Editing Isospectral (section)

== Isospectral manifolds ==

Two closed [[Riemannian manifold]]s are said to be isospectral if the eigenvalues of their [[Laplace–Beltrami operator]] (Laplacians), counted multiplicities, coincide. One of fundamental problems in spectral geometry is to ask to what extent the eigenvalues determine the geometry of a given manifold.

There are many examples of isospectral manifolds which are not isometric. The first example was given in 1964 by [[John Milnor]]. He constructed a pair of flat tori of 16 dimension, using arithmetic lattices first studied by [[Ernst Witt]]. After this example, many isospectral pairs in dimension two and higher were constructed (for instance, by M. F. Vignéras, A. Ikeda, H. Urakawa, C. Gordon). In particular {{harvtxt|Vignéras|1980}}, based on the [[Selberg trace formula]] for PSL(2,'''R''') and PSL(2,'''C'''), constructed examples of isospectral, non-isometric closed hyperbolic 2-manifolds and 3-manifolds as quotients of hyperbolic 2-space and 3-space by arithmetic subgroups, constructed using quaternion algebras associated with quadratic extensions of the rationals by [[class field theory]].<ref>{{harvnb|Maclachlan|Reid|2003}}</ref>  In this case Selberg's trace formula shows that the spectrum of the Laplacian fully determines the ''length spectrum''{{Citation needed|date=May 2010}}, the set of lengths of closed geodesics in each free homotopy class, along with the twist along the geodesic in the 3-dimensional case.<ref>This amounts to knowing the conjugacy class of the corresponding group element in PSL(2,'''R''') or PSL(2,'''C''').
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In 1985 [[Toshikazu Sunada]] found a general method of construction based on a [[covering space]] technique, which, either in its original or certain generalized versions, came to be known as the Sunada method or Sunada construction. Like the previous methods it is based on the trace formula, via the [[Selberg zeta function]]. Sunada noticed that the method of constructing number fields with the same [[Dedekind zeta function]] could be adapted to compact manifolds. His method relies on the fact that if ''M'' is a finite covering of a compact Riemannian manifold
''M''<sub>0</sub> with ''G'' the [[finite group]] of [[deck transformation]]s and ''H''<sub>1</sub>, ''H''<sub>2</sub> are subgroups of ''G'' meeting each conjugacy class of ''G'' in the same number of elements, then the manifolds ''H''<sub>1</sub> \ ''M'' and ''H''<sub>2</sub> \ ''M'' are isospectral
but not necessarily isometric. Although this does not recapture the arithmetic examples of Milnor and Vignéras{{Citation needed|date=May 2010}}, Sunada's method yields many known examples of isospectral manifolds. It led C. Gordon, [[David Webb (mathematician)|D. Webb]] and S. Wolpert to the discovery in 1991 of a counter example to [[Mark Kac]]'s problem "[[Hearing the shape of a drum|Can one hear the shape of a drum?]]" An elementary treatment, based on Sunada's method, was later given in {{harvtxt|Buser|Conway|Doyle|Semmler|1994}}.

Sunada's idea also stimulated the attempt to find isospectral examples which could not be obtained by his technique. Among many examples, the most striking one is a simply connected example of {{harvtxt|Schueth|1999}}. On the other hand, [[Alan Reid (mathematician)|Alan Reid]] proved that certain isospectral arithmetic hyperbolic manifolds in are commensurable. <ref>{{ Cite journal | url = http://doi.org/10.1215/S0012-7094-92-06508-2 | title = Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds| date = 1992| doi = 10.1215/S0012-7094-92-06508-2| last1 = Reid| first1 = Alan W.| journal = Duke Mathematical Journal| volume = 65| issue = 2}}</ref>