The Equivalence of the Existences of Transnormal and Isoparametric Functions on Compact Manifolds

TL;DR

This paper establishes that on compact manifolds, the existence of transnormal and isoparametric functions are equivalent, linking geometric functions to topological structures like vector bundles and disk decompositions.

Contribution

It proves the equivalence of transnormal and isoparametric functions on compact manifolds and characterizes the topological constraints for their existence.

Findings

Transnormal functions require vector bundle or disk bundle structures.

Any such structure can admit a metric with transnormal or isoparametric functions.

Existence of these functions imposes the same topological constraints on compact manifolds.

Abstract

Through exploring the embedded transnormal systems of codimension 1, we show the existence of a transnormal function on a connected complete Riemannian manifold requires the underlying manifold to have a vector bundle structure or a linear double disk bundle decomposition. Conversely, any smooth manifold with either of these structures can be endowed with a Riemannian metric so that it admits a transnormal function, which, under suitable compactness conditions, can become isoparametric. As a corollary, for compact manifolds, the existences of transnormal and isoparametric functions impose the same topological constraints.