Taut Submanifolds and Foliations

TL;DR

This paper characterizes taut submanifolds in complete Riemannian manifolds through their normal exponential maps, explores their $ extbf{Z}_2$-tautness, and studies tautness in singular Riemannian foliations, linking it to quotient properties.

Contribution

It provides an equivalent description of taut submanifolds, constructs generalized Bott-Samelson cycles for their energy functionals, and relates tautness of foliations to their quotients.

Findings

Taut submanifolds are characterized by their normal exponential map properties.

Every taut submanifold is also $ extbf{Z}_2$-taut.

Tautness of a singular Riemannian foliation depends on its quotient.

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 explicitely construct generalized Bott-Samelson cycles for the critical points of the energy functionals on the path spaces of a taut submanifold which, 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.