Almost complex submanifolds of nearly Kähler manifolds. (English) Zbl 1446.53011
Summary: We study almost complex submanifolds of pseudo nearly Kähler manifolds. We show in particular that a 6 dimensional strict nearly Kähler manifold does not admit any 4 dimensional almost complex submanifolds. This generalises results obtained by A. Gray [Proc. Am. Math. Soc. 20, 277–279 (1969; Zbl 0165.55803)] for the nearly Kähler 6-sphere and by F. Podestà and A. Spiro [J. Geom. Phys. 60, No. 2, 156–164 (2010; Zbl 1184.53074)] in the Riemannian case.
MSC:
| 53B25 | Local submanifolds |
| 53B35 | Local differential geometry of Hermitian and Kählerian structures |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
Keywords:
almost complex submanifold; pseudo Riemannian geometry; nearly Kähler geometry; Riemannian submersion; almost pseudo Hermitian manifoldReferences:
| [1] | Alekseevsky, DV; Kruglikov, BS; Winther, H., Homogeneous almost complex strucutres in dimension 6 with semi-simple isotropy, Ann. Global Anal. Geom., 46, 361-387 (2014) · Zbl 1306.53017 · doi:10.1007/s10455-014-9428-y |
| [2] | Belgun, F.; Moroianu, A., Nearly Kähler 6-manifolds with reduced holonomy, Ann. Global Anl. Geom., 19, 307-319 (2001) · Zbl 0992.53037 · doi:10.1023/A:1010799215310 |
| [3] | Butruille, J.-B.: Homogeneous nearly Kähler manifolds. Handbook of Pseudo-Riemannian Geometry and Supersymmetry, RMA Lect. Math. Theor. Phys., vol. 16, pp. 399-423 (2010) · Zbl 1201.53059 |
| [4] | Cortés, V.; Schäfer, L., Flat nearly Kähler manifolds, Ann. Global Anal. Geom., 32, 379-389 (2007) · Zbl 1135.53048 · doi:10.1007/s10455-007-9068-6 |
| [5] | Foscolo, L.; Haskins, M., New \(G_2\) holonomy cones and exotic nearly Kähler structures on \(S^6\) and \(S^3 \times S^3\), Ann. Math., 185, 59-130 (2017) · Zbl 1381.53086 · doi:10.4007/annals.2017.185.1.2 |
| [6] | Gray, A., Almost complex submanifolds of the six sphere, Proc. Am. Math. Soc., 20, 277-279 (1969) · Zbl 0165.55803 · doi:10.1090/S0002-9939-1969-0246332-4 |
| [7] | Gray, A., Nearly Kähler manifolds, J. Differ. Geom., 4, 283-309 (1970) · Zbl 0201.54401 · doi:10.4310/jdg/1214429504 |
| [8] | Gray, A., Riemannian manifolds with geodesic symmetries of order 3, J. Differ. Geom., 7, 343-369 (1972) · Zbl 0275.53026 · doi:10.4310/jdg/1214431159 |
| [9] | Gray, A., The structure of nearly Kähler manifolds, Math. Ann., 223, 233-248 (1976) · Zbl 0345.53019 · doi:10.1007/BF01360955 |
| [10] | González Dávila, JC; Martín Cabrera, F., Homogeneous nearly Kähler manifolds, Ann. Global Anal. Geom., 2, 147-170 (2012) · Zbl 1258.53033 · doi:10.1007/s10455-011-9305-x |
| [11] | Gutowski, J.; Ivanov, S.; Papadopoulos, G., Deformations of generalized calibrations and compact non-Kähler manifolds with vanishing first Chern class, Asian J. Math., 7, 1, 039-080 (2003) · Zbl 1089.53034 · doi:10.4310/AJM.2003.v7.n1.a4 |
| [12] | Kath, I.: Killing Spinors on Pseudo-Riemannian manfiolds, Habiliatationsschrift an der Humboldt-Universiät zu Berlin (1999) |
| [13] | Nagy, PA, On nearly-Kähler geometry, Ann. Global Anal. Geom., 22, 167-178 (2002) · Zbl 1020.53030 · doi:10.1023/A:1019506730571 |
| [14] | Podestà, F.; Spiro, A., Six-dimensional nearly Kähler manifolds of cohomogeneity one, J. Geom. Phys., 60, 156-164 (2010) · Zbl 1184.53074 · doi:10.1016/j.geomphys.2009.09.008 |
| [15] | Russo, G.; Swann, A., Nearly Kähler six-manifolds with two-torus symmetry, J. Geom. Phys., 138, 144-153 (2019) · Zbl 1414.53023 · doi:10.1016/j.geomphys.2018.12.016 |
| [16] | Schäfer, L.: Nearly Pseudo-Kähler Manfiolds and Related Special Holonomies. Lecture Notes in Mathematics 2201 (2017) · Zbl 1380.53004 |
| [17] | Schäfer, L.; Smoczyk, K., Decomposition and minimality of Lagrangian submanifolds in nearly Kähler manifolds, Ann. Global Anal. Geom., 37, 3, 221-240 (2010) · Zbl 1189.53077 · doi:10.1007/s10455-009-9181-9 |
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