On geometry of totally geodesic Riemannian foliations (Q2713912)

The author considers a totally geodesic Riemannian foliation \(F\) on a Riemannian manifold \(M\). He defines a metrical connection \(\widetilde{\nabla}\) for which the foliation \(F\) and its orthogonal distribution \(H\) are parallel. The author shows the importance of this connection by proving the following theorem: The distribution \(H\) is integrable if and only if \(\widetilde{\nabla}\) is symmetric, that is \(\widetilde{\nabla}\) coincides with the Levi-Civita connection.