A general description of totally geodesic foliations (Q1063910)

We give some general concepts and results for totally geodesic foliations \({\mathcal F}\) on complete Riemannian manifolds. In particular, we reduce the problem to that of a generalisation of the theory of principal connections. This enables us to show that the global geometry of F is related to certain sheaves of germs of local Killing vector fields for the Riemannian structure along the leaves. Further, we define a cohomology group \(H^*_{tg}\), and natural mappings from \(H^*_{tg}\) into the de Rham cohomologies of the leaves, such that the characteristic classes in \(H^*_{tg}\) are mapped to the characteristic classes of the leaves.