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Conical Ricci-flat nearly para-Kähler manifolds

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Abstract

In this paper, we classify conical Ricci-flat nearly para-Kähler manifolds having isotropic Nijenhuis tensor. More precisely, we give a bijective correspondence between this class of nearly para-Kähler manifolds and local cones \(M_1 \times (a,b)\) over para-Sasaki-Einstein manifolds \((M_1,g_1,T)\) carrying a parallel 3-form with isotropic support. Moreover, we show that the cone over a para-Sasaki-Einstein five-manifold \((M_1,g_1,T)\) admits a family of parallel 3-forms with isotropic support. As an application our result yields first examples of Ricci-flat (non-flat) nearly para-Kähler structures.

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Notes

  1. Please note that we use another sign convention than [34].

  2. See for example [39] p. 209-211 for the curvature of warped products.

  3. In our convention \(m+1\) is the number of negative directions.

  4. Note that the frame bundle of a para-Kähler manifold can be reduced to the bundle of para-unitary frames \(P_{U(\mathcal P _0,\mathcal G _0)}.\)

  5. Here \(g^{-1}\) is the inverse of the map \(g \,:\, TM \rightarrow T^*M,\; X\mapsto g(X,\cdot )\).

References

  1. Agricola, I., Höll, J.: Cones of G manifolds and killing spinors with skew torsion. arXiv:1303.3601 (2013)

  2. Alekseevsky, D.V., Medori, K., Tomassini, A.: Homogeneous para-Kählerian Einstein manifolds. Uspekhi Mat. Nauk 64(1(385)), 3–50 (2009) (Russian, with Russian summary); English transl., Russ. Math. Surv. 64(1), 1–43 (2009)

  3. Alekseevsky, D.V., Cortés, V., Galaev, A.S., Leistner, Th: Cones over pseudo-Riemannian manifolds and their holonomy. J. Reine Angew. Math. 635, 23–69 (2009)

    MATH  MathSciNet  Google Scholar 

  4. Bär, Ch.: Real killing spinors and holonomy. Commun. Math. Phys. 154(3), 509–521 (1993)

    Article  MATH  Google Scholar 

  5. Bérard Bergery, L., Ikemakhen, A.: Sur l’holonomie des variétés pseudo-riemanniennes de signature \((n, n)\). Bull. Soc. Math. France 125(1), 93–114 (1997) (French, with English and French summaries)

  6. Belgun, F., Moroianu, A.: Nearly Kähler 6-manifolds with reduced holonomy. Ann. Global Anal. Geom. 19(4), 307–319 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Boyer, C., Galicki, K.: Sasakian Geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford (2008)

  8. Butruille, J.-B.: Classification des variétés approximativement kähleriennes homogènes. Ann. Global Anal. Geom. 27(3), 201–225 (2005). doi:10.1007/s10455-005-1581-x (French, with English summary)

  9. Cappelletti Montano, B., Carriazo, A., Martin-Molina, V.: Sasaki-Einstein and paraSasaki-Einstein metrics from \((k,\mu )\)-structures. arXiv:1109.6248

  10. Conti, D., Bruun Madsen, Th.: The odd side of torsion geometry. arXiv:1207.3072 (2012)

  11. Cortés, V., Lawn, M.-A., Schäfer, L.: Affine hyperspheres associated to special para-Kähler manifolds. Int. J. Geom. Methods Mod. Phys. 3(5–6), 995–1009 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cortés, V., Leistner, Th, Schäfer, L., Schulte-Hengesbach, F.: Half-flat structures and special holonomy. Proc. Lond. Math. Soc. 102(3), 113–158 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cortés, V., Schäfer, L.: Flat nearly Kähler manifolds. Ann. Global Anal. Geom. 32(4), 379–389 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cortés, V., Schäfer, L.: Geometric structures on Lie groups with flat bi-invariant metric. J. Lie Theory 19(1), 423–437 (2009)

    MATH  MathSciNet  Google Scholar 

  15. Cruceanu, V., Fortuny, P., Gadea, P.M.: A survey on paracomplex geometry. Rocky Mt. J. Math. 26(1), 83–115 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  16. Cecotti, S., Vafa, C.: Topological-anti-topological fusion. Nuclear Phys. B 367(2), 359–461 (1991). doi:10.1016/0550-3213(91)90021-O

    Article  MATH  MathSciNet  Google Scholar 

  17. Cecotti, S., Vafa, C.: On classification of \(N=2\) supersymmetric theories. Commun. Math. Phys. 158(3), 569–644 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  18. Duff, M.J., Nilsson, B.E.W., Pope, C.N.: Kaluza–Klein supergravity. Phys. Rep. 130(1–2), 1–142 (1986). doi:10.1016/0370-1573(86)90163-8

    Article  MathSciNet  Google Scholar 

  19. Friedrich, Th.: Der erste Eigenwert des Dirac-Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung. Math. Nachr. 97, 117–146 (1980) (German)

    Google Scholar 

  20. Friedrich, Th, Ivanov, Stefan: Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math. 6(2), 303–335 (2002)

    MATH  MathSciNet  Google Scholar 

  21. Friedrich, Th.: On types of non-integrable geometries. In: Proceedings of the 22nd Winter School “Geometry and Physics” (Srní, 2002), pp. 99–113 (2003)

  22. Gallot, S.: Équations différentielles caractéristiques de la sphère. Ann. Sci. École Norm. Sup. (4) 12(2), 235–267 (1979) (French)

    Google Scholar 

  23. Gadea, P.M., Masque, J.M.: Classification of almost para-Hermitian manifolds. Rend. Math. Appl. (7) 11(2), 377–396 (1991) (English, with Italian summary)

    Google Scholar 

  24. Gray, A.: Minimal varieties and almost Hermitian submanifolds. Mich. Math. J. 12, 273–287 (1965)

    Article  MATH  Google Scholar 

  25. Gray, A.: Almost complex submanifolds of the six sphere. Proc. Am. Math. Soc. 20, 277–279 (1969)

    Article  MATH  Google Scholar 

  26. Gray, A.: Nearly Kähler manifolds. J. Differ. Geom. 4, 283–309 (1970)

    MATH  Google Scholar 

  27. Gray, A.: Riemannian manifolds with geodesic symmetries of order \(3\). J. Differ. Geom. 7, 343–369 (1972)

    MATH  Google Scholar 

  28. Gray, A.: The structure of nearly Kähler manifolds. Math. Ann. 223(3), 233–248 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  29. Gray, A., Hervella, L.M.: The sixteen classes of almost Hermitian manifolds and their linear invariants. Ann. Mat. Pura Appl. (4) 123, 35–58 (1980). doi:10.1007/BF01796539

    Article  MATH  MathSciNet  Google Scholar 

  30. Grunewald, R.: Six-dimensional Riemannian manifolds with a real Killing spinor. Ann. Global Anal. Geom. 8(1), 43–59 (1990). doi:10.1007/BF00055017

    Article  MATH  MathSciNet  Google Scholar 

  31. Harvey, F.R., Lawson Jr., H.B.: Split special Lagrangian geometry. arXiv:1007.0450 (2010)

  32. Hitchin, N.: Stable forms and special metrics. (Bilbao, 2000). Contemporary Mathematics, vol. 288, pp. 70–89. American Mathematical Society, Providence (2001)

  33. Hitchin, N.: The geometry of three-forms in six dimensions. J. Differ. Geom. 55(3), 547–576 (2000)

    MATH  MathSciNet  Google Scholar 

  34. Ivanov, S., Zamkovoy, S.: Parahermitian and paraquaternionic manifolds. Differ. Geom. Appl. 23(2), 205–234 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  35. Jensen, G.R.: Imbeddings of Stiefel manifolds into Grassmannians. Duke Math. J. 42(3), 397–407 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  36. Kath, I.: Killing Spinors on Pseudo-Riemannian Manifolds. Habilitationsschrift an der Humboldt-Universität zu Berlin (1999)

  37. Kath, I.: Pseudo-Riemannian \(T\)-duals of compact Riemannian homogeneous spaces. Transform. Groups 5(2), 157–179 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  38. Kirichenko, V.F.: Generalized Gray–Hervella classes and holomorphically projective transformations of generalized almost Hermitian structures. Izv. Math. 69(5), 963–987 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  39. O’Neill, B.: Semi-Riemannian geometry. Pure and Applied Mathematics, vol. 103. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1983). With applications to relativity

  40. Nagy, P.-A.: On nearly-Kähler geometry. Ann. Global Anal. Geom. 22(2), 167–178 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  41. Nagy, P.-A.: Nearly Kähler geometry and Riemannian foliations. Asian J. Math. 6(3), 481–504 (2002)

    MATH  MathSciNet  Google Scholar 

  42. Schäfer, L., Schulte-Hengesbach, F.: Nearly pseudo-Kähler and nearly para-Kähler six-manifolds. In: Handbook of pseudo-Riemannian geometry and supersymmetry, chap. 12. IRMA Lectures in Mathematics and Theoretical Physics, vol. 16. EMS Publishing House, Zürich (2010). arXiv:0907.1222

  43. Schäfer, L.: Para-\(tt^{\ast }\)-bundles on the tangent bundle of an almost para-complex manifold. Ann. Global Anal. Geom. 32(2), 125–145 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  44. Schäfer, L.: \(tt^{*}\)-geometry on the tangent bundle of an almost complex manifold. J. Geom. Phys. 57(3), 999–1014 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  45. Schäfer, L.: On the structure of nearly pseudo-Kähler manifolds. Monatsh. Math. 163(3), 339–371 (2011). doi:10.1007/s00605-009-0184-1

    Article  MATH  MathSciNet  Google Scholar 

  46. Strominger, A.: Superstrings with torsion. Nuclear Phys. B 274(2), 253–284 (1986). doi:10.1016/0550-3213(86)90286-5

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The author thanks Lutz Habermann and the unkown referee for valuable comments on earlier versions of the manuscript and Florin Belgun and Vicente Cortés for interest and remarks during a research stay at Hamburg.

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  1. Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 , Hannover, Germany

    Lars Schäfer

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Schäfer, L. Conical Ricci-flat nearly para-Kähler manifolds. Ann Glob Anal Geom 45, 11–24 (2014). https://doi.org/10.1007/s10455-013-9385-x

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