Harmonicity and minimality of complex and quaternionic radial foliations
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José Carmelo González-Dávila
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.76.168Suche in Google Scholar
[2] J. Berndt, Über Untermannigfaltigkeiten von komplexen Raumformen, Ph.D. Thesis, University of Cologne, 1989. Suche in Google Scholar
[3] J. Berndt, Real hypersurfaces in quaternionic space forms, J. Reine Angew. Math. 419 (1991), 9–26. 10.1515/crll.1991.419.9Suche 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. Suche 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. Suche in Google Scholar
[6] A. L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb. (3) 10, Springer, Berlin, 1987. 10.1007/978-3-540-74311-8Suche 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:1021259000387Suche 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:1010687231629Suche in Google Scholar
[9] J. Bolton, Transnormal systems, Quart. J. Math. Oxford Ser. (2) 24 (1973), 385–395. 10.1093/qmath/24.1.385Suche 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/1113246937Suche 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/BF02803502Suche 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/777Suche in Google Scholar
[13] J. C. González-Dávila, Energy of generalized distributions, Differential Geom. Appl. 49 (2016), 510–528. 10.1016/j.difgeo.2016.09.009Suche 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/S0305004108001709Suche in Google Scholar
[15] A. Gray, Tubes, Addison-Wesley, Reading, 1990. Suche 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.2012.10.005Suche in Google Scholar
[17] P. Molino, Riemannian Foliations, Progr. Math. 73, Birkhäuser, Boston, 1988. 10.1007/978-1-4684-8670-4Suche 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.2016.107Suche 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/s3-29.4.699Suche 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/S0002-9947-1973-0321133-2Suche in Google Scholar
[21] L. Vanhecke, Geometry in normal and tubular neighborhoods, Rend. Sem. Fac. Sci. Univ. Cagliari 58 (1988), 73–176. Suche in Google Scholar
[22] Q. M. Wang, Isoparametric functions on Riemannian manifolds. I, Math. Ann. 277 (1987), no. 4, 639–646. 10.1007/BF01457863Suche 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)90052-3Suche 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)00021-4Suche in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras
- Positive solutions for nonlinear nonhomogeneous parametric Robin problems
- Period relations for cusp forms of GSp4
- A generalization of a graph theory Mertens’ theorem: Galois covering case
- Commutators of the fractional integrals for second-order elliptic operators on Morrey spaces
- Vojta’s conjecture on rational surfaces and the abc conjecture
- The least unramified prime which does not split completely
- The ω-inequality problem for concatenation hierarchies of star-free languages
- Higher weight on GL(3). I: The Eisenstein series
- Fourier transforms of powers of well-behaved 2D real analytic functions
- Parabolic conformally symplectic structures I; definition and distinguished connections
- The quasi-arithmetic means and Cartan barycenters of compactly supported measures
- Estimates of lattice points in the discriminant aspect over abelian extension fields
- On Hecke eigenvalues of Siegel modular forms in the Maass space
- Harmonicity and minimality of complex and quaternionic radial foliations
Artikel in diesem Heft
- Frontmatter
- A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras
- Positive solutions for nonlinear nonhomogeneous parametric Robin problems
- Period relations for cusp forms of GSp4
- A generalization of a graph theory Mertens’ theorem: Galois covering case
- Commutators of the fractional integrals for second-order elliptic operators on Morrey spaces
- Vojta’s conjecture on rational surfaces and the abc conjecture
- The least unramified prime which does not split completely
- The ω-inequality problem for concatenation hierarchies of star-free languages
- Higher weight on GL(3). I: The Eisenstein series
- Fourier transforms of powers of well-behaved 2D real analytic functions
- Parabolic conformally symplectic structures I; definition and distinguished connections
- The quasi-arithmetic means and Cartan barycenters of compactly supported measures
- Estimates of lattice points in the discriminant aspect over abelian extension fields
- On Hecke eigenvalues of Siegel modular forms in the Maass space
- Harmonicity and minimality of complex and quaternionic radial foliations
