Continuous families of isospectral metrics on simply connected manifolds

Details Continuous families of isospectral metrics on simply connected manifolds Continuous families of isospectral metrics on simply connected manifolds

1997-11-13

arxiv.org/abs/dg-ga/9711010v3

Ann. of Math. (2) 149 (1999), no. 1, 287-308

Mathematics

Differential Geometry

22 pages, published version

Scientific paper

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.

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