On the Geometry of Totally Geodesic Riemannian Foliations

In the present article we study a totally geodesic Riemannian foliation $F$ on a complete Riemannian manifold. We introduce a metrical connection $\widetilde{\nabla}$ that is different from the Levi–Civita connection. The distribution defined by the foliation $F$ and its orthogonal complement are parallel. We also study an interrelation between the vertical-horizontal homotopy and the metrical connection $\widetilde{\nabla}$. In the article we prove that the complementary (by orthogonality) distribution to the foliation $F$ is completely integrable if and only if the connection $\widetilde{\nabla}$ coincides with the Levi–Civita connection.