Transversal Jacobi fields for harmonic foliations. (English) Zbl 0629.53031
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.
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.
Reviewer: A.Piatkowski
MSC:
53C12 | Foliations (differential geometric aspects) |
57R30 | Foliations in differential topology; geometric theory |