Abstract
We construct special classes of totally geodesic almost regular foliations, namely, complex radial foliations in Hermitian manifolds and quaternionic radial foliations in quaternionic Kähler manifolds, and we give criteria for their harmonicity and minimality. Then examples of these foliations on complex and quaternionic space forms, which are harmonic and minimal, are presented.
Funding statement: Supported by D.G.I. (Spain) Project MTM2016-77093-P.
References
[1] T. Adachi and S. Maeda, Some characterizations of quaternionic space forms, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), no. 10, 168–172. 10.3792/pjaa.76.168Search in Google Scholar
[2] J. Berndt, Über Untermannigfaltigkeiten von komplexen Raumformen, Ph.D. Thesis, University of Cologne, 1989. Search in Google Scholar
[3] J. Berndt, Real hypersurfaces in quaternionic space forms, J. Reine Angew. Math. 419 (1991), 9–26. 10.1515/crll.1991.419.9Search in Google Scholar
[4] J. Berndt, Riemannian geometry of complex two-plane Grassmannians, Rend. Semin. Mat. Univ. Politec. Torino 55 (1997), no. 1, 19–83. Search in Google Scholar
[5] J. Berndt, On homogeneous hypersurfaces in Riemannian symmetric spaces, Proceedings of the Second International Workshop on Differential Geometry (Taegu 1997), Kyungpook National University, Taegu (1998), 17–34. Search in Google Scholar
[6] A. L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb. (3) 10, Springer, Berlin, 1987. 10.1007/978-3-540-74311-8Search in Google Scholar
[7] E. Boeckx, J. C. González-Dávila and L. Vanhecke, Energy of radial vector fields on compact rank one symmetric spaces, Ann. Global Anal. Geom. 23 (2003), no. 1, 29–52. 10.1023/A:1021259000387Search in Google Scholar
[8] E. Boeckx and L. Vanhecke, Harmonic and minimal radial vector fields, Acta Math. Hungar. 90 (2001), no. 4, 317–331. 10.1023/A:1010687231629Search in Google Scholar
[9] J. Bolton, Transnormal systems, Quart. J. Math. Oxford Ser. (2) 24 (1973), 385–395. 10.1093/qmath/24.1.385Search in Google Scholar
[10] B.-Y. Choi and J.-W. Yim, Distributions on Riemannian manifolds, which are harmonic maps, Tohoku Math. J. (2) 55 (2003), no. 2, 175–188. 10.2748/tmj/1113246937Search in Google Scholar
[11] O. Gil-Medrano, J. C. González-Dávila and L. Vanhecke, Harmonicity and minimality of oriented distributions, Israel J. Math. 143 (2004), 253–279. 10.1007/BF02803502Search in Google Scholar
[12] J. C. González-Dávila, Harmonicity and minimality of distributions on Riemannian manifolds via the intrinsic torsion, Rev. Mat. Iberoam. 30 (2014), no. 1, 247–275. 10.4171/RMI/777Search in Google Scholar
[13] J. C. González-Dávila, Energy of generalized distributions, Differential Geom. Appl. 49 (2016), 510–528. 10.1016/j.difgeo.2016.09.009Search in Google Scholar
[14] J. C. González-Dávila and F. Martín Cabrera, Harmonic G-structures, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 2, 435–459. 10.1017/S0305004108001709Search in Google Scholar
[15] A. Gray, Tubes, Addison-Wesley, Reading, 1990. Search in Google Scholar
[16] R. Miyaoka, Transnormal functions on a Riemannian manifold, Differential Geom. Appl. 31 (2013), no. 1, 130–139. 10.1016/j.difgeo.2012.10.005Search in Google Scholar
[17] P. Molino, Riemannian Foliations, Progr. Math. 73, Birkhäuser, Boston, 1988. 10.1007/978-1-4684-8670-4Search in Google Scholar
[18] K. Niedziałomski, Geometry of G-structures via the intrinsic torsion, SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 107. 10.3842/SIGMA.2016.107Search in Google Scholar
[19] P. Stefan, Accessible sets, orbits, and foliations with singularities, Proc. Lond. Math. Soc. (3) 29 (1974), 699–713. 10.1112/plms/s3-29.4.699Search in Google Scholar
[20] H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171–188. 10.1090/S0002-9947-1973-0321133-2Search in Google Scholar
[21] L. Vanhecke, Geometry in normal and tubular neighborhoods, Rend. Sem. Fac. Sci. Univ. Cagliari 58 (1988), 73–176. Search in Google Scholar
[22] Q. M. Wang, Isoparametric functions on Riemannian manifolds. I, Math. Ann. 277 (1987), no. 4, 639–646. 10.1007/BF01457863Search in Google Scholar
[23] C. M. Wood, A class of harmonic almost-product structures, J. Geom. Phys. 14 (1994), no. 1, 25–42. 10.1016/0393-0440(94)90052-3Search in Google Scholar
[24] C. M. Wood, Harmonic sections of homogeneous fibre bundles, Differential Geom. Appl. 19 (2003), no. 2, 193–210. 10.1016/S0926-2245(03)00021-4Search in Google Scholar
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