Isospectral manifolds with different local geometries (Q2717076)

Two compact Riemannian manifolds are said to be isospectral if the spectra of their associated Laplace operators coincide. Isospectral manifolds share some basic invariants, like the dimension, the volume and, more generally, the heat trace invariants (arising from the asymptotic development of the trace of the heat kernel, which is in fact determined by the spectrum). However, two isospectral manifolds need not be isometric: this basic fact was first proved by \textit{J. Milnor} [Proc. Nat. Acad. Sci. USA 51, 542 (1964; Zbl 0124.31202)]. A general method in the study of isospectrality was given by \textit{T. Sunada} [Ann. Math., II. Ser. 121, 169-186 (1985; Zbl 0585.58047)]: one gets isospectral manifolds as quotients of a fixed manifold by group actions satisfying suitable properties. However, the Sunada principle always yields locally isometric manifolds, hence the question: - If two manifolds are isospectral, are they necessarily locally isometric? The answer is again negative; the first counterexample was given by \textit{Z. Szabó} in 1991 (for manifolds with boundary, and appearing later in [Geom. Funct. Anal. 9, No. 1, 185-214 (1993; Zbl 0964.53026)]). Then, \textit{C. Gordon} extended Szabó's examples into a new general method which produces isospectral closed manifolds which are not locally isometric [see J. Differ. Geom. 37, No. 3, 639-649 (1993; Zbl 0792.53037]. It reads as follows: ``If a torus acts on two Riemannian manifolds freely and isometrically with totally geodesic fibers, and if the quotients of the manifolds by any subtorus of codimension at most one are isospectral when endowed with the submersion metric, then the original manifolds are isospectral''. NEWLINENEWLINENEWLINEThe present paper provides several new examples showing that even imposing somewhat strict conditions, the spectrum is far from determining the local geometry. By employing suitable variations, and simplified versions, of C. Gordon's principle above, the author in fact constructs: -- The first examples of four dimensional isospectral manifolds which are not locally isometric (note that the lowest dimension of previously known examples of this type was six). -- The first examples of isospectral left invariant metrics on compact Lie groups (these examples come in continuous families) and the first examples of isospectral manifolds which are simply connected and irreducible (examples of isospectral, simply connected manifolds were previously given by the author in her paper [Ann. Math. (2) 149, No. 1, 287-308 (1999; Zbl 0964.53027)]). Moreover, the following rigidity result is proved: any continuous isospectral family of left invariant isospectral metrics which contains a bi-invariant metric must be trivial. NEWLINENEWLINENEWLINEThe first examples of conformally equivalent manifolds which are isospectral and not locally isometric (examples of conformally equivalent isospectral manifolds were given by \textit{R. Brooks} and \textit{C. Gordon} in [Bull. Am. Math. Soc. New Ser. 23, No. 2, 433-436 (1990; Zbl 0722.58045)], but they are all locally isometric because they arise from the Sunada method). Hence, even restricting the metric to a fixed conformal class, the local geometry is not determined by the spectrum.