Quadratic harmonic morphisms and O-systems
Annales de l'Institut Fourier, Volume 47 (1997) no. 2, pp. 687-713.

We introduce O-systems (Definition 3.1) of orthogonal transformations of m , and establish correspondences both between equivalence classes of Clifford systems and those of O-systems, and between O-systems and orthogonal multiplications of the form μ: n × m 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.

Nous présentons les O-systèmes (Définition 3.1) des transformations orthogonales de 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 μ: n × m 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.

@article{AIF_1997__47_2_687_0,
     author = {Ou, Ye-Lin},
     title = {Quadratic harmonic morphisms and {O-systems}},
     journal = {Annales de l'Institut Fourier},
     pages = {687--713},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {2},
     year = {1997},
     doi = {10.5802/aif.1578},
     mrnumber = {98j:58038},
     zbl = {0918.58020},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1578/}
}
TY  - JOUR
AU  - Ou, Ye-Lin
TI  - Quadratic harmonic morphisms and O-systems
JO  - Annales de l'Institut Fourier
PY  - 1997
SP  - 687
EP  - 713
VL  - 47
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.1578/
DO  - 10.5802/aif.1578
LA  - en
ID  - AIF_1997__47_2_687_0
ER  - 
%0 Journal Article
%A Ou, Ye-Lin
%T Quadratic harmonic morphisms and O-systems
%J Annales de l'Institut Fourier
%D 1997
%P 687-713
%V 47
%N 2
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.1578/
%R 10.5802/aif.1578
%G en
%F AIF_1997__47_2_687_0
Ou, Ye-Lin. Quadratic harmonic morphisms and O-systems. Annales de l'Institut Fourier, Volume 47 (1997) no. 2, pp. 687-713. doi : 10.5802/aif.1578. http://archive.numdam.org/articles/10.5802/aif.1578/

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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. | EuDML | Numdam | MR | Zbl

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

[8] L. Conlon, Differentiable Manifolds, A first course, Basler Lehrbücher, Berlin, 1993. | MR | Zbl

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

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

[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 | Zbl

[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 | Zbl

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

Cited by Sources: