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Tohoku Mathematical Journal

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

Luc Lemaire and John C. Wood

Source: Tohoku Math. J. (2) Volume 61, Number 2 (2009), 165-204.

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.

Primary Subjects: 58E20
Secondary Subjects: 53C43
Keywords: Harmonic map; Jacobi field; infinitesimal deformation

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Permanent link to this document: http://projecteuclid.org/euclid.tmj/1245849442
Digital Object Identifier: doi:10.2748/tmj/1245849442
Zentralblatt MATH identifier: 05608369
Mathematical Reviews number (MathSciNet): MR2541404

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