Horizontally homothetic harmonic morphisms and stability of totally geodesic submanifolds
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In this article, we study the relations of horizontally homothetic harmonic morphisms with the stability of totally geodesic submanifolds. Let p : (Mn, g) → (Nm, 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 p. 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, p-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.
Abstract
In this article, we study the relations of horizontally homothetic harmonic morphisms with the stability of totally geodesic submanifolds. Let p : (Mn, g) → (Nm, 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 p. 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, p-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.
참고문헌 (11)
- P. Baird, "A Conservation Law for Harmonic Maps, Geometry Symposium, Utrecht 1980 (Utrecht, 1980)", pp.1~25 (1981), Springer
- P. Baird, "p-harmonic maps and minimal submanifolds", Mathematische Annalen, 294(4), pp.611~624 (1992)
- G. D. Choi, "Harmonic morphisms and stable minimal submanifolds, Preprint", ~
- B. Fuglede, "Harmonic morphisms between Riemannian manifolds", Université de Grenoble Annales de l'Institut Fourier Univ Grenoble, 28(2), pp.107~144 (1978)
- S. Gudmundsson, "The Geometry of Harmonic Morphisms", 박사, pp.1992
- T. Ishihara, "A mapping of Riemannian manifolds which preserves harmonic functions", Journal of Mathematics of Kyoto University, 19(2), pp.215~229 (1979)
- A. Kasue, "Growth of equivariant harmonic maps and harmonic morphisms", Osaka Journal of Mathematics, 27(4), pp.899~928 (1990)
- H. B. Lawson, "Lectures on Minimal Submanifolds. Vol. I, Mathematics Lecture Series", pp.9~ (1980), Publish or Perish, Inc.
- E. Loubeau, "On p-harmonic morphisms", Differential Geometry and its Applications, 12(3), pp.219~229 (2000)
- S. Montaldo, "Stability of harmonic morphisms to a surface", International Journal of Mathematics and Mathematical Sciences, 9(7), pp.865~875 (1998)
- B. ONeill, "The fundamental equations of a submersion", Michigan Math Journal, 13, pp.459~469 (1966)