A theorem on the tension field
HTML articles powered by AMS MathViewer
- by Th. Koufogiorgos and Ch. Baikoussis PDF
- Proc. Amer. Math. Soc. 93 (1985), 321-324 Request permission
Abstract:
Suppose $M$ and $N$ are complete Riemannian manifolds; $M$ with Ricci curvature bounded from below and $N$ with sectional curvature bounded from above by a constant ${K_0}$. Let $f:M \to N$ be a smooth map such that $f(M) \subset {B_R}$, where ${B_R}$ is a normal ball in $N$ and furthermore $R < \pi /2\sqrt {{K_0}}$ if ${K_0} > 0$. If the energy density $e(f)$ is bounded below by a positive constant, then there is a point $P \in M$ such that the tension field $\tau (f)$ at $P$ is different from zero.References
- Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
- J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1–68. MR 495450, DOI 10.1112/blms/10.1.1
- Th. Hasanis and D. Koutroufiotis, Immersions of bounded mean curvature, Arch. Math. (Basel) 33 (1979/80), no. 2, 170–171. MR 557750, DOI 10.1007/BF01222742
- Hideki Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan 19 (1967), 205–214. MR 215259, DOI 10.2969/jmsj/01920205
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 93 (1985), 321-324
- MSC: Primary 58E20
- DOI: https://doi.org/10.1090/S0002-9939-1985-0770545-5
- MathSciNet review: 770545