Isospectral Metrics on Projective Spaces

TL;DR

This paper constructs pairs of isospectral but non-isometric metrics on real and complex projective spaces, expanding the understanding of spectral geometry and metric uniqueness.

Contribution

It introduces new isospectral metrics on projective spaces using torus actions and proves their non-isometry, providing explicit examples in spectral geometry.

Findings

Constructed isospectral non-isometric metrics on projective spaces

Demonstrated non-isometry of the constructed metrics

Derived metrics on real projective space from sphere metrics

Abstract

We construct isospectral non isometric metrics on real and complex projective space. We recall the construction using isometric torus actions by Carolyn Gordon in chapter 2. In chapter 3 we will recall some facts about complex projective space. In chapter 4 we build the isospectral metrics. Chapter 5 is devoted to the non isometry proof of the metrics built in chapter 4. In chapter 6 isospectral metrics on real projective space are derived from metrics on the sphere.