Harmonic Morphisms and the Jacobi Operator

We prove that harmonic morphisms preserve the Jacobi operator along harmonic maps. We apply this result to prove infinitesimal and local rigidity (in the sense of Toth) of harmonic morphisms to a sphere. 1. Harmonic morphisms Harmonic maps φ : (M, g) → (N, h) between two smooth Riemannian manifolds are critical points of the energy functional E(φ,Ω) = 1 2 ∫ Ω |dφ| dvg for any compact domain Ω ⊆ M [4], i.e. the first variation of the energy vanishes for any smooth variation of φ. The Euler-Lagrange equation for the energy is the vanishing of the tension field τφ = trace∇dφ, where ∇ denotes the connection on T ∗M ⊗ φ−1TN induced from the Levi-Civita connections ∇ on M and ∇ on N . If {ei}i=1 is a local orthonormal frame on M we have τφ = ∑m i=1 { ∇φei ( dφ(ei) ) − dφ ( ∇Mei ei )} where ∇ denotes the pull-back connection on φ−1TN . Let φ : (M, g) → (N, h) be a smooth map between two Riemannian manifolds. The tangent space at a point x ∈ M can be decomposed as TxM = Hx ⊕ Vx where Vx = ker(dφx) and Hx = Vx. The spaces Vx and Hx are called the vertical and horizontal space at the point x ∈M , respectively. Date: October 1999. 1991 Mathematics Subject Classification. 58E20.