Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-26T20:28:58.577Z Has data issue: false hasContentIssue false
  • English
  • Français

SOME REMARKS ON TRANSVERSALLY HARMONIC MAPS

Published online by Cambridge University Press:  01 January 2008

JERZY J. KONDERAK  and
ROBERT WOLAK
JERZY J. KONDERAK
Affiliation:
Instutyt Matematyki, Uniwersytet Jagielloński, ul. Reymonta 4, 30059 Kraków, Poland e-mail: robert.wolak@im.uj.edu.pl
ROBERT WOLAK
Affiliation:
Instutyt Matematyki, Uniwersytet Jagielloński, ul. Reymonta 4, 30059 Kraków, Poland e-mail: robert.wolak@im.uj.edu.pl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider transversally harmonic foliated maps between two Riemannian manifolds equipped with Riemannian foliations. We give various characterisations of such maps and we study the relation between the properties ‘harmonic’ and ‘transversally harmonic’ for a given map. We also consider these problems for particular classes of manifolds: manifolds with transversally almost Hermitian foliations and Riemannian flows.

Keywords

53C1258E20
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 1 , January 2008 , pp. 1 - 16
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Baird, P. and Wood, J. C., Harmonic morphisms between Riemannian manifolds (Oxford Univesity Press, 2003).CrossRefGoogle Scholar
2.Barletta, E. and Dragomir, S., On transversally holomorphic maps of Kählerian foliations, Acta Appl. Math. 54 (1998), 121–134.CrossRefGoogle Scholar
3.Bierstone, E., The structure of orbit spaces and the singularities of equivariant mappings, Mon. de Mat. 35 (IMPA, Rio de Janeiro, 1980).Google Scholar
4.Black, M.: Harmonic maps into homogenous spaces, Pitman research notes in math., Longman Scientific and Technical, Harlow, 1991Google Scholar
5.Cairns, G., A general description of totally geodesic foliations, That o huku Math. J. 38 (1986), 37–55.Google Scholar
6.Cairns, G. and Ghys, E., Totally geodesic foliations on 4-manifolds, J. Diff. Geom. 23 (1986), 241–254.Google Scholar
7.Blair, D.E.: Riemannian geometry of contact and sym-plec-tic ma-ni-folds, Birk-häu-ser, Basel, 2001CrossRefGoogle Scholar
8.Boualem, H., Molino, P.: Modèle locaux saturés de feuilletages riemanniens singuliers, C. R. Acad. Sci. Paris 316 (1993), 913–916Google Scholar
9.Cannas, A. da Silva: Symplectic Geometry, LNM 1764, Springer 2001Google Scholar
10.Cordero, L. A., Wolak, R.: Examples of foliations with foliated geometric structures, Pacific J. Math. 142, 2 1990, 265-276CrossRefGoogle Scholar
11.Dominguez, D., Finiteness and tenseness theorems for Riemannian foliations, Amer. J. Math. 120 (1998), 1237–1276.CrossRefGoogle Scholar
12.Dombrowski, P., On the geometry of the tangent bundles, J. Reine Angew. Math. 210 (1962), 73–88.CrossRefGoogle Scholar
13.Edwards, R., Millett, K. and Sullivan, D., Foliations with all leaves compact, Topology 16 (1977), 13–32.CrossRefGoogle Scholar
14.Eells, J. and Fuglede, B., Harmonic maps between Riemannian polyhedra (Cambridge University Press, 2001).Google Scholar
15.Eells, J. and Lemaire, L., A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1–68.CrossRefGoogle Scholar
16.Eells, J. and Lemaire, L., Selected topics in harmonic maps, C.B.M.S., Regional Conference Series, 57 (AMS., Providence, R.I. 1983).CrossRefGoogle Scholar
17.Eells, J. and Lemaire, L., Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385–524.CrossRefGoogle Scholar
18.Eells, J., Verjovsky, A.: Harmonica and Riemannian foliations, Bol. Soc. Mat. Mexicana (3) (1998), 1–12Google Scholar
19.Eells, J. and Sampson, J.H., Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160.CrossRefGoogle Scholar
20.Goresky, M. and MacPherson, R., Intersection homology theory, Topology 19 (1980), 135–165.CrossRefGoogle Scholar
21.Kacimi–Alaoui, A. El: Opèrateur transversalement elliptic sur un feuilletage riemannien et applications, Comp. Math. 73, (1990), 57–106Google Scholar
22.Fujimoto, A.: Theory of G–structures, Tokyo, 1972Google Scholar
23.Gauduchon, P.: Alcuni spunti di geometria quasi hermitiana e hermitiana, Quaderni del seminario di Top. Algerbrica e Diff., Univ. Rome, 1983Google Scholar
24.Ghys, E., Classification des feuilletages totalement géodésiques de codimension un, Comm. Math. Helv. 58 (1983), 543–572.CrossRefGoogle Scholar
25.Gray, A. and Hervella, L., The sixteen classes of almost Hermitian manifolds and their linear invariants, Annali Mat. Pura Appl. 123 (1980), 35–58.CrossRefGoogle Scholar
26.Helgason, S., Differential geometry of symmetric spaces (Academic Press, 1962).Google Scholar
27.Ianuš, S.: Sulle strutture canoniche dello spazio fibrato tangente di una varietà riemanniana, Rendiconti di Matematica, Univ. di Roma, VI, 6, fasc.1 (1973), 75–96Google Scholar
28.Hermann, R., A sufficient condition that a mapping of Riemannian manifolds be a fiber bundle, Proc. AMS 11 (1960), 236–242Google Scholar
29.Ishihara, T., A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ. 19 (1979), 215–229.Google Scholar
30.Józefowicz, M. and Wolak, R., A few remarks on the geometry of the space of leaf closures of a Riemannian foliation, in Proceedings of a year in Differential Geometry University of Maryland (Banach Center Publications), Vol. 76 (2007), 395–409.Google Scholar
31.Kamber, F. and Tondeur, Ph., Foliations and metrics, Proceedings of a Year in Differential Geometry, University of Maryland (Birkhäuser, 1983), 103–152Google Scholar
32.Kamber, F. and Tondeur, Ph., Duality theorems for foliations, Astérisque 116 (1984), 108–116.Google Scholar
33.Kobayashi, S., Transformation groups in differential geometry (Springer, Verlag, 1972).CrossRefGoogle Scholar
34.Konderak, J.J., On natural first order Lagrangians, Bull. London Math. Soc. 23 (1991), 169–174CrossRefGoogle Scholar
35.Konderak, J. and Wolak, R., On transversally harmonic maps between manifolds with Riemannian foliations, Quart. J. Math. Oxford Ser. (2) 54 (2003), 335–354.CrossRefGoogle Scholar
36.Kowalski, O., Curvature of the Riemannian metric on the tangent bundle of a Riemannian manifold, J. Reine und Angew. Math. 220 (1971), 124–129.Google Scholar
37.Lichnerowicz, A., Applications harmoniques et variétés kähleriennes, Sympos. Math. 3 Bologna (1970), 341–402.Google Scholar
38.Masa, X., Duality and minimality in Riemannian foliations, Comm. Math. Helv. 67 (1992), 17–27.CrossRefGoogle Scholar
39.Miquel, V. and Wolak, R., Minimal singular Riemannian foliations, CRAS 342 (2006), 33–37.Google Scholar
40.Molino, P., Riemannian foliations (Birkhäuser, 1988).CrossRefGoogle Scholar
41.Molino, P.: Orbit–like foliations, Geometric Study of Foliations, Proc. Tokyo 1993, World Scientific, Singapore, (1994), 97–119Google Scholar
42.Moore, C. and Schochet, C., Global analysis on foliated spaces, (Springer-Verlag, 1988).CrossRefGoogle Scholar
43.Mostow, M., Continuous cohomology of spaces with two topologies, Mem. Amer Math. Soc. 7 (1976), no. 175.Google Scholar
44.Pierrot, M.: Orbites des champs feuilletés pour un feuilletages riemanniens sur une variété compacte, C. R. Acad. Sc. Paris 301 (1985), 443–445Google Scholar
45.Rawnsley, J. H., f–structures, f–twistor spaces and harmonic maps, in Geometry Seminar Luigo Bianchi II, 1984, Lecture Notes in Mathematics 1164 (Springer-Verlag, 1985), 85–159CrossRefGoogle Scholar
46.Sanini, A., Applicazioni armoniche tra i fibrati tangenti di varietá riemanniane, Bol. U.M.I. 6 (2A), (1983), 55–63.Google Scholar
47.Tondeur, P., Geometry of foliations (Birkhäuser, 1997).CrossRefGoogle Scholar
48.Wolak, R., Foliated and associated geometric structures on foliated manifolds, Ann. Fac. Sc. Toulouse 10 (1989), 337–360.CrossRefGoogle Scholar
49.Wolak, R.: The structure tensor of a transverse G-structure on a foliated manifold, Boll. U.M.I. 4-A, (1990), 1–15Google Scholar
50.Wolak, R., Geometric structures on foliated manifolds (Santiago de Compostela, 1989).Google Scholar
51.Wolak, R., Pierrot's theorem for singular Riemannian foliations, Publ. Mat. 38 (1994), 433–439.CrossRefGoogle Scholar
52.Xin, Y. L., Riemannian submersion and equivariant harmonic maps, in Proceedings of the Symposium an Differential Geometry in honour of S.Buchin (World Scientific Publ., Singapore, 1993), 272–287.Google Scholar
53.Xin, Y.L., Geometry of harmonic maps (Birkhäuser, 1996).CrossRefGoogle Scholar
54.Yano, K.: On a structure defined by a tensor field of type (1, 1) satisfying f 3 + f = 0, Tensor, N.S., (1963), 99–109Google Scholar
55.Yano, K., Ishihara, S.: On integrability conditions of a structure f satysfying f 3 + f=0, Quart. J. Math., 15 (1964), 217–222CrossRefGoogle Scholar