A Note on Zoll Manifolds with Entire Grauert Tubes

TL;DR

This paper investigates Zoll manifolds with entire Grauert tubes, computing algebraic indices and proving that certain types are uniquely isometric to canonical models, enhancing understanding of their geometric structure.

Contribution

It introduces algebraic indices for Zoll manifolds with entire Grauert tubes and proves their uniqueness in the case of type .

Findings

Algebraic indices of tangent bundles are computed.

Zoll manifolds of type  with entire Grauert tubes are isometric to .

Provides new insights into the geometry of Zoll manifolds with Grauert tubes.

Abstract

We compute several algebraic indices of the algebraization of the tangent bundle of a Zoll manifold that admits an entire Grauert tube. As an application, we prove that any Zoll manifold of type $\mathbb{HP}^2$ with an entire Grauert tube is isometric to the canonical $\mathbb{HP}^2$.