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Uspekhi Mat. Nauk, 2009, Volume 64, Issue 1(385), Pages 3–50 (Mi umn9262)  

This article is cited in 49 scientific papers (total in 49 papers)

Homogeneous para-Kähler Einstein manifolds

D. V. Alekseevskya, C. Medorib, A. Tomassinib

a University of Edinburgh
b Università degli Studi di Parma

Abstract: 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.

DOI: https://doi.org/10.4213/rm9262

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English version:
Russian Mathematical Surveys, 2009, 64:1, 1–43

Bibliographic databases:

UDC: 514.747+514.76
MSC: Primary 53C25, 53C26; Secondary 53B35, 53C55, 53C15
Received: 09.06.2008

Citation: D. V. Alekseevsky, C. Medori, A. Tomassini, “Homogeneous para-Kähler Einstein manifolds”, Uspekhi Mat. Nauk, 64:1(385) (2009), 3–50; Russian Math. Surveys, 64:1 (2009), 1–43

Citation in format AMSBIB
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    1. Caldarella A.V., “On paraquaternionic submersions between paraquaternionic Kähler manifolds”, Acta Appl. Math., 112:1 (2010), 1–14  crossref  mathscinet  zmath  isi  elib  scopus
    2. Chursin M., Schäfer L., Smoczyk K., “Mean curvature flow of space-like Lagrangian submanifolds in almost para-Kähler manifolds”, Calc. Var., 41:1-2 (2011), 111–125  crossref  mathscinet  zmath  isi  scopus
    3. Bajo I., Benayadi S., “Abelian para-Kähler structures on Lie algebras”, Differential Geom. Appl., 29:2 (2011), 160–173  crossref  mathscinet  zmath  isi  scopus
    4. Calvaruso G., “Symplectic, complex and Kähler structures on four-dimensional generalized symmetric spaces”, Differential Geom. Appl., 29:6 (2011), 758–769  crossref  mathscinet  zmath  isi  elib  scopus
    5. Medori C., Tomassini A., “On small deformations of paracomplex manifolds”, J. Noncommut. Geom., 5:4 (2011), 507–522  crossref  mathscinet  zmath  isi  scopus
    6. Schäfer L., “Foliations of semi-Riemannian manifolds”, Results Math., 61:1-2 (2012), 97–126  crossref  mathscinet  zmath  isi  scopus
    7. Brozos-Vázquez M., García-Río E., Gilkey P., Hervella L., “Geometric realizability of covariant derivative Kähler tensors for almost pseudo-Hermitian and almost para-Hermitian manifolds”, Ann. Mat. Pura Appl. (4), 191:3 (2012), 487–502  crossref  mathscinet  zmath  isi  scopus
    8. Rossi F.A., “On deformations of D-manifolds and CR D-manifolds”, J. Geom. Phys., 62:2 (2012), 464–478  crossref  mathscinet  zmath  adsnasa  isi  scopus
    9. Alekseevsky D., Alonso-Blanco R., Manno G., Pugliese F., “Monge-Ampère equations on (para-)Kähler manifolds: from characteristic subspaces to special Lagrangian submanifolds”, Acta Appl. Math., 120 (2012), 3–27  crossref  mathscinet  zmath  isi  scopus
    10. Vaisman I., “On the geometry of double field theory”, J. Math. Phys., 53:3 (2012), 033509, 21 pp.  crossref  mathscinet  zmath  adsnasa  isi  scopus
    11. Manno G., Metafune G., “On the extendability of conformal vector fields of 2-dimensional manifolds”, Differential Geom. Appl., 30:4 (2012), 365–369  crossref  mathscinet  zmath  isi  scopus
    12. Vaccaro M., “(Para-)Hermitian and (para-)Kähler submanifolds of a para-quaternionic Kähler manifold”, Differential Geom. Appl., 30:4 (2012), 347–364  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    13. Vîlcu G.E., Voicu R.C., “Curvature properties of pseudo-sphere bundles over paraquaternionic manifolds”, Int. J. Geom. Methods Mod. Phys., 9:3 (2012), 1250024, 23 pp.  crossref  mathscinet  isi  scopus
    14. Qing Ding, Xiangping Liu, Wei Wang, “The vortex filament in the Minkowski 3-space and generalized bi-Schrödinger maps”, J. Phys. A, 45:45 (2012), 455201  crossref  mathscinet  zmath  isi  scopus
    15. Bajo I., Sanmartín E., “Pseudo-Kähler Lie algebras with Abelian complex structures”, J. Phys. A, 45:46 (2012), 465205, 21 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    16. Batalin I.A., Bering K., “A triplectic bi-Darboux theorem and para-hypercomplex geometry”, J. Math. Phys., 53:12 (2012), 123507, 25 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    17. Druţǎ-Romaniuc S.-L., “Natural diagonal Riemannian almost product and para-Hermitian cotangent bundles”, Czechoslovak Math. J., 62:4 (2012), 937–949  crossref  mathscinet  zmath  isi  scopus
    18. Angella D., Rossi F.A., “Cohomology of D-Complex Manifolds”, Differ. Geom. Appl., 30:5 (2012), 530–547  crossref  mathscinet  zmath  isi  scopus
    19. Druţǎ-Romaniuc S.-L., “General natural Riemannian almost product and para-Hermitian structures on tangent bundles”, Taiwan. J. Math., 16:2 (2012), 497–510  crossref  mathscinet  zmath  isi  scopus
    20. S. L. Druţă-Romaniuc, “Para-Kähler tangent bundles of constant para-holomorphic sectional curvature”, Bull. Iranian Math. Soc., 38:4 (2012), 955–972  mathscinet  zmath  isi
    21. Druţă-Romaniuc S.L., “Riemannian almost product and para-Hermitian cotangent bundles of general natural lift type”, Acta Math. Hung., 139:3 (2013), 228–244  crossref  mathscinet  isi  scopus
    22. Calvaruso G., Fino F., “Complex and paracomplex structures on homogeneous pseudo-Riemannian four-manifolds”, Int. J. Math., 24:1 (2013), 1250130, 28 pp.  crossref  mathscinet  zmath  isi  scopus
    23. Izu Vaisman, “Towards a double field theory on para-Hermitian manifolds”, J. Math. Phys., 54:12 (2013), 123507  crossref  mathscinet  zmath  isi  scopus
    24. L. Schäfer, “Conical Ricci-flat nearly para-Kähler manifolds”, Ann. Glob. Anal. Geom., 45:1 (2014), 11–24  crossref  mathscinet  zmath  isi  scopus
    25. Qing Ding, ZhiZhou He, “The noncommutative KdV equation and its para-Kähler structure”, Sci. China Math., 57:7 (2014), 1505–1516  crossref  mathscinet  zmath  isi  scopus
    26. H. Anciaux, “Spaces of geodesics of pseudo-Riemannian space forms and normal congruences of hypersurfaces”, Trans. Amer. Math. Soc., 366:5 (2014), 2699–2718  crossref  mathscinet  zmath  isi  scopus
    27. H. Anciaux, “Marginally trapped submanifolds in space forms with arbitrary signature”, Pacific J. Math., 272:2 (2014), 257–274  crossref  mathscinet  zmath  isi  scopus
    28. Calvaruso G., Zaeim A., “Geometric Structures Over Non-Reductive Homogeneous 4-Spaces”, Adv. Geom., 14:2 (2014), 191–214  crossref  mathscinet  zmath  isi  scopus
    29. Angella D., “Cohomological Aspects in Complex Non-Kahler Geometry Introduction”: Angella, D, Cohomological Aspects in Complex Non-Kahler Geometry, Lect. Notes Math., 2095, Springer Int Publishing Ag, 2014, VII+  mathscinet  isi
    30. Giovanni Calvaruso, “A complete classification of four-dimensional paraKähler Lie algebras”, Complex Manifolds, 2:1 (2015)  crossref  mathscinet
    31. Calvaruso G., “Four-Dimensional Parakahler Lie Algebras: Classification and Geometry”, 41, no. 3, 2015, 733–748  mathscinet  zmath  isi
    32. Iscan M., Caglar G., “Para-Kähler–Einstein structures on Walker 4-manifolds”, Int. J. Geom. Methods Mod. Phys., 13:2 (2016), 1650006  crossref  mathscinet  zmath  isi  elib  scopus
    33. Krynski W., “Webs and the Plebański equation”, Math. Proc. Camb. Philos. Soc., 161:3 (2016), 455–468  crossref  mathscinet  zmath  isi  scopus
    34. Ida C., Ionescu A., Manea A., “A note on para-holomorphic Riemannian–Einstein manifolds”, Int. J. Geom. Methods Mod. Phys., 13:9 (2016), 1650107  crossref  mathscinet  zmath  isi  elib  scopus
    35. Bejan C.-L., Druta-Romaniuc S.-L., “Harmonic functions and quadratic harmonic morphisms on Walker spaces”, Turk. J. Math., 40:5 (2016), 1004–1019  crossref  mathscinet  isi  scopus
    36. Chudecki A., “On Geometry of Congruences of Null Strings in 4-Dimensional Complex and Real Pseudo-Riemannian Spaces”, J. Math. Phys., 58:11 (2017), 112502  crossref  mathscinet  zmath  isi  scopus
    37. Sahin B., Sahin F., “Generalized Almost Para-Contact Manifolds”, Int. J. Geom. Methods Mod. Phys., 14:10 (2017), 1750147  crossref  mathscinet  zmath  isi  scopus
    38. Petrecca D., Schaefer L., “Second Variation For l-Minimal Legendrian Submanifolds in Pseudo-Sasakian Manifolds”, Mon.heft. Math., 184:2 (2017), 273–289  crossref  mathscinet  zmath  isi  scopus
    39. Ida C., Manea A., “On the Integrability of Generalized Almost Para-Norden and Para-Hermitian Structures”, Mediterr. J. Math., 14:4 (2017), UNSP 173  crossref  mathscinet  isi  scopus
    40. Kruglikov B.S., Winther H., “Non-Degenerate Para-Complex Structures in 6D With Large Symmetry Groups”, Ann. Glob. Anal. Geom., 52:3 (2017), 341–362  crossref  mathscinet  zmath  isi  scopus
    41. Schaefer L., “Nearly Pseudo-Kahler Manifolds and Related Special Holonomies Introduction”: Schafer, L, Nearly Pseudo-Kahler Manifolds and Related Special Holonomies, Lect. Notes Math., Lecture Notes in Mathematics, 2201, Springer International Publishing Ag, 2017, 1+  crossref  mathscinet  isi  scopus
    42. Bor G., Hernandez Lamoneda L., Nurowski P., “The Dancing Metric, G(2)-Symmetry and Projective Rolling”, Trans. Am. Math. Soc., 370:6 (2018), 4433–4481  crossref  mathscinet  zmath  isi  scopus
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    44. N. K. Smolentsev, “O pochti (para) kompleksnykh strukturakh Keli na sferakh $\mathbf{S}^{2,4}$ i $\mathbf{S}^{3,3}$”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2018, no. 53, 22–38  mathnet  crossref  elib
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    46. Druta-Romaniuc S.L., “General Natural (Alpha, Epsilon)-Structures”, Mediterr. J. Math., 15:6 (2018), 228  crossref  mathscinet  zmath  isi  scopus
    47. Svoboda D., “Algebroid Structures on Para-Hermitian Manifolds”, J. Math. Phys., 59:12 (2018), 122302  crossref  mathscinet  zmath  isi  scopus
    48. Conti D., Rossi F.A., “Einstein Nilpotent Lie Groups”, J. Pure Appl. Algebr., 223:3 (2019), 976–997  crossref  mathscinet  zmath  isi  scopus
    49. Smolentsev N.K., “Left-Invariant Almost Para-Hermitian Structures on Some Six- Dimensional Nilpotent Lie Groups”, Vestn. Tomsk. Gos. Univ.-Mat. Mek., 2019, no. 58, 41–55  crossref  isi
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