Quadratic harmonic morphisms and O-systems

Ou, Ye-Lin

Annales de l'Institut Fourier, Tome 47 (1997), p. 687-713 / Harvested from Numdam

Résumé

Nous présentons les O-systèmes (Définition 3.1) des transformations orthogonales de ${\Bbb R}^m$ et nous établissons des correspondances à la fois entre les classes d’équivalence des systèmes de Clifford et celles des O-systèmes et les multiplications orthogonales de la forme $\mu :{\Bbb R}^n\times {\Bbb R}^m\rightarrow {\Bbb R}^m$, ce qui nous permet de résoudre les problèmes d’existence simultanément pour les O-systèmes et pour les morphismes harmoniques quadratiques ombilicaux. Le problème d’existence pour les morphismes quadratiques harmoniques généraux est alors résolu par le “Splitting Lemma” . Nous avons également étudié les propriétés possédées par tous les morphismes harmoniques quadratiques pour les paires fixes d’espaces de domaines et co-domaines.

Abstract

We introduce O-systems (Definition 3.1) of orthogonal transformations of ${\Bbb R}^m$, and establish ${\Bbb R}$ correspondences both between equivalence classes of Clifford systems and those of O-systems, and between O-systems and orthogonal multiplications of the form $\mu : {\Bbb R}^n\times {\Bbb R}^m\rightarrow {\Bbb R}^m$, which allow us to solve the existence problems both for $O$-systems and for umbilical quadratic harmonic morphisms simultaneously. The existence problem for general quadratic harmonic morphisms is then solved by the Splitting Lemma. We also study properties possessed by all quadratic harmonic morphisms for fixed pairs of domain and range spaces.

Détail

Publié le : 1997-01-01

MR 98j:58038 | Zbl 0918.58020

DOI : https://doi.org/10.5802/aif.1578

Bibliographie

[1] P. Baird, Harmonic maps with symmetry, harmonic morphisms, and deformation of metrics, Pitman Res. Notes Math. Ser., vol. 87, Pitman, Boston, London, Melbourne, 1983. | MR 85i:58038 | Zbl 0515.58010

[2] P. Baird and J.C. Wood, Bernstein theorems for harmonic morphisms from ℝ3 and S3, Math. Ann., 280 (1988), 579-603. | MR 90e:58027 | Zbl 0621.58011

[3] P. Baird and J.C. Wood, Harmonic morphisms and conformal foliations by geodesics of three-dimensional space forms, J. Austral. Math. Soc., Ser. A, 51 (1991), 118-153. | MR 92k:53048 | Zbl 0744.53013

[4] P. Baird and J.C. Wood, Harmonic morphisms, Seifert fibre spaces and conformal foliations, Proc. London Math. Soc., 64 (1992), 170-196. | Zbl 0755.58019

[5] P. Baird and J.C. Wood, Hermitian structures and harmonic morphisms on higher dimensional Euclidean spaces, Internat. J. Math., 6 (1995), 161-192. | MR 96a:58068 | Zbl 0823.58010

[6] A. Bernard, E.A. Campbell, and A.M. Davie, Brownian motion and generalized analytic and inner functions, Ann. Inst. Fourier (Grenoble), 29-1 (1979), 207-228. | Numdam | MR 81b:30088 | Zbl 0386.30029

[7] E. Cartan, Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann. Mat. Pura Appl., 17 (1938), 177-191. | JFM 64.1361.02 | Zbl 0020.06505

[8] L. Conlon, Differentiable Manifolds, A first course, Basler Lehrbücher, Berlin, 1993. | MR 94d:58001 | Zbl 0770.57001

[9] B. Eckmann, Beweis des Satzes von Hurwitz-Radon, Comment. Math. Helvet., 15 (1952), 358-366. | Zbl 0028.10402

[10] J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc., 10 (1978), 1-68. | MR 82b:58033 | Zbl 0401.58003

[11] J. Eells and L. Lemaire, Selected topics in harmonic maps, CBMS Regional Conf. Ser. in Math., vol. 50, Amer. Math. Soc., Providence, R.I., 1983. | MR 85g:58030 | Zbl 0515.58011

[12] J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc., 20 (1988), 385-524. | MR 89i:58027 | Zbl 0669.58009

[13] J. Eells and A. Ratto, Harmonic maps and minimal immersions with symmetries, Ann. of Math. Stud., vol. 130, Princeton University Press, 1993. | Zbl 0783.58003

[14] J. Eells and P. Yiu, Polynomial harmonic morphisms between Euclidean spheres, Proc. Amer. Math. Soc., vol. 123, 9 (1995), 2921-2925. | MR 95k:58048 | Zbl 0853.58036

[15] D. Ferus, H. Karcher, and H.F. Münzner, Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z., 177 (1981), 479-502. | Zbl 0452.53032

[16] B. Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble), 28-2 (1978), 107-144. | Numdam | MR 80h:58023 | Zbl 0339.53026

[17] S. Gudmundsson, Harmonic morphisms from quaternionic projective spaces, Geom. Dedicata, 56 (1995), 327-332. | MR 96d:58033 | Zbl 0834.58016

[18] S. Gudmundsson, Harmonic morphisms between spaces of constant curvature, Proc. Edinburgh Math. Soc., 36 (1992), 133-143. | MR 93j:58034 | Zbl 0790.58012

[19] S. Gudmundsson, Harmonic morphisms from complex projective spaces, Geom. Dedicata, 53 (1994), 155-161. | MR 95j:58034 | Zbl 0826.53028

[20] S. Gudmundsson and R. Sigurdsson, A note on the classification of holomorphic harmonic morphisms, Potential Analysis, 2 (1993), 295-298. | MR 94i:58043 | Zbl 0783.58015

[21] A. Hurwitz, Über die Komposition der quadratischen Formen, Math. Ann., 88 (1923), 1-25. | JFM 48.1164.03

[22] D. Husemoller, Fibre Bundles, McGraw Hill, New York, 1966. | MR 37 #4821 | Zbl 0144.44804

[23] T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ., 19 (1979), 215-229. | MR 80k:58045 | Zbl 0421.31006

[24] P. Levy, Processus Stochastiques et Mouvement Brownien, Gauthier-Villard, Paris, 1948. | Zbl 0034.22603

[25] H.F. Münzner, Isoparametrische Hyperflächen in Sphären, I, Math. Ann., 251 (1980), 57-71. | Zbl 0432.53033

[26] K. Nomizu, Elie Cartan's work on isoparametric families of hypersurfaces, in Differential Geometry, S.S. Chern and R. Osserman ed., Proc. Sympos. Pure Math., vol. 27, Amer. Math. Soc., Providence, R.I., 1975, 191-200. | MR 54 #11240 | Zbl 0318.53052

[27] Y.-L. Ou, Complete lifts of harmonic maps and morphisms between Euclidean spaces, Contributions to Algebra and Geometry, vol. 37 (1996), 31-40. | MR 97h:58050 | Zbl 0859.53037

[28] Y.-L. Ou, O-systems, orthogonal multiplications and isoparametric functions, Guangxi University for Nationalities, preprint, 1996.

[29] Y.-L. Ou, On constructions of harmonic morphisms into Euclidean spaces, J. Guangxi University for Nationalities, vol. 2 (1996), 1-6.

[30] Y.-L. Ou and J.C. Wood, On the classification of quadratic harmonic morphisms between Euclidean spaces, Algebras, Groups and Geometries, vol. 13 (1996), 41-53. | MR 97d:58063 | Zbl 0872.58022

[31] H. Ozeki and M. Takeuchi, On some types of isoparametric hypersurfaces in spheres, I, Tôhoku Math. J., 27 (1975), 515-559. | Zbl 0359.53011

[32] H. Ozeki and M. Takeuchi, On some types of isoparametric hypersurfaces in spheres, II, Tôhoku Math. J., 28 (1976), 7-55. | Zbl 0359.53012

[33] J. Radon, Lineare Scharen orthogonalar Matrizen, Abh. Math. Semin. Univ. Hamburg, I (1922), 1-14. | JFM 48.0092.06

[34] R.T. Smith, Harmonic mappings of spheres, Thesis, Warwick University, 1972. | Zbl 0279.53055

[35] R. Takagi, A class of hypersurfaces with constant principal curvatures in a sphere, J. Diff. Geom., 11 (1976), 225-233. | MR 54 #13798 | Zbl 0337.53003

[36] R. Takagi and T. Takahashi, On the principal curvatures of homogeneous hypersurfaces in a sphere, in Differential Geometry in honour of K. Yano, Tokyo, 1972, 469-481. | MR 48 #12413 | Zbl 0244.53042

[37] J.C. Wood, Harmonic morphisms, foliations and Gauss maps, in Complex differential geometry and nonlinear partial differential equations, Y.T. Siu ed., Contemp. Math., vol. 49, Amer. Math. Soc., Providence, R.I., 1986, 145-184. | MR 87i:58045 | Zbl 0592.53020

[38] J.C. Wood, Harmonic morphisms and Hermitian structures on Einstein 4-manifolds, Internat. J. Math., 3 (1992), 415-439. | MR 94a:58054 | Zbl 0763.53051