Jacobi fields, Riccati equation and Riemannian foliations
If \((M,{\mathcal F})\) is a foliated Riemannian manifold and \(\gamma\) is a geodesic orthogonal to the leaves of \(\mathcal F\), then a vector field \(V\) along \(\gamma\) is \(\mathcal F\)-Jacobi when the orthogonal to \(\mathcal F\) component of \(V'+R(\dot\gamma,V)\dot\gamma\), \(R\) being the curvature tensor on \(M\) and \(V'\) being the covariant derivative of \(V\) in the direction of \(\dot\gamma\), vanishes. The Jacobi equation defining \(\mathcal F\)-Jacobi fields can be expressed in terms of the curvature and the integrability tensor of the normal bundle of \(\mathcal F\). This leads to a Riccati type equation for the shape operator \(S\) of \(\mathcal F\) along \(\gamma\). This equation is studied in detail and applied to Riemannian and harmonic foliations under some curvature restrictions on \(M\).
scientific article; zbMATH DE number 926191