Abstract
We review some basic facts and examples of harmonic (1,1)- and (1,2)-tensor fields. This leads to the consideration of vector fields defining harmonic sections of the tangent bundle. Finally, by the study of the musical isomorphisms, a correspondence is obtained between vector fields and one-forms defining harmonic sections of the tangent and the cotangent bundle, respectively.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ambrose, W. and Singer, I.M., On homogeneous Riemannian manifolds, Duke Math. J. 25 (1958), 647–669.
Besse, A.L., Einstein manifolds, Ergeb. Math. Grenzgeb. 3. Folge 10, Springer-Verlag, Berlin, Heidelberg, New York, 1982.
Bonome, A., Castro, R., García-Río, E., Hervella, L. and Matsushita, Y., Almost complex manifolds with holomorphic distributions, Rend. Mat. 14 (1994), 567–589.
Cendán-Verdes, J., García-Río, E. and Vázquez-Abal, M.E., On the semi-Riemannian structure of the tangent bundle of a two-point homogeneous space, Riv. Mat. Univ. Parma (5) 3 (1994), 253–270.
Chen, B.Y. and Nagano, T, Harmonic metrics, harmonic tensors, and Gauss maps, J. Math. Soc. Japan 36 (1984), 295–313.
Chen, B.Y. and Vanhecke, L., Differential geometry of geodesic spheres, J. Reine Angew. Math. 325 (1981), 28–67.
Eells, J. and Lemaire, L., A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1–68.
Eells, J. and Lemaire, L., Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385–524.
Eells, J. and Sampson, J.H., Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160.
Gadea, P.M. and Oubiña, J.A., Homogenous pseudo-Riemannian structures and homogeneous almost para-Hermitian structures, Houston J. Math. 18 (1992), 449–465.
Gadea, P.M. and Oubiña, J.A., Reductive homogenous pseudo-Riemannian manifolds, Monatsh. Math. 124 (1997), 17–34.
García-Río, E., Vanhecke, L. and Vázquez-Abal, M.E., Tangent bundles of order r and harmonicity of induced maps, Boll. Un. Mat. Ital., to appear.
García-Río, E., Vanhecke, L. and Vázquez-Abal, M.E., Harmonic endomorphism fields, Illinois J. Math. 41 (1997), 23–30.
García-Río, E., Vanhecke, L. and Vázquez-Abal, M.E., Harmonic connections, Acta Sci. Math. (Szeged) 62 (1996), 61–83.
Gil-Medrano, O., Geometric properties of some classes of almost-product manifolds, Rend. Circ. Mat. Palermo 32 (1983), 315–329.
Gray, A. and Hervella, L., The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. 123 (1980), 35–58.
Grifone, J. Structure presque-tangente et connexions I, II, Ann. Inst. Fourier (Grenoble) 22, 1 (1972), 287–334 and 22, 3 (1972), 291-338.
Kamber, F. and Tondeur, Ph., Harmonic foliations, Lecture Notes in Math. 949, Springer-Verlag, Berlin, Heidelberg, New York, 1982, 87–121.
Konderak, J.J., Construction of harmonic maps between pseudo-Riemannian spheres and hyperbolic spaces, Proc. Amer. Math. Soc. 109 (1990), 469–476.
de León, M. and Rodriguez, P., Methods of differential geometry in analytical mechanics, Mathematics Studies 159, North-Holland, Amsterdam, 1989.
Nouhaud, O., Applications harmoniques d’une variété riemannienne dans son fibré tangent. Généralisation, C. R. Acad. Sci. Paris 284 (1977), 815–818.
Ohnita, Y. and Udagawa, S., Complex-analyticity of pluriharmonic maps and their constructions, in Prospects in Complex Geometry, Proc. Katata/Kyoto 1989, (Eds. J. Noguchi, T. Ohsawa), Lecture Notes in Math. 1468, Springer-Verlag, Berlin, Heidelberg, New York, 1991, 371–407.
Oproiu, V., On the harmonic sections of cotangent bundles, Rend. Sem. Fac. Sci. Univ. Cagliari 59 (1989), 177–184
Oproiu, V., Harmonic maps between tangent bundles, Rend. Sem. Mat. Sci. Univ. Politecn. Torino 47 (1989), 47–55
Oproiu, V., Some aspects from the geometry of the cotangent bundle, preprint, 1996.
Patterson, E.M. and Walker, A.G., Riemann extensions, Quart. J. Math. Oxford 2 (1952), 85–94.
Poor, W.A., Differential geometric structures, McGraw-Hill Inc., New York, 1981.
Reinhart, B.L., Differential geometry of foliations, Ergeb. Math. Grenzgeb. 99, Springer-Verlag, Berlin, Heidelberg, New York, 1983.
Tondeur, Ph., Foliations on Riemannian manifolds, Universitext, Springer-Verlag, Berlin, Heidelberg, New York, 1988.
Tondeur, Ph. and Vanhecke, L., Harmonicity of a foliation and of an associated map, Bull. Austral. Math. Soc. 54 (1996), 241–246.
Tricerri, F. and Vanhecke, L., Homogeneous structures on Riemannian manifolds, London Math. Soc. Lecture Note Series 83, Cambridge Univ. Press, Cambridge, 1983.
Tricerri, F. and Vanhecke, L., Special homogeneous structures on Riemannian manifolds, Topics in Differential Geometry I,II(Eds. J. Szenthe and L. Tamassy), Colloquia Math. Societatis J. Bolyai 64, North-Holland, Amsterdam, 1988, 1211–1246.
Vázquez-Abal, M.E., Harmonicity on the tangent bundle of order r, C. R. Acad. Sci. Paris 312 (1991), 131–136.
Watanabe, Y. and Dohira, R., Remarks on harmonic metrics, harmonic tensors and holomorphic vector fields in a Kähler manifold, Math. J. Toyama Univ. 18 (1995), 137–146.
Wood, C.M., A class of harmonic almost-product structures, J. Geom. Phys. 14 (1994), 25–42.
Wood, C.M., Harmonic almost-complex structures, Compositio Math. 99 (1995), 183–212.
Yano, K., Integral formulas in Riemannian geometry, Pure and Appl. Math. 16, Marcel Dekker, New York, 1973.
Yano, K. and Ishihara, S., Tangent and cotangent bundles, Pure and Appl. Math. 16, Marcel Dekker, New York, 1973.
Yano, K. and Nagano, T., On geodesic vector fields in a compact orientable Riemannian space, Comm. Math. Helv. 35 (1961), 55–64.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
García-Río, E., Vanhecke, L., Vázquez-Abal, M.E. (1999). Notes on Harmonic Tensor Fields. In: Szenthe, J. (eds) New Developments in Differential Geometry, Budapest 1996. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5276-1_8
Download citation
DOI: https://doi.org/10.1007/978-94-011-5276-1_8
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6220-6
Online ISBN: 978-94-011-5276-1
eBook Packages: Springer Book Archive