A Note on Zoll Manifolds with Entire Grauert Tubes

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 ${\mathbb{H}\mathbb{P}}^2$ with an entire Grauert tube is isometric to the canonical ${\mathbb{H}\mathbb{P}}^2$.