Tohoku Mathematical Journal
2009
June
SECOND SERIES VOL. 61, NO. 2

Tohoku Math. J.
61 (2009), 165-204

Title JACOBI FIELDS ALONG HARMONIC 2-SPHERES IN 3- AND 4-SPHERES ARE NOT ALL INTEGRABLE

Author Luc Lemaire and John C. Wood

(Received December 26, 2007, revised September 1, 2008)
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 asmooth 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.

2000 Mathematics Subject Classification. Primary 58E20; Secondary 53C43.
Key words and phrases. Harmonic map, Jacobi field, infinitesimal deformation.

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