Homogeneous para-Kähler Einstein manifolds

D. V. Alekseevsky ; C. Medori ; A. Tomassini

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 64 (2009) no. 1, pp. 1-43

Résumé

A para-Kähler manifold can be defined as a pseudo-Riemannian manifold $(M,g)$ with a parallel skew-symmetric para-complex structure $K$, that is, a parallel field of skew-symmetric endomorphisms with $K^2=\operatorname{Id}$ or, equivalently, as a symplectic manifold $(M,\omega)$ with a bi-Lagrangian structure $L^\pm$, that is, two complementary integrable Lagrangian distributions. A homogeneous manifold $M = G/H$ of a semisimple Lie group $G$ admits an invariant para-Kähler structure $(g,K)$ if and only if it is a covering of the adjoint orbit $\operatorname{Ad}_Gh$ of a semisimple element $h$. A description is given of all invariant para-Kähler structures $(g,K)$ on such a homogeneous manifold. With the use of a para-complex analogue of basic formulae of Kähler geometry it is proved that any invariant para-complex structure $K$ on $M=G/H$ defines a unique para-Kähler Einstein structure $(g,K)$ with given non-zero scalar curvature. An explicit formula for the Einstein metric $g$ is given. A survey of recent results on para-complex geometry is included.

Bibliographie

[1] V. Cortés, Ch. Mayer, T. Mohaupt, F. Saueressig, “Special geometry of Euclidean supersymmetry. I. Vector multiplets”, J. High Energy Phys., 2004, no. 3, 028, 73 pp. | DOI | MR

[2] Z. Hou, Sh. Deng, S. Kaneyuki, “Dipolarizations in compact Lie algebras and homogeneous para-Kähler manifolds”, Tokyo J. Math., 20:2 (1997), 381–388 | MR | Zbl

[3] Z. Hou, S. Deng, S. Kaneyuki, K. Nishiyama, “Dipolarizations in semisimple Lie algebras and homogeneous para-Kähler manifolds”, J. Lie Theory, 9:1 (1999), 215–232 | MR | Zbl

[4] P. K. Rashevskii, “Skalyarnye polya v rassloennom prostranstve”, Trudy sem. po vekt. tenzor. analizu, 6, 1948, 225–248 | MR

[5] P. Libermann, “Sur les structures presque paracomplexes”, C. R. Acad. Sci. Paris, 234 (1952), 2517–2519 | MR | Zbl

[6] P. Libermann, “Sur le problème d'équivalence de certaines structures infinitésimales”, Ann. Mat. Pura Appl. (4), 36:1 (1954), 27–120 | DOI | MR | Zbl

[7] V. Cruceanu, P. Fortuny, P. M. Gadea, “A survey on paracomplex geometry”, Rocky Mountain J. Math., 26:1 (1996), 83–115 | DOI | MR | Zbl

[8] V. Cruceanu, P. M. Gadea, J. Munoz Masque, “Para-Hermitian and para-Kähler manifolds”, Quaderni Istit. Mat. Univ. Messina, 1 (1995), 1–72

[9] F. Etayo, R. Santamaría, U. R. Tríias, “The geometry of a bi-Lagrangian manifold”, Differential Geom. Appl., 24:1 (2006), 33–59 | DOI | MR | Zbl

[10] V. F. Kirichenko, “Methods of generalized Hermitian geometry in the theory of almost contact manifolds”, J. Soviet Math., 42:5 (1988), 1885–1919 | DOI | MR | Zbl

[11] M.-A. Lawn, L. Schäfer, “Decompositions of para-complex vector bundles and para-complex affine immersions”, Results Math., 48:3–4 (2005), 246–274 | MR | Zbl

[12] L. Schäfer, “$tt^*$-bundles in para-complex geometry, special para-Kähler manifolds and para-pluriharmonic maps”, Differential Geom. Appl., 24:1 (2006), 60–89 | DOI | MR | Zbl

[13] S. Cecotti, C. Vafa, “Topological–anti-topological fusion”, Nuclear Phys. B, 367:2 (1991), 359–461 | DOI | MR | Zbl

[14] B. Dubrovin, “Geometry and integrability of topological-antitopological fusion”, Comm. Math. Phys., 152:3 (1993), 539–564 | DOI | MR | Zbl

[15] L. Schäfer, “Para-$tt^*$-bundles on the tangent bundle of an almost para-complex manifold”, Ann. Global Anal. Geom., 32:2 (2007), 125–145 | DOI | MR | Zbl

[16] L. Schäfer, “Harmonic bundle solutions of topological-antitopological fusion in para-complex geometry”, Differential Geom. Appl., 26:1 (2008), 97–105 | DOI | MR | Zbl

[17] A. Wade, “Dirac structures and paracomplex manifolds”, C. R. Math. Acad. Sci. Paris, 338:11 (2004), 889–894 | MR | Zbl

[18] I. Vaisman, “Reduction and submanifolds of generalized complex manifolds”, Differential Geom. Appl., 25:2 (2007), 147–166 | DOI | MR | Zbl

[19] M. Gualtieri, Generalized complex geometry, Ph. D. thesis, University of Oxford, 2003; arXiv: math/0401221

[20] N. Hitchin, “Generalized Calabi–Yau manifolds”, Quarterly J. Math., 54:3 (2003), 281–308 | DOI | MR | Zbl

[21] O. Ben-Bassat, “Mirror symmetry and generalized complex manifolds. Part I: The transform on vector bundles, spinors, and branes”, J. Geom. Phys., 56:4 (2006), 533–558 | DOI | MR | Zbl

[22] A. Andrada, “Complex product structures and affine foliations”, Ann. Global Anal. Geom., 27:4 (2005), 377–405 | DOI | MR | Zbl

[23] A. Andrada, “Hypersymplectic Lie algebras”, J. Geom. Phys., 56:10 (2006), 2039–2067 | DOI | MR | Zbl

[24] A. Andrada, M. L. Barberis, I. G. Dotti, G. P. Ovando, “Product structures on four dimensional solvable Lie algebras”, Homology Homotopy Appl., 7:1 (2005), 9–37 | MR | Zbl

[25] A. Andrada, I. G. Dotti, “Double products and hypersymplectic structures on $\mathbb{R}^{4n}$”, Comm. Math. Phys., 262:1 (2006), 1–16 | DOI | MR | Zbl

[26] A. Andrada, S. Salamon, “Complex product structures on Lie algebras”, Forum Math., 17:2 (2005), 261–295 | DOI | MR | Zbl

[27] N. Blažić, S. Vukmirović, “Para-hypercomplex structures on a four-dimensional Lie group”, Contemporary geometry and related topics, World Sci. Publ., River Edge, NJ, 2004, 41–56 | MR | Zbl

[28] S. Marchiafava, P. Nagy, (Anti-)hypercomplex structures and 3-webs on a manifold, Preprint 38/03, Università degli Studi di Roma “La Sapienza”, 2003

[29] M. A. Akivis, A. M. Shelekhov, Geometry and algebra of multidimensional three-webs, Math. Appl. (Soviet Ser.), 82, Kluwer Acad. Publ., Dordrecht, 1992 | MR | Zbl

[30] V. F. Kirichenko, V. V. Konnov, “Almost Kählerian manifolds of hyperbolic type”, Izv. Math., 67:4 (2003), 655–694 | DOI | MR | Zbl

[31] O. E. Arsen'eva, V. F. Kirichenko, “Self-dual geometry of generalized Hermitian surfaces”, Sb. Math., 189:1 (1998), 19–41 | DOI | MR | Zbl

[32] P. M. Gadea, J. M. Masque, “Classification of almost para-Hermitian manifolds”, Rend. Mat. Appl. (7), 11:2 (1991), 377–396 | MR | Zbl

[33] S. Ianus, G. E. Vîlcu, Some constructions of almost para-hyperhermitian structures on manifolds and tangent bundles, , 2007 arXiv: 0707.3360v1

[34] V. Oproiu, N. Papaghiuc, G. Mitric, “Some classes of para-Hermitian structures on cotangent bundles”, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), 43:1 (1997), 7–22 | MR | Zbl

[35] A. Bonome, R. Castro, E. García-Río, L. Hervella, R. Vázquez-Lorenzo, “On the paraholomorphic sectional curvature of almost para-Hermitian manifolds”, Houston J. Math., 24:2 (1998), 277–300 | MR | Zbl

[36] P. M. Gadea, J. A. Oubiña, “Homogeneous almost para-Hermitian structures”, Indian J. Pure Appl. Math., 26:4 (1995), 351–362 | MR | Zbl

[37] Z. Olszak, “On para-Hermitian structures on twisted products on Lie groups”, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 43(91):3–4 (2000), 313–323 | MR | Zbl

[38] Z. Olszak, “On 4-dimensional conformally flat left invariant para-Hermitian structures”, Geometry and topology manifolds (Krynica, 1999), Univ. Iagel. Acta Math., 38, Jagiellonian Univ. Press, Krakow, 2000, 29–40 | MR | Zbl

[39] Z. Olszak, “Left invariant para-Hermitian structures on semidirect products of Lie groups”, Tensor (N.S.), 62:1 (2000), 1–11 | MR | Zbl

[40] Z. Olszak, “On natural para-Hermitian structures on twisted products of Lie groups”, PDEs, submanifolds and affine differential geometry (Warsaw, 2000), Banach Center Publ., 57, Polish Acad. Sci., Warsaw, 2002, 203–209 | MR | Zbl

[41] F. Etayo, M. Fioravanti, “CR-submanifolds of the paracomplex projective space”, Publ. Math. Debrecen, 47:1–2 (1995), 151–160 | MR | Zbl

[42] F. Etayo, M. Fioravanti, U. R. Trías, “On the submanifolds of an almost para-Hermitian manifold”, Acta Math. Hungar., 85:4 (1999), 277–286 | DOI | MR | Zbl

[43] Y. Matsushita, “On Euler characteristics and Hirzebruch indices of four-dimensional almost para-Hermitian manifolds”, JP J. Geom. Topol., 5:2 (2005), 115–120 | MR | Zbl

[44] C.-L. Bejan, M. Benyounes, “Harmonic maps between almost para-Hermitian manifolds”, New developments in differential geometry (Budapest, 1996), Kluwer Acad. Publ., Dordrecht, 1999, 67–97 | MR | Zbl

[45] J. J. Konderak, “A symplectic reduction for pseudo-Riemannian manifolds with compatible almost product structures”, Beiträge Algebra Geom., 45:2 (2004), 465–479 | MR | Zbl

[46] A. Weinstein, “Symplectic manifolds and their Lagrangian submanifolds”, Adv. Math., 6:3 (1971), 329–346 | DOI | MR | Zbl

[47] I. Vaisman, “Basics of Lagrangian foliations”, Publ. Mat., 33:3 (1989), 559–575 | MR | Zbl

[48] J.-M. Morvan, “Quelques invariants topologiques en géométrie symplectique”, Ann. Inst. H. Poincaré Sect. A (N.S.), 38:4 (1983), 349–370 | MR | Zbl

[49] J. F. Cariñena, L. A. Ibort, “Bilagrangian connections and the reduction of Lax equations”, Lett. Math. Phys., 8:5 (1984), 359–365 | DOI | MR | Zbl

[50] S. Ivanov, S. Zamkovoy, “Parahermitian and paraquaternionic manifolds”, Differential Geom. Appl., 23:2 (2005), 205–234 | DOI | MR | Zbl

[51] E. García-Río, Y. Matsushita, R. Vázquez-Lorenzo, “Paraquaternionic Kähler manifolds”, Rocky Mountain J. Math., 31:1 (2001), 237–260 | DOI | MR | Zbl

[52] R. L. Bryant, “Bochner–Kähler metrics”, J. Amer. Math. Soc., 14:3 (2001), 623–715 | DOI | MR | Zbl

[53] S. Tabachnikov, “Geometry of Lagrangian and Legendrian 2-web”, Differential Geom. Appl., 3:3 (1993), 265–284 | DOI | MR | Zbl

[54] Th. Krantz, Holonomy representations which are a diagonal direct sum of two faithful representations, , 2008 arxiv: 0704.2776v3

[55] P. M. Gadea, A. Montesinos, “The paracomplex projective spaces as symmetric and natural spaces”, Indian J. Pure Appl. Math., 23:4 (1992), 261–275 | MR | Zbl

[56] G. Santos-Garcia, P. M. Gadea, “A Weitzenböck formula for the Laplacian of para-Kähler space forms”, Rend. Sem. Mat. Messina Ser. II, 2(16) (1993), 81–89 | MR | Zbl

[57] S. Erdem, “Paracomplex projective models and harmonic maps into them”, Beiträge Algebra Geom., 40:2 (1999), 385–398 | MR | Zbl

[58] A. Al-Aqeel, A. Bejancu, “On the geometry of paracomplex submanifolds”, Demonstratio Math., 34:4 (2001), 919–932 | MR | Zbl

[59] R. Rosca, “CR-sous-variétés co-isotropes d'une variété parakählérienne”, C. R. Acad. Sci. Paris Sér. I Math., 298:7 (1984), 149–151 | MR | Zbl

[60] S. Donato, “CICR submanifolds of a para-Kähler manifold having the self-orthogonal Killing property”, Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur., 66 (1988), 283–292 | MR | Zbl

[61] G. B. Rizza, “Some remarks on para-Kähler manifolds”, Proceedings of the Second International Workshop on Differential Geometry and its Applications (Constanţa, 1995), An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat., 3, no. 2, 1995, 113–120 | MR | Zbl

[62] S. Ianus, G. B. Rizza, “Some submanifolds of a para-Kähler manifold”, Rend. Circ. Mat. Palermo (2), 47:1 (1998), 71–80 | DOI | MR | Zbl

[63] D. Alekseevsky, “Pseudo-Kähler and para-Kähler symmetric spaces”, Handbook of Pseudo-Riemannian Geometry and Supersymmetry, IRMA Lect. Math. Theor. Phys., eds. V. Cortes, V. Batyrev, Eur. Math. Soc., Zurich (to appear)

[64] S. Kaneyuki, “On classification of para-Hermitian symmetric spaces”, Tokyo J. Math., 8:2 (1985), 473–482 | MR | Zbl

[65] S. Kaneyuki, “Compactification of parahermitian symmetric spaces and its applications. II: Stratifications and automorphism groups”, J. Lie Theory, 13:2 (2003), 535–563 | MR | Zbl

[66] S. Kaneyuki, M. Kozai, “Paracomplex structures and affine symmetric spaces”, Tokyo J. Math., 8:1 (1985), 81–98 | MR | Zbl

[67] W. Bertram, The geometry of Jordan and Lie structures, Lecture Notes in Math., 1754, Springer-Verlag, Berlin, 2000 | DOI | MR | Zbl

[68] F. Defever, R. Deszcz, L. Verstraelen, “On pseudosymmetric para-Kähler manifolds”, Colloq. Math., 74:2 (1997), 253–260 | MR | Zbl

[69] F. Defever, R. Deszcz, L. Verstraelen, “On semisymmetric para-Kähler manifolds”, Acta Math. Hungar., 74:1–2 (1997), 7–17 | DOI | MR | Zbl

[70] D. V. Alekseevsky, A. F. Spiro, “Homogeneous bi-Lagrangian manifolds and invariant Monge–Ampère equations”, Symmetries and perturbation theory, eds. G. Gaeta, R. Vitolo, S. Walcher, World Sci. Publ., Hackensack, NJ, 2008, 3–12 | MR | Zbl

[71] Zi Xin Hou, “On homogeneous paracomplex and para-Kähler manifolds”, Chinese J. Contemp. Math., 15:2 (1994), 193–206 | MR | Zbl

[72] Z. Hou, S. Deng, “Recent progress on dipolarizations in Lie algebras and homogeneous para-Kähler manifolds”, Adv. Math. (China), 30:6 (2001), 489–494 | MR

[73] D. V. Alekseevsky, C. Medori, “Bi-isotropic decompositions of semisimple Lie algebras and homogeneous bi-Lagrangian manifolds”, J. Algebra, 313:1 (2007), 8–27 | DOI | MR | Zbl

[74] V. F. Molchanov, “Quantization on para-Hermitian symmetric spaces”, Contemporary mathematical physics, Amer. Math. Soc. Transl. Ser. 2, 175, Amer. Math. Soc., Providence, RI, 1996, 81–95 | MR | Zbl

[75] V. F. Molchanov, G. Van Dijk, “The Berezin form for rank one para-Hermitian symmetric spaces”, J. Math. Pures Appl. (9), 77:8 (1998), 747–799 | DOI | MR | Zbl

[76] V. F. Molchanov, N. B. Volotova, “Polynomial quantization on rank one para-Hermitian symmetric spaces”, Acta Appl. Math., 81:1–3 (2004), 215–232 | DOI | MR | Zbl

[77] H. Hess, “Connections on symplectic manifolds and geometric quantization”, Differential geometrical methods in mathematical physics (Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math., 836, Springer, Berlin, 1980, 153–166 | DOI | MR | Zbl

[78] V. Cortés, Ch. Mayer, Th. Mohaupt, F. Saueressig, “Special geometry of Euclidean supersymmetry. II. Hypermultiplets and the $c$-map”, J. High Energy Phys., 2005, no. 6, 37 pp. | DOI | MR

[79] G. W. Gibbons, M. B. Green, M. J. Perry, “Instantons and seven-branes in type IIB superstring theory”, Phys. Lett. B, 370:1–2 (1996), 37–44 | DOI | MR

[80] V. Cortés, M.-A. Lawn, L. Schäfer, “Affine hyperspheres associated to special para-Kähler manifolds”, Int. J. Geom. Methods Mod. Phys., 3:5–6 (2006), 995–1009 | DOI | MR | Zbl

[81] D. V. Alekseevsky, N. Blažić, V. Cortés, S. Vukmirović, “A class of Osserman spaces”, J. Geom. Phys., 53:3 (2005), 345–353 | DOI | MR | Zbl

[82] A. Fino, H. Pedersen, Y. S. Poon, M. S. Weye, M. W. Sorensen, “Neutral Calabi–Yau structures on Kodaira manifolds”, Comm. Math. Phys., 248:2 (2004), 255–268 | DOI | MR | Zbl

[83] A. Dancer, A. Swann, “Toric hypersymplectic quotients”, Trans. Amer. Math. Soc., 359:3 (2007), 1265–1284 | DOI | MR | Zbl

[84] S. Ivanov, V. Tsanov, S. Zamkovoy, “Hyper-parahermitian manifolds with torsion”, J. Geom. Phys., 56:4 (2006), 670–690 | DOI | MR | Zbl

[85] A. S. Dancer, H. R. Jorgensen, A. F. Swann, “Metric geometry over the split quaternions”, Rend. Sem. Mat. Univ. Politec. Torino, 63:2 (2005), 119–139 | MR | Zbl

[86] T. N. Bailey, M. G. Eastwood, “Complex paraconformal manifolds – their differential geometry and twistor theory”, Forum Math., 3:1 (1991), 61–103 | MR | Zbl

[87] D. V. Alekseevsky, V. Cortés, “Classification of pseudo-Riemannian symmetric spaces of quaternionic Kähler type”, Lie groups and invariant theory, Amer. Math. Soc. Transl. Ser. 2, 213, Amer. Math. Soc., Providence, RI, 2005, 33–62 | MR | Zbl

[88] D. E. Blair, “A product twistor space”, Serdica Math. J., 28:2 (2002), 163–174 | MR | Zbl

[89] D. E. Blair, J. Davidov, O. Mus̆karov, “Hyperbolic twistor spaces”, Rocky Mountain J. Math., 35:5 (2005), 1437–1465 | DOI | MR | Zbl

[90] D. V. Alekseevsky, V. Cortés, “The twistor spaces of a para-quaternionic Kähler manifold”, Osaka J. Math., 45:1 (2008), 215–251 | MR | Zbl

[91] S. Ianuş, R. Mazzocco, G. E. Vîlcu, “Real lightlike hypersurfaces of paraquaternionic Kähler manifolds”, Mediterr. J. Math., 3:3–4 (2006), 581–592 | DOI | MR

[92] S. Zamkovoy, “Geometry of paraquaternionic Kähler manifolds with torsion”, J. Geom. Phys., 57:1 (2006), 69–87 | DOI | MR | Zbl

[93] D. V. Alekseevsky, C. Medori, A. Tomassini, “Maximally homogeneous para-CR-manifolds”, Ann. Global Anal. Geom., 30:1 (2006), 1–27 | DOI | MR | Zbl

[94] P. Nurowski, G. A. J. Sparling, “Three-dimensional Cauchy–Riemann structures and second-order ordinary differential equations”, Classical Quantum Gravity, 20:23 (2003), 4995–5016 | DOI | MR | Zbl

[95] D. Alekseevsky, Y. Kamishima, “Quaternionic and para-quaternionic CR structure on $(4n+3)$-dimensional manifolds”, Cent. Eur. J. Math., 2:5 (2004), 732–753 | DOI | MR | Zbl

[96] D. Alekseevsky, Y. Kamishima, “Pseudo-conformal quaternionic CR structure on $(4n+3)$-dimensional manifolds”, Ann. Mat. Pura Appl. (4), 187:3 (2008), 487–529 | DOI | MR | Zbl

[97] J. L. Koszul, “Sur la forme hermitienne canonique des espaces homogènes complexes”, Canad. J. Math., 7 (1955), 562–576 | MR | Zbl

[98] Y. Benoist, F. Labourie, “Sur les difféomorphismes d'Anosov affines à feuilletages stable et instable différentiables”, Invent. Math., 111:1 (1993), 285–308 | DOI | MR | Zbl

[99] L. Benoist, F. Labourie, “Sur les espaces homogènes modèles de variétés compactes”, Inst. Hautes Études Sci. Publ. Math., 76:1 (1992), 99–109 | DOI | MR | Zbl

[100] V. V. Gorbatsevic, A. L. Onishchik, E. B. Vinberg, Structure of Lie groups and Lie algebras, Encyclopaedia Math. Sci., 41, Springer-Verlag, Berlin, 1994 | MR | MR | Zbl | Zbl

[101] D. Ž. Djoković, “Classification of $\mathbb{Z}$-graded real semisimple Lie algebras”, J. Algebra, 76:2 (1982), 367–382 | DOI | MR | Zbl

[102] D. V. Alekseevskii, A. M. Perelomov, “Invariant Kähler–Einstein metrics on compact homogeneous spaces”, Funct. Anal. Appl., 20:3 (1986), 171–182 | DOI | MR | Zbl

[103] J. E. Humpreys, Introduction to Lie algebras and representation theory, Grad. Texts in Math., 9, Springer-Verlag, New York–Berlin, 1978 | MR | Zbl