Harmonic conformal flows on manifolds of constant curvature

Amine Fawaz

Open Mathematics (2007)

  • Volume: 5, Issue: 3, page 493-504
  • ISSN: 2391-5455

Abstract

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We compute the energy of conformal flows on Riemannian manifolds and we prove that conformal flows on manifolds of constant curvature are critical if and only if they are isometric.

How to cite

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Amine Fawaz. "Harmonic conformal flows on manifolds of constant curvature." Open Mathematics 5.3 (2007): 493-504. <http://eudml.org/doc/269154>.

@article{AmineFawaz2007,
abstract = {We compute the energy of conformal flows on Riemannian manifolds and we prove that conformal flows on manifolds of constant curvature are critical if and only if they are isometric.},
author = {Amine Fawaz},
journal = {Open Mathematics},
keywords = {basic form; energy; foliation; geodesic curvature; harmonic; mean curvature; projectable vector field; symmetric functions; umbilical; conformal flow; isometric flow; constant curvature},
language = {eng},
number = {3},
pages = {493-504},
title = {Harmonic conformal flows on manifolds of constant curvature},
url = {http://eudml.org/doc/269154},
volume = {5},
year = {2007},
}

TY - JOUR
AU - Amine Fawaz
TI - Harmonic conformal flows on manifolds of constant curvature
JO - Open Mathematics
PY - 2007
VL - 5
IS - 3
SP - 493
EP - 504
AB - We compute the energy of conformal flows on Riemannian manifolds and we prove that conformal flows on manifolds of constant curvature are critical if and only if they are isometric.
LA - eng
KW - basic form; energy; foliation; geodesic curvature; harmonic; mean curvature; projectable vector field; symmetric functions; umbilical; conformal flow; isometric flow; constant curvature
UR - http://eudml.org/doc/269154
ER -

References

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  3. [3] F. Brito and P. Chacon: “Energy of Global Frames”, To appear in the J. Aust. Math. Soc.. 
  4. [4] M. Berger M. and D. Ebin, “Some decompositions of the space of symmetric tensors on a Riemannian manifold”, J. Differ. Geom., Vol. 3, (1969), pp. 379–392. Zbl0194.53103
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  7. [7] Y. Carrière: “Flots Riemanniens, in “Structure Transverse des Feuilletages”, Astéerisque, Vo. 116 (1984), pp. 31–52. 
  8. [8] J. Eells and J. Sampson: “Harmonic mappings of Riemannian manifolds”, Amer. J. Math., Vol. 86, (1964), pp. 109–160. http://dx.doi.org/10.2307/2373037 Zbl0122.40102
  9. [9] A. Fawaz: “Energy and Riemannian Flows”, To appear in Geometriae Dedicata. Zbl1179.53036
  10. [10] A. Fawaz: “Energy and Foliations on Riemann Surfaces”, Ann. Glob. Anal. Geom., Vol. 28 (2005), pp. 75–89. http://dx.doi.org/10.1007/s10455-005-4405-0 Zbl1079.53042
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  12. [12] W. Poor: Differential Geometric Structures, McGraw Hill Book Company, New York etc. 1981. 
  13. [13] P. Tondeur: Geometry of Foliations, Monographs in Math. Vol. 90, Birkhäuser, 1997. Zbl0905.53002
  14. [14] G. Wiegmink: “Total bending of vector fields on the sphere S 3”, Differ. Geome. Appl., Vol. 6, (1996), pp. 219–236 http://dx.doi.org/10.1016/0926-2245(96)82419-3 
  15. [15] G. Wiegmink: “Total bending of vector fields on Riemannian manifolds”, Math. Ann., Vol. 303, (1995), pp. 325–344. http://dx.doi.org/10.1007/BF01460993 Zbl0834.53034
  16. [16] C.M. Wood: “On the energy of a unit vector field”, Geometria Dedicata, Vol. 64 (1997), pp. 319–330. http://dx.doi.org/10.1023/A:1017976425512 
  17. [17] K. Yano: Integral Formulas in Riemannian Geometry, Marcel-Decker Inc., New York, 1970. Zbl0213.23801

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