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.


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
Full Text: DOI arXiv Euclid


[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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