View source for Spectral geometry

To any [[Riemannian manifold|Riemannian manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s1202201.png" /> one can associate a number of natural elliptic differential operators which arise from the geometric structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s1202202.png" />. Usually, these operators act in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s1202203.png" /> of smooth sections of some [[Vector bundle|vector bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s1202204.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s1202205.png" /> equipped with a positive-definite Riemannian inner product. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s1202206.png" /> is a complete Riemannian manifold (cf. [[Complete Riemannian space|Complete Riemannian space]]), then many of these operators give rise to self-adjoint operators (cf. also [[Self-adjoint operator|Self-adjoint operator]]) in the [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s1202207.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s1202208.png" />-sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s1202209.png" />. Examples of such operators are the Hodge–de Rham Laplacians <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022010.png" /> (cf. also [[Laplace operator|Laplace operator]]) acting on the space of differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022011.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022013.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022014.png" /> this is just the Laplace–Beltrami operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022015.png" /> acting on the space of smooth functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022016.png" />; cf. also [[Laplace–Beltrami equation|Laplace–Beltrami equation]]) and, more generally, second-order self-adjoint operators of Laplace type on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022017.png" />, that is, with leading symbol given by the metric tensor.

Spectral geometry deals with the study of the influence of the spectra of such operators on the geometry and topology of a Riemannian manifold (possibly with boundary; cf. also [[Spectrum of an operator|Spectrum of an operator]]). Everything started with the classical Weyl asymptotic formula, and was later translated into the colloquial question  "Can one hear the shape of a manifold?"  by several authors, most notably M. Kac [[#References|[a12]]], because of the analogy with the [[Wave equation|wave equation]].

The case which has been studied most is that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022018.png" /> on a compact Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022019.png" />, perhaps with boundary. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022020.png" />, one imposes Dirichlet or Neumann boundary conditions to get a self-adjoint extension. Then the corresponding self-adjoint extension has countably many eigenvalues and these form a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022021.png" /> (each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022022.png" /> being repeated with its multiplicity), which accumulate only at infinity, see [[#References|[a6]]]. The entire collection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022023.png" />'s with their finite multiplicities is called the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022024.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022025.png" />. Two compact Riemannian manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022027.png" /> are said to be isospectral if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022028.png" />. Two isometric compact Riemannian manifolds are necessarily isospectral. The inverse problem, namely to what extent does <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022029.png" /> determine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022030.png" />, up to isometry (see also below), has been studied quite extensively. In particular, the answer to the question whether isospectral Riemannian manifolds are necessarily isometric is now (1998) well known to be negative. The first counterexample was a pair of isospectral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022031.png" />-dimensional flat tori given by J. Milnor in 1964. Until 1980, however, the only other examples discovered were a few additional pairs of flat tori or twisted products with tori. Starting in 1980, many examples as well as fairly general techniques for constructing examples of isospectral Riemannian manifolds that are not isometric have appeared, see [[#References|[a2]]], [[#References|[a4]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]], [[#References|[a13]]], [[#References|[a23]]], and [[#References|[a19]]]. Among these examples are pairs of manifolds with non-isomorphic fundamental groups, locally symmetric spaces both of rank one and higher rank, Riemann surfaces of every genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022032.png" />, continuous families of isospectral Riemannian manifolds, lens spaces, non-locally isometric Riemannian manifolds and other examples. T. Sunada introduced a systematic method for constructing pairs of non-isometric isospectral Riemannian manifolds (see [[#References|[a22]]]), and H. Pesce succeeded in giving a major strengthening of this method (see [[#References|[a20]]], [[#References|[a21]]]).

Since it is difficult to study <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022033.png" /> directly, instead one introduces certain functions of the eigenvalues which can be used to extract geometric information from the spectrum. Some useful such functions having interesting applications to spectral geometry are discussed below.

===Heat coefficients.===
The connecting link between the heat equation approach to index theory and spectral geometry is the asymptotic expansion of the heat kernel. For simplicity, assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022034.png" />. S. Minakshisundaram and A. Pleijel have proved that for every closed Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022035.png" /> there exists an asymptotic expansion

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022036.png" /></td> </tr></table>

as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022037.png" />. The numerical invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022038.png" /> are the heat coefficients and they determine and are uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022039.png" />. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022040.png" /> are locally computable from the metric. In fact, they are universal polynomials in the curvature of the [[Levi-Civita connection|Levi-Civita connection]] associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022041.png" /> and their covariant derivatives. In particular, it follows from this expansion that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022042.png" /> determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022044.png" /> and the total [[Scalar curvature|scalar curvature]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022045.png" /> (and hence, by the [[Gauss–Bonnet theorem|Gauss–Bonnet theorem]], the [[Euler characteristic|Euler characteristic]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022046.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022047.png" /> is a surface). There are more results along these lines; for instance, the fact that standard spheres and real projective spaces in dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022048.png" /> are characterized by their spectra can be deduced from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022051.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022052.png" />'s get more and more complicated as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022053.png" /> increases, but at least the leading terms (i.e. the terms with a maximal number of derivatives) in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022054.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022055.png" /> can be described. General references for this area are [[#References|[a1]]], [[#References|[a3]]], [[#References|[a7]]], [[#References|[a8]]], and [[#References|[a14]]].

===The regularized determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022056.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022057.png" />.===
This regularized determinant is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022058.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022059.png" /> is the meromorphic extension to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022060.png" /> of the zeta-function associated to the non-zero eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022061.png" />. It is a global spectral invariant, that is, it cannot be computed locally from the metric. There are very interesting results concerning the extremal points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022062.png" /> as a function of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022063.png" /> on a given closed surface. References are [[#References|[a17]]], [[#References|[a16]]] and [[#References|[a18]]].

===The regularized characteristic determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022064.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022065.png" />.===
This regularized characteristic determinant is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022066.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022067.png" /> is the [[Analytic continuation|analytic continuation]] to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022068.png" /> of the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022069.png" /></td> </tr></table>

where the sum ranges over the non-zero eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022070.png" />. This function generalizes the concept of a characteristic polynomial in finite dimension. For a closed surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022071.png" /> of constant negative curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022073.png" /> is closely related to the Selberg zeta-function. A reference is [[#References|[a5]]].

===Topological aspects.===
If one takes into account the spectra of other natural geometric operators, then global topological aspects come into play. For instance, a linear combination of the values at zero of the analytic continuations to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022074.png" /> of the zeta-functions associated respectively to the non-zero eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022075.png" /> gives the Euler characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120220/s12022076.png" /> and another linear combination of the derivatives at zero of these functions gives the [[Reidemeister torsion|Reidemeister torsion]]. A general reference for this area as well as for spectral problems on non-compact Riemannian manifolds is [[#References|[a15]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.H. Bérard,   "Spectral geometry: Direct and inverse problems" , ''Lecture Notes Math.'' , '''1207''' , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.H. Bérard,   "Variétés Riemanniennes isospectrales non isométriques"  ''Astérisque'' , '''177–178'''  (1989)  pp. 127–154</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Berger,   P. Gauduchon,   E. Mazet,   "Le spectre d'une variété Riemannienne" , ''Lecture Notes Math.'' , '''194''' , Springer  (1971)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Buser,   "Geometry and spectra of compact Riemann surfaces" , ''Progr. Math.'' , '''106''' , Birkhäuser  (1992)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Cartier,   A. Voros,   "Une nouvelle interprétation de la formule des traces de Selberg" , ''The Grothendieck Festschrift: II'' , ''Progr. Math.'' , '''87''' , Birkhäuser  (1990)  pp. 1–67</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I. Chavel,   "Eigenvalues in Riemannian geometry" , Acad. Press  (1984)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P.B. Gilkey,   "Spectral geometry of Riemannian manifolds"  ''Contemp. Math.'' , '''101'''  (1989)  pp. 147–153</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  P.B. Gilkey,   "Invariance theory, the heat equation, and the Atiyah–Singer index theorem" , CRC  (1995)  (Edition: Second)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  C.S. Gordon,   "You can't hear the shape of a manifold"  J. Tirao and N. Wallach (ed.) , ''New Developments in Lie Theory and Their Applications. Proc. 3rd Workshop Represent. Th. Lie Groups Appl. (Cordoba, 1989)'' , ''Progr. Math.'' , '''105''' , Birkhäuser  (1992)  pp. 129–146</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  C.S. Gordon,   "Isospectral closed Riemannian manifolds which are not locally isometric I"  ''J. Diff. Geom.'' , '''37'''  (1993)  pp. 639–649</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  C.S. Gordon,   "Isospectral closed Riemannian manifolds which are not locally isometric II"  ''Contemp. Math.'' , '''173'''  (1994)  pp. 121–131</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  M. Kac,   "Can one hear the shape of a drum?"  ''Amer. Math. Monthly'' , '''73'''  (1966)  pp. 1–23</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  C.S. Gordon,   D.L. Webb,   S. Wolpert,   "One can not hear the shape of a drum"  ''Bull. Amer. Math. Soc.'' , '''27''' :  1  (1992)  pp. 134–138</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  H.P. McKean,   I.M. Singer,   "Curvature and the eigenvalues of the Laplacian"  ''J. Diff. Geom.'' , '''1'''  (1967)  pp. 43–69</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  W. Müller,   "Spectral theory and geometry" , ''First European Congress of Mathematics (Paris, July 6-10, 1992)'' , '''I''' , Birkhäuser  (1994)  pp. 153–185</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  B. Osgood,   R. Phillips,   P. Sarnak,   "Extremals of determinants of Laplacians"  ''J. Funct. Anal.'' , '''80'''  (1988)  pp. 148–211</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  B. Osgood,   R. Phillips,   P. Sarnak,   "Compact isospectral sets of surfaces"  ''J. Funct. Anal.'' , '''80'''  (1988)  pp. 212–234</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  B. Osgood,   R. Phillips,   P. Sarnak,   "Moduli spaces, heights and isospectral sets of plain domains"  ''Ann. of Math.'' , '''129'''  (1989)  pp. 293–362</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  H. Pesce,   "Variétés isospectrales et représentations de groupes"  ''Contemp. Math.'' , '''173'''  (1994)  pp. 231–240</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  H. Pesce,   "Représentations relativement équivalentes et variétés Riemanniennes isospectrales"  ''Comment. Math. Helvetici'' , '''71'''  (1996)  pp. 243–268</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  H. Pesce,   "Une réciproque générique du théorème de Sunada"  ''Compositio Math.'' , '''109'''  (1997)  pp. 357–365</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top">  T. Sunada,   "Riemannian coverings and isospectral manifolds"  ''Ann. of Math.'' , '''121'''  (1985)  pp. 169–186</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top">  M.F. Vignéras,   "Variétés Riemanniennes isospectrales et non isométriques"  ''Ann. of Math.'' , '''112'''  (1980)  pp. 21–32</TD></TR></table>