Geometric structures and the Laplace spectrum, Part II

Lin, Samuel; Schmidt, Benjamin; Sutton, Craig

doi:10.1090/tran/8417

Geometric structures and the Laplace spectrum, Part II
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by Samuel Lin, Benjamin Schmidt and Craig Sutton PDF

Trans. Amer. Math. Soc. 374 (2021), 8483-8530 Request permission

Abstract:

We classify the locally homogeneous Riemannian metrics on elliptic three-manifolds, which contributes to our classification of the compact homogeneous Riemannian three-manifolds. Then, we apply the former result to our continued exploration of the extent to which the spectrum encodes the local geometry of a locally homogeneous three-manifold. Specifically, using the first four heat invariants, we find that any collection of isospectral locally homogeneous metrics on an elliptic three-manifold $\Gamma \backslash S^3$ contains at most two isometry classes and these classes are necessarily locally isometric. In fact, if the elliptic three-manifold is $S^3$, $\mathbb {R}P^3$ or has non-cyclic fundamental group, then (up to isometry) its locally homogeneous Riemannian metrics can be mutually distinguished via their spectra. Currently, there is no example of an isospectral pair consisting of locally homogeneous elliptic three-manifolds with non-isometric universal cover; however, we show that if such a pair exists, then it satisfies certain restrictive geometric conditions. Finally, we note that our classification of locally homogeneous elliptic three-manifolds shows that, for $q \geq 3$, the lens space $L(q;1,1)$ admits pairs of locally isometric locally homogeneous metrics where only one of the metrics is homogeneous. While this phenomenon cannot occur in dimension two, these lens spaces account for all such examples in dimension three.

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