Polynomial harmonic morphisms between Euclidean spheres
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- by James Eells and Paul Yiu
- Proc. Amer. Math. Soc. 123 (1995), 2921-2925
- DOI: https://doi.org/10.1090/S0002-9939-1995-1273489-4
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Abstract:
A characterization is given of the harmonic morphisms between euclidean spheres whose component functions are harmonic homogeneous polynomials of the same degree, and also of polynomial harmonic morphisms between euclidean spaces which map spheres into spheres. These turn out to be isometric to the classical Hopf fibrations.References
- Paul Baird, Harmonic maps with symmetry, harmonic morphisms and deformations of metrics, Research Notes in Mathematics, vol. 87, Pitman (Advanced Publishing Program), Boston, MA, 1983. MR 716320
- Paul Baird, Harmonic morphisms and circle actions on $3$- and $4$-manifolds, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 1, 177–212 (English, with French summary). MR 1056781, DOI 10.5802/aif.1210
- Paul Baird and Andrea Ratto, Conservation laws, equivariant harmonic maps and harmonic morphisms, Proc. London Math. Soc. (3) 64 (1992), no. 1, 197–224. MR 1132860, DOI 10.1112/plms/s3-64.1.197
- Paul Baird and John C. Wood, Bernstein theorems for harmonic morphisms from $\textbf {R}^3$ and $S^3$, Math. Ann. 280 (1988), no. 4, 579–603. MR 939920, DOI 10.1007/BF01450078
- William Browder, Higher torsion in $H$-spaces, Trans. Amer. Math. Soc. 108 (1963), 353–375. MR 155326, DOI 10.1090/S0002-9947-1963-0155326-8
- James Eells Jr. and Nicolaas H. Kuiper, An invariant for certain smooth manifolds, Ann. Mat. Pura Appl. (4) 60 (1962), 93–110. MR 156356, DOI 10.1007/BF02412768
- 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 —, Topics in harmonic maps, CBMS Regional Conf. Ser. in Math., vol. 50, Amer. Math. Soc., Providence, RI, 1983.
- J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), no. 5, 385–524. MR 956352, DOI 10.1112/blms/20.5.385
- James Eells and Andrea Ratto, Harmonic maps and minimal immersions with symmetries, Annals of Mathematics Studies, vol. 130, Princeton University Press, Princeton, NJ, 1993. Methods of ordinary differential equations applied to elliptic variational problems. MR 1242555, DOI 10.1515/9781400882502
- Bent Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 2, vi, 107–144 (English, with French summary). MR 499588 G. Gigante, A note on harmonic morphisms, preprint, 1983.
- Agnes Chi Ling Hsu, A characterization of the Hopf map by stretch, Math. Z. 129 (1972), 195–206. MR 312519, DOI 10.1007/BF01187348
- Tôru Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ. 19 (1979), no. 2, 215–229. MR 545705, DOI 10.1215/kjm/1250522428
- H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, and R. Remmert, Numbers, Graduate Texts in Mathematics, vol. 123, Springer-Verlag, New York, 1990. With an introduction by K. Lamotke; Translated from the second German edition by H. L. S. Orde; Translation edited and with a preface by J. H. Ewing; Readings in Mathematics. MR 1066206, DOI 10.1007/978-1-4612-1005-4
- R. Wood, Polynomial maps from spheres to spheres, Invent. Math. 5 (1968), 163–168. MR 227999, DOI 10.1007/BF01425547
- Paul Y. H. Yiu, Quadratic forms between spheres and the nonexistence of sums of squares formulae, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 3, 493–504. MR 857724, DOI 10.1017/S0305004100066226
Bibliographic Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 2921-2925
- MSC: Primary 58E20; Secondary 55R25
- DOI: https://doi.org/10.1090/S0002-9939-1995-1273489-4
- MathSciNet review: 1273489