Taut Submanifolds and Foliations

We give an equivalent description of taut submanifolds of complete Riemannian manifolds as exactly those submanifolds whose normal exponential map has integrable fibers. It turns out that every taut submanifold is also $\mathbb Z_2$-taut, so that tautness is essentially the same as $\mathbb Z_2$-tautness. In the case where the normal exponential map of a submanifold has integrable fibers, we explicitely construct generalized Bott-Samelson cycles for the critical points of the energy functionals on the path spaces which, generically, represent a basis for the $\mathbb Z_2$-cohomology. We also consider singular Riemannian foliations all of whose leaves are taut and discuss some of their main features. 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