A general description of totally geodesic foliations

We give some general concepts and results for totally geodesic foliations &~ on complete Riemannian manifolds. In particular, we reduce the problem to that of a generalization of the theory of principal connections. This enables us to show that the global geometry of ^~ 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? g , and natural mappings from H t * into the de Rham cohomologies of the leaves, such that the characteristic classes in H t * g are mapped to the characteristic classes of the leaves.

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