The Euler characteristic of hypersurfaces in space forms and applications to isoparametric hypersurfaces

TL;DR

This paper revisits a classical formula for the Euler characteristic of hypersurfaces in space forms, re-proves it with modern techniques, and applies it to deepen understanding of isoparametric hypersurfaces.

Contribution

It provides a new proof of Allendoerfer-Weil's formula and applies it to analyze the geometry of isoparametric hypersurfaces in constant curvature spaces.

Findings

Re-derivation of Allendoerfer-Weil's formula using modern techniques

New insights into the structure of isoparametric hypersurfaces

Enhanced understanding of topological invariants in space forms

Abstract

We revisit Allendoerfer-Weil's formula for the Euler characteristic of embedded hypersurfaces in constant sectional curvature manifolds, first taking some time to re-prove it while demonstrating techniques of [2] and then applying it to gain new understanding of isoparametric hypersurfaces.