Some Remarks on Isoparametric Functions in Closed 4-Manifolds

TL;DR

This paper investigates the topological and geometric restrictions of isoparametric functions on closed 4-manifolds, revealing limitations on negative curvature and insights into their fundamental groups.

Contribution

It establishes fundamental restrictions on the topology and geometry of 4-manifolds admitting isoparametric functions, including non-existence of negatively curved metrics.

Findings

Closed 4-manifolds with isoparametric functions cannot have negatively curved metrics

Provides descriptions of fundamental groups in certain cases

Contributes to understanding the global structure of isoparametric foliations

Abstract

We study transnormal and isoparametric functions on closed Riemannian 4-manifolds and establish fundamental restrictions on their topology and geometry. In particular, we show that such manifolds cannot be endowed with negatively curved metrics, contrasting with known results in the compact simply connected case. Moreover, in certain cases, we provide a description of their fundamental groups. These findings contribute to a better understanding of the global structure of isoparametric foliations.