Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

HORIZONTALLY HOMOTHETIC HARMONIC MORPHISMS AND STABILITY OF TOTALLY GEODESIC SUBMANIFOLDS

  • Yun, Gab-Jin (DEPARTMENT OF MATHEMATICS MYONG JI UNIVERSITY) ;
  • Choi, Gun-Don (GARC AND DEPARTMENT OF MATHEMATICS SEOUL NATIONAL UNIVERSITY)
  • Published : 2008.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this article, we study the relations of horizontally homothetic harmonic morphisms with the stability of totally geodesic submanifolds. Let $\varphi:(M^n,g)\rightarrow(N^m,h)$ be a horizontally homothetic harmonic morphism from a Riemannian manifold into a Riemannian manifold of non-positive sectional curvature and let T be the tensor measuring minimality or totally geodesics of fibers of $\varphi$. We prove that if T is parallel and the horizontal distribution is integrable, then for any totally geodesic submanifold P in N, the inverse set, $\varphi^{-1}$(P), is volume-stable in M. In case that P is a totally geodesic hypersurface the condition on the curvature can be weakened to Ricci curvature.

Keywords

References

  1. P. Baird and J. Eells, A Conservation Law for Harmonic Maps, Geometry Symposium, Utrecht 1980 (Utrecht, 1980), pp. 1-25, Lecture Notes in Math., 894, Springer, Berlin-New York, 1981
  2. P. Baird and S. Gudmundsson, p-harmonic maps and minimal submanifolds, Math. Ann. 294 (1992), no. 4, 611-624 https://doi.org/10.1007/BF01934344
  3. G. D. Choi and Gabjin Yun, Harmonic morphisms and stable minimal submanifolds, Preprint
  4. B. Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 2, vi, 107-144
  5. S. Gudmundsson, The Geometry of Harmonic Morphisms, Ph. D Thesis, 1992
  6. T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ. 19 (1979), no. 2, 215-229 https://doi.org/10.1215/kjm/1250522428
  7. A. Kasue and T. Washio, Growth of equivariant harmonic maps and harmonic morphisms, Osaka J. Math. 27 (1990), no. 4, 899-928
  8. H. B. Lawson, Lectures on Minimal Submanifolds. Vol. I. Second edition. Mathematics Lecture Series, 9. Publish or Perish, Inc., Wilmington, Del., 1980
  9. E. Loubeau, On p-harmonic morphisms, Differential Geom. Appl. 12 (2000), no. 3, 219-229 https://doi.org/10.1016/S0926-2245(00)00013-9
  10. S. Montaldo, Stability of harmonic morphisms to a surface, Internat. J. Math. 9 (1998), no. 7, 865-875 https://doi.org/10.1142/S0129167X98000361
  11. B. O'Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459-469 https://doi.org/10.1307/mmj/1028999604

Cited by

  1. Stability and integrability of horizontally conformal maps and harmonic morphisms 2013, https://doi.org/10.1002/mana.201200183