A characterization of harmonic foliations by variations of the metric

Hvidsten, Michael D.; Tondeur, Philippe

doi:10.1090/S0002-9939-1986-0854047-4

A characterization of harmonic foliations by variations of the metric
HTML articles powered by AMS MathViewer

by Michael D. Hvidsten and Philippe Tondeur

Proc. Amer. Math. Soc. 98 (1986), 359-362

DOI: https://doi.org/10.1090/S0002-9939-1986-0854047-4

PDF | Request permission

Abstract:

In this paper it is shown that a harmonic Riemannian foliation of a Riemannian manifold can be characterized as being a critical point of the energy of the foliation under certain variations of the manifold’s Riemannian metric. These variations are those induced by the flows of vector fields on the manifold.

References

M. Berger and D. Ebin, Some decompositions of the space of symmetric tensors on a Riemannian manifold, J. Differential Geometry 3 (1969), 379–392. MR 266084
David G. Ebin, The manifold of Riemannian metrics, Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 11–40. MR 0267604
J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1–68. MR 495450, DOI 10.1112/blms/10.1.1
James Eells Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 164306, DOI 10.2307/2373037
Franz W. Kamber and Philippe Tondeur, Harmonic foliations, Harmonic maps (New Orleans, La., 1980) Lecture Notes in Math., vol. 949, Springer, Berlin-New York, 1982, pp. 87–121. MR 673585
Bruce L. Reinhart, Foliated manifolds with bundle-like metrics, Ann. of Math. (2) 69 (1959), 119–132. MR 107279, DOI 10.2307/1970097
Dennis Sullivan, A homological characterization of foliations consisting of minimal surfaces, Comment. Math. Helv. 54 (1979), no. 2, 218–223. MR 535056, DOI 10.1007/BF02566269