Tohoku Mathematical Journal

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

Luc Lemaire and John C. Wood

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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.

Article information

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

First available in Project Euclid: 24 June 2009

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Zentralblatt MATH identifier

Primary: 58E20: Harmonic maps [See also 53C43], etc.
Secondary: 53C43: Differential geometric aspects of harmonic maps [See also 58E20]

Harmonic map Jacobi field infinitesimal deformation


Lemaire, Luc; Wood, John C. Jacobi fields along harmonic 2-spheres in 3- and 4-spheres are not all integrable. Tohoku Math. J. (2) 61 (2009), no. 2, 165--204. doi:10.2748/tmj/1245849442.

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