Zoll Manifolds of Type $\mathbb{CP}^n$ with Entire Grauert Tubes

TL;DR

This paper proves that Zoll manifolds of type P^n with entire Grauert tubes are essentially isometric to the standard complex projective space with its canonical metric, highlighting a rigidity property.

Contribution

It establishes a rigidity result showing such Zoll manifolds are uniquely determined as P^n with the Fubini-Study metric, up to scaling.

Findings

Zoll manifolds of this type are isometric to P^n with the Fubini-Study metric

Entire Grauert tubes imply a strong geometric rigidity

The result characterizes the geometry of these manifolds uniquely.

Abstract

We show that a Zoll manifold of type $\mathbb{CP}^n$ with an entire Grauert tube is isometric to $\mathbb{CP}^n$ with the canonical Fubini-Study metric, up to constant multiplication.