  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.
