Isospectral pairs of metrics on balls, spheres, and other manifolds with different local geometries

• 成果类型: Article
• 署名作者: Szabó, ZI
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/3062103
• 发表日期: 2001
• 页码: 437-475
• 关键词: closed riemannian-manifolds; continuous families; deformations; nilpotent; SURFACES; spectrum; SPACES

摘要

The first isospectral pairs of metrics are constructed on the most simple simply connected domains, namely, on balls and spheres. This long-standing problem, concerning the existence of such pairs, has been solved by a new method called anticommutator technique. Among the wide range of such pairs, the most striking examples are provided on the spheres S4k-1, where k greater than or equal to 3. One of these metrics is homogeneous (since it is the metric on the geodesic sphere of a 2-point homogeneous space), while the other is locally inhomogeneous. These examples demonstrate the surprising fact that no information about the isometrics is encoded in the spectrum of the Laplacian acting on functions. In other words, the group of isometries, even the local homogeneity property, is lost to the nonaudible in the debate of audible versus nonaudible geometry.