Continuous families of isospectral metrics on simply connected manifolds

TL;DR

This paper constructs continuous families of isospectral, non-isometric metrics on simply connected manifolds, providing the first such examples without boundary and exploring their geometric properties.

Contribution

It introduces the first known continuous isospectral, non-isometric metrics on simply connected manifolds, expanding the understanding of spectral geometry.

Findings

Constructed continuous isospectral families on S^4×S^3×S^3

Metrics are not locally homogeneous

Scalar curvature critical values vary during deformation

Abstract

We construct continuous families of Riemannian metrics on certain simply connected manifolds with the property that the resulting Riemannian manifolds are pairwise isospectral for the Laplace operator acting on functions. These are the first examples of simply connected Riemannian manifolds without boundary which are isospectral, but not isometric. For example, we construct continuous isospectral families of metrics on the product of spheres S^4\times S^3\times S^3. The metrics considered are not locally homogeneous. For a big class of such families, the set of critical values of the scalar curvature function changes during the deformation. Moreover, the manifolds are in general not isospectral for the Laplace operator acting on 1-forms.