Taut submanifolds and foliations

Abstract

We give an equivalent description of taut submanifolds of complete Riemannian manifolds as exactly those submanifolds whose normal exponential map has the property that every preimage of a point is a union of submanifolds. It turns out that every taut submanifold is also $\mathbb{Z}_2$-taut. We explicitly construct generalized Bott-Samelson cycles for the critical points of the energy functionals on the path spaces of a taut submanifold that, generically, represent a basis for the $\mathbb{Z}_2$-cohomology. We also consider singular Riemannian foliations all of whose leaves are taut. Using our characterization of taut submanifolds, we are able to show that tautness of a singular Riemannian foliation is actually a property of the quotient.