Quadratic harmonic morphisms and O-systems

Ye-Lin Ou

Annales de l'institut Fourier (1997)

  • Volume: 47, Issue: 2, page 687-713
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Ou, Ye-Lin. "Quadratic harmonic morphisms and O-systems." Annales de l'institut Fourier 47.2 (1997): 687-713. <http://eudml.org/doc/75242>.

@article{Ou1997,
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.},
author = {Ou, Ye-Lin},
journal = {Annales de l'institut Fourier},
keywords = {harmonic applications; quadratic harmonic morphisms; -systems; Clifford systems},
language = {eng},
number = {2},
pages = {687-713},
publisher = {Association des Annales de l'Institut Fourier},
title = {Quadratic harmonic morphisms and O-systems},
url = {http://eudml.org/doc/75242},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Ou, Ye-Lin
TI - Quadratic harmonic morphisms and O-systems
JO - Annales de l'institut Fourier
PY - 1997
PB - Association des Annales de l'Institut Fourier
VL - 47
IS - 2
SP - 687
EP - 713
AB - 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.
LA - eng
KW - harmonic applications; quadratic harmonic morphisms; -systems; Clifford systems
UR - http://eudml.org/doc/75242
ER -

References

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  1. [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. Zbl0515.58010MR85i:58038
  2. [2] P. BAIRD and J.C. WOOD, Bernstein theorems for harmonic morphisms from ℝ3 and S3, Math. Ann., 280 (1988), 579-603. Zbl0621.58011MR90e:58027
  3. [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. Zbl0744.53013MR92k:53048
  4. [4] P. BAIRD and J.C. WOOD, Harmonic morphisms, Seifert fibre spaces and conformal foliations, Proc. London Math. Soc., 64 (1992), 170-196. Zbl0755.58019
  5. [5] P. BAIRD and J.C. WOOD, Hermitian structures and harmonic morphisms on higher dimensional Euclidean spaces, Internat. J. Math., 6 (1995), 161-192. Zbl0823.58010MR96a:58068
  6. [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. Zbl0386.30029MR81b:30088
  7. [7] E. CARTAN, Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann. Mat. Pura Appl., 17 (1938), 177-191. Zbl0020.06505JFM64.1361.02
  8. [8] L. CONLON, Differentiable Manifolds, A first course, Basler Lehrbücher, Berlin, 1993. Zbl0770.57001MR94d:58001
  9. [9] B. ECKMANN, Beweis des Satzes von Hurwitz-Radon, Comment. Math. Helvet., 15 (1952), 358-366. Zbl0028.10402
  10. [10] J. EELLS and L. LEMAIRE, A report on harmonic maps, Bull. London Math. Soc., 10 (1978), 1-68. Zbl0401.58003MR82b:58033
  11. [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. Zbl0515.58011MR85g:58030
  12. [12] J. EELLS and L. LEMAIRE, Another report on harmonic maps, Bull. London Math. Soc., 20 (1988), 385-524. Zbl0669.58009MR89i:58027
  13. [13] J. EELLS and A. RATTO, Harmonic maps and minimal immersions with symmetries, Ann. of Math. Stud., vol. 130, Princeton University Press, 1993. Zbl0783.58003
  14. [14] J. EELLS and P. YIU, Polynomial harmonic morphisms between Euclidean spheres, Proc. Amer. Math. Soc., vol. 123, 9 (1995), 2921-2925. Zbl0853.58036MR95k:58048
  15. [15] D. FERUS, H. KARCHER, and H.F. MÜNZNER, Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z., 177 (1981), 479-502. 
  16. [16] B. FUGLEDE, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble), 28-2 (1978), 107-144. Zbl0339.53026MR80h:58023
  17. [17] S. GUDMUNDSSON, Harmonic morphisms from quaternionic projective spaces, Geom. Dedicata, 56 (1995), 327-332. Zbl0834.58016MR96d:58033
  18. [18] S. GUDMUNDSSON, Harmonic morphisms between spaces of constant curvature, Proc. Edinburgh Math. Soc., 36 (1992), 133-143. Zbl0790.58012MR93j:58034
  19. [19] S. GUDMUNDSSON, Harmonic morphisms from complex projective spaces, Geom. Dedicata, 53 (1994), 155-161. Zbl0826.53028MR95j:58034
  20. [20] S. GUDMUNDSSON and R. SIGURDSSON, A note on the classification of holomorphic harmonic morphisms, Potential Analysis, 2 (1993), 295-298. Zbl0783.58015MR94i:58043
  21. [21] A. HURWITZ, Über die Komposition der quadratischen Formen, Math. Ann., 88 (1923), 1-25. JFM48.1164.03
  22. [22] D. HUSEMOLLER, Fibre Bundles, McGraw Hill, New York, 1966. Zbl0144.44804MR37 #4821
  23. [23] T. ISHIHARA, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ., 19 (1979), 215-229. Zbl0421.31006MR80k:58045
  24. [24] P. LEVY, Processus Stochastiques et Mouvement Brownien, Gauthier-Villard, Paris, 1948. Zbl0034.22603
  25. [25] H.F. MÜNZNER, Isoparametrische Hyperflächen in Sphären, I, Math. Ann., 251 (1980), 57-71. 
  26. [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. Zbl0318.53052MR54 #11240
  27. [27] Y.-L. OU, Complete lifts of harmonic maps and morphisms between Euclidean spaces, Contributions to Algebra and Geometry, vol. 37 (1996), 31-40. Zbl0859.53037MR97h:58050
  28. [28] Y.-L. OU, O-systems, orthogonal multiplications and isoparametric functions, Guangxi University for Nationalities, preprint, 1996. 
  29. [29] Y.-L. OU, On constructions of harmonic morphisms into Euclidean spaces, J. Guangxi University for Nationalities, vol. 2 (1996), 1-6. 
  30. [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. Zbl0872.58022MR97d:58063
  31. [31] H. OZEKI and M. TAKEUCHI, On some types of isoparametric hypersurfaces in spheres, I, Tôhoku Math. J., 27 (1975), 515-559. Zbl0359.53011
  32. [32] H. OZEKI and M. TAKEUCHI, On some types of isoparametric hypersurfaces in spheres, II, Tôhoku Math. J., 28 (1976), 7-55. Zbl0359.53012
  33. [33] J. RADON, Lineare Scharen orthogonalar Matrizen, Abh. Math. Semin. Univ. Hamburg, I (1922), 1-14. JFM48.0092.06
  34. [34] R.T. SMITH, Harmonic mappings of spheres, Thesis, Warwick University, 1972. Zbl0279.53055
  35. [35] R. TAKAGI, A class of hypersurfaces with constant principal curvatures in a sphere, J. Diff. Geom., 11 (1976), 225-233. Zbl0337.53003MR54 #13798
  36. [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. Zbl0244.53042MR48 #12413
  37. [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. Zbl0592.53020MR87i:58045
  38. [38] J.C. WOOD, Harmonic morphisms and Hermitian structures on Einstein 4-manifolds, Internat. J. Math., 3 (1992), 415-439. Zbl0763.53051MR94a:58054

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