Transversal Jacobi fields for harmonic foliations (Q1093926)

Theorem 1. Let \({\mathcal F}\) be a transversally orientable harmonic Riemannian foliation of a compact orientable Riemannian manifold M, and Y an infinitesimal automorphism of \({\mathcal F}\). Then the following conditions are equivalent: (i) \(\pi\) (Y) is a transversal Killing field; (ii) \(\pi\) (Y) is a transversally divergence-free Jacobi field; (iii) \(\pi\) (Y) is transversally affine. Here \(\pi\) denotes the projection onto the Lie algebra \(\Gamma\) Q of sections of the normal bundle Q, invariant under the action of \(\Gamma\) \({\mathcal F}\) by Lie derivatives. Theorem 2. Let \({\mathcal F}\) and Y be as above with codim \({\mathcal F}=2\). Then the following conditions are equivalent: (i) \(\pi\) (Y) is a transversal conformal field; (ii) \(\pi\) (Y) is a transversal Jacobi field.