No CrossRef data available.
Article contents
ENERGY OF GLOBAL FRAMES
Published online by Cambridge University Press: 01 April 2008
Abstract
Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
The energy of a unit vector field X on a closed Riemannian manifold M is defined as the energy of the section into T1M determined by X. For odd-dimensional spheres, the energy functional has an infimum for each dimension 2k+1 which is not attained by any non-singular vector field for k>1. For k=1, Hopf vector fields are the unique minima. In this paper we show that for any closed Riemannian manifold, the energy of a frame defined on the manifold, possibly except on a finite subset, admits a lower bound in terms of the total scalar curvature of the manifold. In particular, for odd-dimensional spheres this lower bound is attained by a family of frames defined on the sphere minus one point and consisting of vector fields parallel along geodesics.
MSC classification
- Type
- Research Article
- Information
- Copyright
- Copyright © 2008 Australian Mathematical Society
Footnotes
The first author is supported by CNPq (Brazil) Grant No. 301207/80 and by Fapesp (Brazil) Proj. Tem. No. 1999/02684-5. The second author is partially supported by MEC/FEDER Grant No. MTM2004-04934-C04-02 (Spain). This work has been carried out during a post-doctoral stay of the second author supported by DGU (Spain) No. HBE2002-008.
References
[1]Blair, D. E., Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathematics, 203 (Birkhäuser, Boston, 2002).CrossRefGoogle Scholar
[2]Boeckx, E. and Vanhecke, L., ‘Energy of radial vector fields on compact rank one symmetric spaces’, Ann. Global Anal. Geom. 23 (2003), 29–52.CrossRefGoogle Scholar
[3]Borrelli, V., Brito, F. and Gil-Medrano, O., ‘The infimum of the energy of unit vector fields on odd-dimensional spheres’, Ann. Global Anal. Geom. 23 (2003), 129–140.CrossRefGoogle Scholar
[4]Brito, F. G. B., ‘Total bending of flows with mean curvature correction’, Differential Geom. Appl. 12 (2000), 157–163.CrossRefGoogle Scholar
[5]Brito, F. G. B. and Walczak, P., ‘On the energy of unit vector fields with isolated singularities’, Ann. Math. Polon. 73 (2000), 269–274.CrossRefGoogle Scholar
[6]Chacón, P. M. and Naveira, A. M., ‘Corrected energy of distributions on Riemannian manifolds’, Osaka J. Math. 41 (2004), 97–105.Google Scholar
[7]Eells, J. and Lemaire, L., ‘A report on harmonic maps’, Bull. London Math. Soc. 10 (1978), 1–68.CrossRefGoogle Scholar
[8]Gil-Medrano, O., ‘Volume and energy of vector fields on spheres. A survey’, in: Differential Geometry, Valencia, 2001 (World Scientific, Singapore, 2002), pp. 167–178.CrossRefGoogle Scholar
[9]Gil-Medrano, O., González-Dávila, J. C. and Vanhecke, L., ‘Harmonicity and minimality of oriented distributions’, Israel J. Math. 143 (2004), 253–279.CrossRefGoogle Scholar
[10]Pedersen, S., ‘Volume of vector fields on spheres’, Trans. Amer. Math. Soc. 336 (1993), 69–78.CrossRefGoogle Scholar
[11]Sakai, T., Riemannian Geometry, Translations of the Mathematical Monographs, 149 (American Mathematical Society, Providence, RI, 1996).CrossRefGoogle Scholar
[12]Salvai, M., ‘On the energy of sections of parallelizable sphere bundles’, Rend. Sem. Mat. Univ. Politec. Torino 60 (2002), 147–155.Google Scholar
[13]Wiegmink, G., ‘Total bending of vector fields on Riemannian manifolds’, Math. Ann. 303 (1995), 325–344.CrossRefGoogle Scholar
[14]Wiegmink, G., ‘Total bending of vector fields on the sphere S 3’, Differential Geom. Appl. 6 (1996), 219–236.CrossRefGoogle Scholar
[15]Wood, C. M., ‘The energy of Hopf vector fields’, Manuscripta Math. 101 (2000), 71–88.CrossRefGoogle Scholar
You have
Access