Spheres and projective spaces determined by the one-form spectrum

The ordinary one-form spectrum determines the standard spheres, real, complex and quaternionic projective spaces, and the Cayley plane among closed connected Riemannian manifolds. The same conclusion holds in the complementary degree without an orientability assumption; for the complex projective plane it holds in each degree 0 ≤ p ≤ 4. Above dimension four, partial spectral matchings suffice in degree one and in the complementary degree under quantitative bounds on omitted eigenvalue copies and matching cost. In sufficiently large dimension, a matching after independent finite omissions with error o(ν⁻¹), uniformly as the model eigenvalue ν → ∞, determines every d-dimensional model in a common set of form degrees. As d → ∞, the lower limit of their proportion among 0, …, d is at least √(8/21). Heat identities and geodesic returns give averaged curvature bounds; for nonreal projective targets of dimension greater than four, equality in a periodic index estimate then determines the Jacobi operators.