Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 
 

 

Restrictions on harmonic morphisms


Author: M. T. Mustafa
Journal: Conform. Geom. Dyn. 3 (1999), 102-115
MSC (1991): Primary 58E20, 53C20
DOI: https://doi.org/10.1090/S1088-4173-99-00026-0
Published electronically: August 16, 1999
MathSciNet review: 1716571
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider horizontally (weakly) conformal maps $\phi$ between Riemannian manifolds and calculate a formula for the Laplacian of the dilation of $\phi$, using the language of moving frames. Applying this formula to harmonic horizontally (weakly) conformal maps or equivalently to harmonic morphisms we obtain a Weitzenböck formula similar to an earlier result of the author (J. London Math. Soc. (2) 57 (1998), 746-756), and hence vanishing results for harmonic morphisms from compact manifolds of positive curvature. Further, a method is developed to obtain restrictions on harmonic morphisms from some non-compact and non-positively curved domains. Finally, a discussion of restrictions on harmonic morphisms between simply connected space forms is given.


References [Enhancements On Off] (What's this?)

  • 1. Baird, P.: Harmonic maps with symmetry, harmonic morphisms, and deformation of metrics, Pitman Research Notes in Mathematics Series 87, Pitman, Boston, London, Melbourne, 1983. MR 85i:58038
  • 2. Baird, P. and Wood, J. C.: 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
  • 3. Bryant, R. L.: Harmonic morphisms with fibres of dimension one, Communications in Analysis and Geometry (to appear).
  • 4. Eells, J. and Lemaire, L.: A report on harmonic maps, Bull. London Math. Soc. 10 (1978) 1-68. MR 82b:58033
  • 5. Eells, J. and Lemaire, L.: Another report on harmonic maps, Bull. London Math. Soc. 20 (1988) 385-524. MR 89i:58027
  • 6. Eells, J. and Sampson, J. H.: Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964) 109-160. MR 29:1603
  • 7. Eells, J. and Yiu, P.: Polynomial harmonic morphisms between Euclidean spheres, Proc. Amer. Math. Soc. 123 (1995) 2921-2925. MR 95k:58048
  • 8. Fuglede, B.: Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble) 28 (1978) 107-144. MR 80h:58023
  • 9. Goldberg, S. I., Ishihara, T. and Petridis N. C.: Mappings of bounded dilatation of Riemannian manifolds, J. Diff. Geometry 10 (1975) 619-630. MR 52:11787
  • 10. Gudmundsson, S.: Harmonic morphisms between spaces of constant curvature, Proc. Edinburgh Math. Soc. 36 (1993) 133-143. MR 93j:58034
  • 11. Gudmundsson, S.: Harmonic morphisms from complex projective spaces, Geom. Dedicata 53 (1994) 155-161. MR 95j:58034
  • 12. Gudmundsson, S.: The Bibliography of Harmonic Morphisms,
    http://www.maths.lth.se/matematiklu/personal/sigma/harmonic/bibliography.html.
  • 13. Ishihara, T.: A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ. 19 (1979) 215-229. MR 80k:58045
  • 14. Kasue, A. and Washio, T.: Growth of equivariant harmonic maps and harmonic morphisms, Osaka J. Math. 29 (1992) 899-928. MR 92d:58043; MR 93e:58038
  • 15. Lohkamp, J.: Metrics of negative Ricci curvature, Ann. of Math. 140 (1994) 655-683. MR 95i:53042
  • 16. Montaldo, S.: Harmonic maps and morphisms via moving frames, Lecture notes, University of Leeds (1997).
  • 17. Mustafa, M. T.: A Bochner technique for harmonic morphisms, J. London Math. Soc. (2) 57 (1998) 746-756. CMP 99:05
  • 18. Mustafa, M. T. and Wood, J. C.: Harmonic morphisms from three-dimensional Euclidean and spherical space forms, Algebras, Groups and Geometries 15 (1998) 155-172. CMP 99:09
  • 19. Ou, Y. L. and Wood, J. C.: On the classification of quadratic harmonic morphisms between Euclidean spaces, Algebras, Groups and Geometries 13 (1996) 41-53. MR 97d:58063
  • 20. Willmore, T. J.: Riemannian geometry, Oxford University Press (1993). MR 95e:53002
  • 21. Wood, J. C.: Harmonic morphisms, foliations and Gauss maps, Complex differential geometry and nonlinear partial differential equations (Providence, R.I.) (Y.T. Siu, ed.), Contemp. Math. 49, Amer. Math. Soc., Providence, R.I., 1986, 145-184. MR 87i:58045
  • 22. Wood, J. C.: Harmonic maps and morphisms in 4 dimensions, Geometry, Topology and Physics, Proceedings of the First Brazil-USA Workshop, Campinas, Brazil, June 30-July 7, 1996, B. N. Apanasov, S. B. Bradlow, K. K. Uhlenbeck (Editors), Walter de Gruyter $\&$ Co., Berlin, New York (1997), 317-333. MR 99b:58067

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (1991): 58E20, 53C20

Retrieve articles in all journals with MSC (1991): 58E20, 53C20


Additional Information

M. T. Mustafa
Affiliation: Assistant Professor, Faculty of Engineering Sciences, GIK Institute of Engineering Sciences and Technology, Topi, Distt. Swabi, N.W.F.P., Pakistan
Email: mustafa@giki.edu.pk

DOI: https://doi.org/10.1090/S1088-4173-99-00026-0
Keywords: Harmonic morphisms, harmonic maps, Bochner technique
Received by editor(s): December 29, 1997
Received by editor(s) in revised form: June 8, 1999
Published electronically: August 16, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society