We introduce O-systems (Definition 3.1) of orthogonal transformations of , 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 , which allow us to solve the existence problems both for -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 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 , 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://www.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://www.numdam.org/articles/10.5802/aif.1578/ DO - 10.5802/aif.1578 LA - en ID - AIF_1997__47_2_687_0 ER -
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://www.numdam.org/articles/10.5802/aif.1578/
[1] 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] Bernstein theorems for harmonic morphisms from ℝ3 and S3, Math. Ann., 280 (1988), 579-603. | EuDML | MR | Zbl
and ,[3] Harmonic morphisms and conformal foliations by geodesics of three-dimensional space forms, J. Austral. Math. Soc., Ser. A, 51 (1991), 118-153. | MR | Zbl
and ,[4] Harmonic morphisms, Seifert fibre spaces and conformal foliations, Proc. London Math. Soc., 64 (1992), 170-196. | Zbl
and ,[5] Hermitian structures and harmonic morphisms on higher dimensional Euclidean spaces, Internat. J. Math., 6 (1995), 161-192. | MR | Zbl
and ,[6] Brownian motion and generalized analytic and inner functions, Ann. Inst. Fourier (Grenoble), 29-1 (1979), 207-228. | EuDML | Numdam | MR | Zbl
, , and ,[7] Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann. Mat. Pura Appl., 17 (1938), 177-191. | JFM | Zbl
,[8] Differentiable Manifolds, A first course, Basler Lehrbücher, Berlin, 1993. | MR | Zbl
,[9] Beweis des Satzes von Hurwitz-Radon, Comment. Math. Helvet., 15 (1952), 358-366. | EuDML | Zbl
,[10] A report on harmonic maps, Bull. London Math. Soc., 10 (1978), 1-68. | MR | Zbl
and ,[11] Selected topics in harmonic maps, CBMS Regional Conf. Ser. in Math., vol. 50, Amer. Math. Soc., Providence, R.I., 1983. | MR | Zbl
and ,[12] Another report on harmonic maps, Bull. London Math. Soc., 20 (1988), 385-524. | MR | Zbl
and ,[13] Harmonic maps and minimal immersions with symmetries, Ann. of Math. Stud., vol. 130, Princeton University Press, 1993. | Zbl
and ,[14] Polynomial harmonic morphisms between Euclidean spheres, Proc. Amer. Math. Soc., vol. 123, 9 (1995), 2921-2925. | MR | Zbl
and ,[15] Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z., 177 (1981), 479-502. | Zbl
, , and ,[16] Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble), 28-2 (1978), 107-144. | Numdam | MR | Zbl
,[17] Harmonic morphisms from quaternionic projective spaces, Geom. Dedicata, 56 (1995), 327-332. | MR | Zbl
,[18] Harmonic morphisms between spaces of constant curvature, Proc. Edinburgh Math. Soc., 36 (1992), 133-143. | MR | Zbl
,[19] Harmonic morphisms from complex projective spaces, Geom. Dedicata, 53 (1994), 155-161. | MR | Zbl
,[20] A note on the classification of holomorphic harmonic morphisms, Potential Analysis, 2 (1993), 295-298. | MR | Zbl
and ,[21] Über die Komposition der quadratischen Formen, Math. Ann., 88 (1923), 1-25. | JFM
,[22] Fibre Bundles, McGraw Hill, New York, 1966. | MR | Zbl
,[23] A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ., 19 (1979), 215-229. | MR | Zbl
,[24] Processus Stochastiques et Mouvement Brownien, Gauthier-Villard, Paris, 1948. | Zbl
,[25] Isoparametrische Hyperflächen in Sphären, I, Math. Ann., 251 (1980), 57-71. | Zbl
,[26] 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] Complete lifts of harmonic maps and morphisms between Euclidean spaces, Contributions to Algebra and Geometry, vol. 37 (1996), 31-40. | MR | Zbl
,[28] O-systems, orthogonal multiplications and isoparametric functions, Guangxi University for Nationalities, preprint, 1996.
,[29] On constructions of harmonic morphisms into Euclidean spaces, J. Guangxi University for Nationalities, vol. 2 (1996), 1-6.
,[30] On the classification of quadratic harmonic morphisms between Euclidean spaces, Algebras, Groups and Geometries, vol. 13 (1996), 41-53. | MR | Zbl
and ,[31] On some types of isoparametric hypersurfaces in spheres, I, Tôhoku Math. J., 27 (1975), 515-559. | Zbl
and ,[32] On some types of isoparametric hypersurfaces in spheres, II, Tôhoku Math. J., 28 (1976), 7-55. | Zbl
and ,[33] Lineare Scharen orthogonalar Matrizen, Abh. Math. Semin. Univ. Hamburg, I (1922), 1-14. | JFM
,[34] Harmonic mappings of spheres, Thesis, Warwick University, 1972. | Zbl
,[35] A class of hypersurfaces with constant principal curvatures in a sphere, J. Diff. Geom., 11 (1976), 225-233. | MR | Zbl
,[36] On the principal curvatures of homogeneous hypersurfaces in a sphere, in Differential Geometry in honour of K. Yano, Tokyo, 1972, 469-481. | MR | Zbl
and ,[37] 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] Harmonic morphisms and Hermitian structures on Einstein 4-manifolds, Internat. J. Math., 3 (1992), 415-439. | MR | Zbl
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