Rigidity of compact rank one symmetric spaces

TL;DR

This paper proves that compact rank one symmetric spaces are rigid under certain curvature and metric conditions, showing that any metric agreeing outside a convex subset must be isometric, and highlights the necessity of nonnegative curvature.

Contribution

It establishes a new rigidity theorem for compact rank one symmetric spaces under bounded curvature and local metric agreement conditions.

Findings

Metrics agreeing outside a convex subset are isometric to the original.

Nonnegativity of curvature is essential for the rigidity result.

The work extends previous rigidity results to rank one symmetric spaces.

Abstract

We consider rigidity properties of compact symmetric spaces $X$ with metric $g_0$ of rank one. Suppose $g$ is another Riemannian metric on $X$ with sectional curvature $\kappa$ bounded by $0 \leq \kappa \leq 1$. If $g$ equals $g_0$ outside a convex proper subset of $X$, then $g$ is isometric with $g_0$. We also exhibit examples of surfaces showing that the nonnegativity of the curvature is needed. Our main result complements earlier results on other symmetric spaces by Gromov and Schroeder-Ziller.