Non-isometric Riemannian G -Manifolds with Equal Equivariant Spectra
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Non-isometric Riemannian G-Manifolds with Equal Equivariant Spectra Yuguo Qin1 Received: 16 May 2018 / Revised: 14 June 2018 / Accepted: 19 June 2018 / Published online: 1 September 2018 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract In this paper, the author examines the two methods that people used to systematically construct isospectral non-isometric Riemannian manifolds, the Sunada–Pesce–Sutton method and the torus action method, and shows that both methods can be used to produce equivariantly isospectral non-isometric Riemannian G-manifolds. The author also shows that the Milnor’s isospectral pair is not equivariantly isospectral. Keywords Laplacian · Equivariant spectrum · Equivariantly isospectral Mathematics Subject Classification 58J53 · 58J70
1 Introduction One fundamental problem in spectral geometry is to construct pairs and/or families of isospectral but non-isometric Riemannian manifolds. Such examples not only provide the non-uniqueness (and even the abundance of solutions) to the inverse spectral problem, but also single out the geometric/topological properties of the underline manifolds that are not determined by the spectral data. The first pair of isospectral nonisometric Riemannian manifolds was discovered by Milnor [13] in 1964, after which more and more isospectral pairs of Riemannian manifolds were found. The first systematical way to construct isospectral pairs of Riemannian manifolds was developed by Sunada [16] via representation theory in 1985. His method was further developed by many people, including Pesce [14] and Sutton [17], and sometimes is referred to
The author is partially supported by the NSFC Grant No. 11571331 and “the Fundamental Research Funds for the Central Universities”.
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Yuguo Qin [email protected] School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, People’s Republic of China
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as the Sunada–Pesce–Sutton technique now. For further recent developments, c.f. [2] and [1]. We remark that the Sunada–Pesce approach gives only locally isometric pairs, while Sutton’s generalization can be applied to produce locally non-isometric pairs of isospectral manifolds. In the other direction, people is also working on the problem of finding continuous families of isospectral non-isometric Riemannian metrics on a given manifold, or in other words, an isospectral deformation, of a given Riemannian manifold. The first example of this kind was constructed in 1984 by Gordon and Wilson [9]. Several years later, Deturck and Gordon demonstrated in [3] a way to construct isospectral deformation by Riemannian covering that admits a two-step nilpotent Lie group action. This approach was further developed by Gordon [8] and Schueth [15] to the so-called method of torus action to construct isospectral families of metrics with different local geometry on manifolds including spheres. According to the examples people have
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