Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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MODULI SPACES OF ORIENTED TYPE ${\mathcal{A}}$ MANIFOLDS OF DIMENSION AT LEAST 3

  • Received : 2016.10.06
  • Accepted : 2017.02.28
  • Published : 2017.11.01
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Abstract

We examine the moduli space of oriented locally homogeneous manifolds of Type ${\mathcal{A}}$ which have non-degenerate symmetric Ricci tensor both in the setting of manifolds with torsion and also in the torsion free setting where the dimension is at least 3. These exhibit phenomena that is very different than in the case of surfaces. In dimension 3, we determine all the possible symmetry groups in the torsion free setting.

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References

  1. I. Agricola, S. Chiossi, T. Friedrich, and J. Holl, Spinorial description of SU(3) and $G_2$-manifolds, J. Geom. Phys. 98 (2015), 535-555. https://doi.org/10.1016/j.geomphys.2015.08.023
  2. I. Agricola, S. Chiossi, G. Simon, and A. Fino, Solvmanifolds with integrable and nonintegrable $G_2$ structures, Differential Geom. Appl. 25 (2007), no. 2, 125-135. https://doi.org/10.1016/j.difgeo.2006.05.002
  3. I. Agricola and H. Kim, A note on generalized Dirac Eigenvalues for split holonomy and torsion, Bull. Korean Math. Soc. 51 (2014), no. 6, 1579-1589. https://doi.org/10.4134/BKMS.2014.51.6.1579
  4. T. Arias-Marco and O. Kowalski, Classification of locally homogeneous affine connections with arbitrary torsion on 2-dimensional manifolds, Monatsh. Math. 153 (2008), no. 1, 1-18. https://doi.org/10.1007/s00605-007-0494-0
  5. X. Bekaert and K. Morand, Connections and dynamical trajectories in generalized Newton-Cartan gravity I. An intrinsic view, J. Math. Phys. 57 (2016), no. 2, 022507, 44 pp. https://doi.org/10.1063/1.4937445
  6. W. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, New York, 1975.
  7. M. Brozos-Vazquez, E. Garcia-Rio, and P. Gilkey, Homogeneous ane surfaces: Affine killing vector fields and gradient Ricci solitons, http://arxiv.org/abs/1512.05515 (to appear J. Math. Soc. Japan).
  8. M. Brozos-Vazquez, E. Garcia-Rio, and P. Gilkey, Homogeneous affine surfaces: Moduli spaces, J. Math. Anal. Appl. 444 (2016), no. 2, 1155-1184. https://doi.org/10.1016/j.jmaa.2016.07.005
  9. E. Calvino-Louzao, E. Garcia-Rio, P. Gilkey, and R. Vazquez-Lorenzo, The geometry of modified Riemannian extensions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009), no. 2107, 2023-2040. https://doi.org/10.1098/rspa.2009.0046
  10. E. Calvino-Louzao, E. Garcia-Rio, and R. Vazquez-Lorenzo, Riemann extensions of torsion-free connections with degenerate Ricci tensor, Canad. J. Math. 62 (2010), no. 5, 1037-1057. https://doi.org/10.4153/CJM-2010-059-2
  11. A. Coimbra, C. Strickland-Constable, and D. Waldram, Supersymmetric backgrounds and generalized special holonomy, Classical Quantum Gravity 33 (2016), no. 12, 125026, 27 pp. https://doi.org/10.1088/0264-9381/33/12/125026
  12. A. Derdzinski, Noncompactness and maximum mobility of type III Ricci-flat self-dual neutral Walker four-manifolds, Q. J. Math. 62 (2011), no. 2, 363-395. https://doi.org/10.1093/qmath/hap033
  13. S. Deser, S. Ertl, and D. Grumiller, Canonical bifurcation in higher derivative, higher spin theories, J. Phys. A 46 (2013), no. 21, 214018, 9 pp. https://doi.org/10.1088/1751-8113/46/21/214018
  14. G. Dileo and A. Lotta, Some Einstein nilmanifolds with skew torsion arising in CR geometry, Int. J. Geom. Methods Mod. Phys. 12 (2015), no. 8, 1560017, 6 pp. https://doi.org/10.1142/S0219887815600178
  15. S. Dumitrescu, Locally homogeneous rigid geometric structures on surfaces, Geom. Dedicata 160 (2012), 71-90. https://doi.org/10.1007/s10711-011-9670-4
  16. S. Fedoruk, E. Ivanov, and A. Smilga, N = 4 mechanics with diverse (4,4,0) multiplets: explicit examples of hyper-Kahler with torsion, Clifford Kahler with torsion, and octonionic Kahler with torsion geometries, J. Math. Phys. 55 (2014), no. 5, 052302, 29pp. https://doi.org/10.1063/1.4871440
  17. Th. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math. 6 (2002), no. 2, 303-335. https://doi.org/10.4310/AJM.2002.v6.n2.a5
  18. Th. Friedrich and S. Ivanov, Almost contact manifolds, connections with torsion, and parallel spinors, J. Reine Angew. Math. 559 (2003), 217-236.
  19. S. Gallot, D. Hulin, and J. Lafontaine, Riemannian Geometry 3rd ed, Springer Universitext. Springer-Verlag, Berlin, 2004.
  20. J. Gegenberg, A. Day, H. Liu, and S. Seahra, An instability of hyperbolic space under the Yang-Mills flow, J. Math. Phys. 55 (2014), no. 4, 042501, 9 pp. https://doi.org/10.1063/1.4869870
  21. P. Gilkey, The moduli space of Type A surfaces with torsion and non-singular symmetric Ricci tensor, J. Geom. Phys. 110 (2016), 69-77. https://doi.org/10.1016/j.geomphys.2016.07.012
  22. P. Gilkey, J. H. Park, and R. Vazquez-Lorenzo, Aspects of Differential Geometry I, (Synthesis lectures on mathematics and statistics), Morgan and Claypool, 2015.
  23. A. Guillot and A. Sanchez-Godinez, A classification of locally homogeneous ane connections on compact surfaces, Ann. Global Anal. Geom. 46 (2014), no. 4, 335-349. https://doi.org/10.1007/s10455-014-9426-0
  24. S. Ivanov, Connections with torsion, parallel spinors, and geometry of spin(7) manifolds, Math. Res. Lett. 11 (2004), no. 2-3, 171-186. https://doi.org/10.4310/MRL.2004.v11.n2.a3
  25. S. Ivanov and M. Zlatanovic, Connections on a non-symmetric (generalized) Riemannian manifold and gravity, Classical Quantum Gravity 33 (2016), no. 7, 075016, 23 pp. https://doi.org/10.1088/0264-9381/33/7/075016
  26. M. Ivanova and M. Manev, A classification of the torsion tensors on almost contact manifolds with B-metric, Cent. Eur. J. Math. 12 (2014), no. 10, 1416-1432.
  27. M. Kassuba, Eigenvalue estimates for Dirac operators in geometries with torsion, Ann. Glob. Anal. Geom. 37 (2010), no. 1, 33-71. https://doi.org/10.1007/s10455-009-9172-x
  28. D. Klemm and M. Nozawa, Geometry of Killing spinors in neutral signature, Classical Quantum Gravity 32 (2015), no. 18, 185012, 36 pp. https://doi.org/10.1088/0264-9381/32/18/185012
  29. O. Kowalski, B. Opozda, and Z. Vlasek, A classification of locally homogeneous ane connections with skew-symmetric Ricci tensor on 2-dimensional manifolds, Monatsh. Math. 130 (2000), no. 2, 109-125. https://doi.org/10.1007/s006050070041
  30. O. Kowalski, B. Opozda, and Z. Vlasek, On locally nonhomogeneous pseudo-Riemannian manifolds with locally homogeneous Levi-Civita connections, Internat. J. Math. 14 (2003), no. 5, 559-572. https://doi.org/10.1142/S0129167X03001971
  31. O. Kowalski and M. Sekizawa, The Riemann extensions with cyclic parallel Ricci tensor, Math. Nachr. 287 (2014), no. 8-9, 955-961. https://doi.org/10.1002/mana.201200299
  32. M. Manev, A connection with parallel torsion on almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics, J. Geom. Phys. 61 (2011), no. 1, 248-259. https://doi.org/10.1016/j.geomphys.2010.09.018
  33. M. Manev and K. Gribachev, A connection with parallel totally skew-symmetric torsion on a class of almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics, Int. J. Geom. Methods Mod. Phys. 8 (2011), no. 1, 115-131. https://doi.org/10.1142/S0219887811005026
  34. D. Mekerov, Natural connection with totally skew-symmetric torsion on Riemannian almost product manifolds, Int. J. Geom. Methods Mod. Phys. 09 (2012), no. 1, 1250003, 14 pp. https://doi.org/10.1142/S021988781250003X
  35. B. Opozda, A classification of locally homogeneous connections on 2-dimensional manifolds, Differential Geom. Appl. 21 (2004), no. 2, 173-198. https://doi.org/10.1016/j.difgeo.2004.03.005
  36. B. Opozda, Locally homogeneous ane connections on compact surfaces, Proc. Amer. Math. Soc. 132 (2004), no. 9, 2713-2721. https://doi.org/10.1090/S0002-9939-04-07402-7
  37. J. Wang, Y. Wang, and C. Yang, Dirac operators with torsion and the non-commutative residue for manifolds with boundary, J. Geom. Phys. 81 (2014), 92-111. https://doi.org/10.1016/j.geomphys.2014.03.007