On Rigidity of Harmonic Mappings into Spheres

M is stable to first order with respect to variations of / [2]. Though, by the work of T. Sunada [14], harmonic maps into nonpositively curved codomains are globally rigid, an essential obstruction to proving rigidity in the case when Riem ^ 0 is the lack of Hartman's uniqueness [3]. On the other hand it has long been noticed that, in all cases, harmonic maps behave nicely with respect to infinitesimal deformations preserving harmonicity up to second order, that is, the second variation formula [8, 13] and Jacobi fields along harmonic maps have been proved to be useful in showing rigidity of harmonic maps (see [2, 6, 9, 10, 11, 12, 13]). The purpose of this paper is to study harmonic maps into spheres with various rigidity properties when higher order terms of the expansion of the energy functional along a variation are also taken into account. In Section 2 we define the (geodesic) variations of a harmonic map given by translating the map along geodesies of a prescribed Jacobi field along this map. The concept of harmonic variation is introduced [15] and its close relationship with the Jacobi fields is indicated (Theorem 1). In Section 3 infinitesimal and local rigidity of harmonic maps are studied in detail; for example, by reducing the problem to that of the linear algebra, we show that harmonic embeddings / : S -*• S" with energy density e(f) = w/2 are rigid (Theorem 2). (For examples of nonrigid harmonic embeddings, see [17].) In Section 4 certain metric spaces of locally rigid harmonic maps are introduced and their classification is reduced to that of the canonical inclusion map i: S -• S". Throughout this note all manifolds, maps, bundles, etc. will be smooth, that is, of class C°°. The report [2] is our general reference, adopting the sign conventions of [5]. We thank Prof. J. Eells and Prof. A. Lichnerowicz for their valuable suggestions and encouragement at the conference on invariant metrics, harmonic mappings and related topics held in Rome in 1981. We also thank Prof. A. Lee for useful discussions on the matrix calculus used in this paper and for giving a proof of Lemma 2.