Critical mappings of Riemannian manifolds

Bleecker, David D.

doi:10.1090/S0002-9947-1979-0539921-1

Critical mappings of Riemannian manifolds
HTML articles powered by AMS MathViewer

by David D. Bleecker

Trans. Amer. Math. Soc. 254 (1979), 319-338

DOI: https://doi.org/10.1090/S0002-9947-1979-0539921-1

PDF | Request permission

Abstract:

We consider maps, from one Riemannian manifold to another, which are critical for all invariantly defined functionals on the space of maps. There are many such critical mappings, perhaps too numerous to suitably classify, although a characterization of sorts is provided. They are proven to have constant rank, with the image being a homogeneous minimal submanifold of the target manifold. Critical maps need not be Riemannian submersions onto their images. Also, there are homogeneous spaces for which the identity map is not critical. Many open problems remain.

References

Le spectre d’une variété riemannienne

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, DOI 10.4310/jdg/1214429060
David D. Bleecker, Critical Riemannian manifolds, J. Differential Geometry 14 (1979), no. 4, 599–608 (1981). MR 600616
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
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
Richard H. Escobales Jr., Riemannian submersions with totally geodesic fibers, J. Differential Geometry 10 (1975), 253–276. MR 370423
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
Wu-yi Hsiang and H. Blaine Lawson Jr., Minimal submanifolds of low cohomogeneity, J. Differential Geometry 5 (1971), 1–38. MR 298593
J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331, DOI 10.1515/9781400881802
Barrett O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469. MR 200865
Joseph A. Wolf, The goemetry and structure of isotropy irreducible homogeneous spaces, Acta Math. 120 (1968), 59–148. MR 223501, DOI 10.1007/BF02394607