1987), 853-874. · Zbl 0663.53050 [34] C. R. Graham and J. M. Lee, Einstein metrics with prescribed conformal infinity on the ball , Adv. Math. 87 (1991), 186-225. · Zbl 0765.53034 [35] C. Guillarmou and J. Qing, Spectral characterization of Poincaré-Einstein manifolds with infinity of positive Yamabe type , Int. Math. Res. Not. IMRN 9 (2010), 1720-1740. · Zbl 1191.53030 [36] K. Habermann, Twistor spinors and their zeroes , J. Geom. Phys. 14 (1994), 1-24. · Zbl 0807.53037 [37] M. Hammerl and K. Sagerschnig, Conformal Structures Associated to Generic Rank 2 Distributions on 5-manifolds-Characterization and Killing-field Decomposition , SIGMA Symmetry Integrability Geom. Methods Appl. 5 , Natl. Acad. Sci. Ukr., Inst. Math., Dep. Appl. Res., Kiev, 2009. · Zbl 1191.53016 [38] M. Hammerl and K. Sagerschnig, The twistor spinors of generic 2- and 3-distributions , Ann. Global Anal. Geom. 39 (2011), 403-425. · Zbl 1229.53058 [39] I. Kolář, P. W. Michor, and J. Slovák, Natural Operations in Differential Geometry , Springer, Berlin, 1993. · Zbl 0782.53013 [40] W. Kühnel and H.-B. Rademacher, Twistor spinors with zeros , Internat. J. Math. 5 (1994), 877-895. · Zbl 0818.53054 [41] J. M. Lee, The Fefferman metric and pseudo-Hermitian invariants , Trans. Amer. Math. Soc. 296 (1986), 411-429. · Zbl 0595.32026 [42] J. M. Lee, The spectrum of an asymptotically hyperbolic Einstein manifold , Comm. Anal. Geom. 3 (1995), 253-271. · Zbl 0934.58029 [43] T. Leistner and P. Nurowski, Ambient metrics with exceptional holonomy , Ann. Sc. Norm. Super. Pisa Cl. Sci. 11 (2012), 407-436. · Zbl 1255.53018 [44] F. Leitner, Conformal Killing forms with normalisation condition , Rend. Circ. Mat. Palermo (2) Suppl. 75 (2005), 279-292. · Zbl 1101.53040 [45] F. Leitner, On transversally symmetric pseudo-Einstein and Fefferman-Einstein spaces , Math. Z. 256 (2007), 443-459. · Zbl 1147.32021 [46] F. Leitner, “A remark on unitary conformal holonomy” in Symmetries and Overdetermined Systems of Partial Differential Equations , IMA Vol. Math. Appl. 144 , Springer, New York, 2008, 445-460. · Zbl 1142.53040 [47] F. Leitner, About Twistor Spinors with Zero in Lorentzian Geometry , SIGMA Symmetry Integrability Geom. Methods Appl. 5 , Natl. Acad. Sci. Ukr., Inst. Math., Dep. Appl. Res., Kiev, 2009. · Zbl 1189.53044 [48] F. Leitner, The collapsing sphere product of Poincaré-Einstein spaces , J. Geom. Phys. 60 (2010), 1558-1575. · Zbl 1197.53058 [49] F. Leitner, Multiple almost Einstein structures with intersecting scale singularities , Monatsh. Math. 165 (2012), 15-39. · Zbl 1244.53054 [50] V. Matveev and S. Rosemann, Proof of the Yano-Obata conjecture for h-projective transformations , J. Differential Geom. 92 (2012), 221-261. · Zbl 1277.53073 [51] S. Merkulov and L. Schwachhöfer, Classification of irreducible holonomies of torsion-free affine connections , Ann. of Math. (2) 150 (1999), 77-149. · Zbl 0992.53038 [52] P. Nurowski, Differential equations and conformal structures , J. Geom. Phys. 55 (2005), 19-49. · Zbl 1082.53024 [53] P. Nurowski and G. A. Sparling, Three-dimensional Cauchy-Riemann structures and second-order ordinary differential equations , Classical Quantum Gravity 20 (2003), 4995-5016. · Zbl 1051.32019 [54] S. Sasaki, On the spaces with normal conformal connexions whose groups of holonomy fix a point or a hypersphere, II , Jap. J. Math. 18 (1943), 623-633. · Zbl 0060.38905 [55] R. W. Sharpe, Differential Geometry-Cartan’s Generalisation of Klein’s Erlangen Program , Springer, New York, 1997. · Zbl 0876.53001
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