On totally geodesic foliations with bundle-like metric

Abstract

Let $$\mathcal{F}$$ be a totally geodesic foliation of dimension n and codimension p on a Riemannian manifold (M, g). Suppose that g is a bundle-like metric for $$\mathcal{F}$$ and M has at least one point at which none of its mixed sectional curvatures vanishes. Under these conditions we prove that n ≤ p − 1. We show that this inequality is optimal, and none of the above conditions can be removed.