Jacobi fields along harmonic 2-spheres in $S^3$ and $S^4$ are not all integrable
Differential Geometry
2007-12-20 v2
Abstract
In a previous paper, we showed that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (i.e., is tangent to a smooth variation through harmonic maps). In this paper, in contrast, we show that there are (non-full) harmonic maps from the 2-sphere to the 3-sphere and 4-sphere which have non-integrable Jacobi fields. This is particularly surprising in the case of the 3-sphere where the space of harmonic maps of any degree is a smooth manifold, each map having image in a totally geodesic 2-sphere.
Keywords
Cite
@article{arxiv.0709.1417,
title = {Jacobi fields along harmonic 2-spheres in $S^3$ and $S^4$ are not all integrable},
author = {Luc Lemaire and John C Wood},
journal= {arXiv preprint arXiv:0709.1417},
year = {2007}
}
Comments
43 pages. Some typos corrected; introduction expanded