Abstract
We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of isospectral but non-homeomorphic nilmanifolds with respect to the sub-Laplacian. We observe that these pairs are also isospectral with respect to the Laplacian. More generally, our method allows us to construct an arbitrary number of isospectral but mutually non-homeomorphic nilmanifolds. Finally, we present two nilmanifolds of different dimensions such that the short time heat trace expansions of the corresponding sub-Laplace operators coincide up to a term which vanishes to infinite order as time tends to zero.
Highlights
In 1966 Mark Kac’s famous paper [22] asked the question “Can one hear the shape of a drum?”
Generalizing the last example the isospectrality problem may be considered for quotients \G of nilpotent Lie groups G of step k ≥ 2 by a lattice
We recall the integral form of the heat kernel for a sub-Laplacian on connected two step nilpotent Lie groups given in [5,6], see [9,12]
Summary
In 1966 Mark Kac’s famous paper [22] asked the question “Can one hear the shape of a drum?”. Based on an explicit heat trace formula for the sub-Laplacian combined with the recent classification of pseudo H -type algebras in [14,15] we can give the negative answer to Kac’s question in this non-standard setting and present a list of new examples. In a second step we need to classify non-homeomorphic nilmanifolds 1\G1 and 2\G2 of the same dimension We reduce this task to a classification of pseudo H -type Lie algebras up to isomorphisms (cf Corollary 7.2). 5 we study the eigenvalues of a matrix-valued function which encodes the structure constants of the pseudo H -type Lie algebra These data are essential in the calculation of the heat kernel of the sub-Laplacian in Sect. We present two nilmanifolds of different dimensions such that the short time heat trace expansions of the corresponding sub-Laplace operators coincide up to a term vanishing to infinite order as time tends to zero
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