Jacobi fields along harmonic 2-spheres in $S^3$ and $S^4$ are not all integrable

Details Jacobi fields along harmonic 2-spheres in $S^3$ and $S^4$ are not all integrable Jacobi fields along harmonic 2-spheres in $S^3$ and $S^4$ are not all integrable

2007-09-10

arxiv.org/abs/0709.1417v2

Mathematics

Differential Geometry

43 pages. Some typos corrected; introduction expanded

Scientific paper

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.

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