Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Marked Length Spectrum vs. The Laplace Spectrum on Forms on
  Riemannian Nilmanifolds   The subject of this paper is the relationship among the marked length
spectrum, the length spectrum, the Laplace spectrum on functions, and the
Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show
that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in
this class has the same marked length spectrum, they necessarily share the same
Laplace spectrum on functions. In contrast, we present the first example of a
pair of isospectral Riemannian manifolds with the same marked length spectrum
but not the same spectrum on one-forms. Outside of the standard spheres vs. the
Zoll spheres, which are not even isospectral, this is the only example of a
pair of Riemannian manifolds with the same marked length spectrum, but not the
same spectrum on forms. This partially extends and partially contrasts the work
of Eberlein, who showed that on two-step nilmanifolds, the same marked length
spectrum implies the same Laplace spectrum both on functions and on forms.

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The categories of this paper are: hep-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: math-ph
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The categories of this paper are: physics
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The categories of this paper are: cmp-lg
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The categories of this paper are: atom-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: eess
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-th
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The categories of this paper are: mtrl-th