It is well-known that a map F: M→S~N of a Riemannian manifold M into the Euclidean N-sphere S~N is contained in R~(N+1) is harmonic iff the induced vector-valued function F: M→R~(N+1) satisfies the equation Δ~MF=μF, (1) where Δ~M is the Laplacian on M and the scalar μ is uniquely determined by F, in fact, μ is nothing but the energy density e(F)=trace ||F_*||~2 of F. (We work in the C~∞-category, i.e., we assume that all manifolds, maps, bundles etc. are of class C~∞.)
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