Spectral geometry

To any Riemannian manifold $( M , g )$ one can associate a number of natural elliptic differential operators which arise from the geometric structure of $( M , g )$. Usually, these operators act in the space $C ^ { \infty } ( E )$ of smooth sections of some vector bundle $E$ over $M$ equipped with a positive-definite Riemannian inner product. If $( M , g )$ is a complete Riemannian manifold (cf. Complete Riemannian space), then many of these operators give rise to self-adjoint operators (cf. also Self-adjoint operator) in the Hilbert space $L ^ { 2 } ( E )$ of $L^{2}$-sections of $E$. Examples of such operators are the Hodge–de Rham Laplacians $\Delta ^ { ( p ) }$ (cf. also Laplace operator) acting on the space of differential $p$-forms on $M$, $0 \leq p \leq \operatorname { dim } M$ (for $p = 0$ this is just the Laplace–Beltrami operator $\Delta ^ { ( 0 ) } = \Delta$ acting on the space of smooth functions on $M$; cf. also Laplace–Beltrami equation) and, more generally, second-order self-adjoint operators of Laplace type on $C ^ { \infty } ( E )$, 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). 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 [a12], because of the analogy with the wave equation.

The case which has been studied most is that of $\Delta$ on a compact Riemannian manifold $( M , g )$, perhaps with boundary. When $\partial M \neq \emptyset$, 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 $0 \leq \lambda _ { 0 } \leq \lambda _ { 1 } \leq \ldots$ (each $\lambda$ being repeated with its multiplicity), which accumulate only at infinity, see [a6]. The entire collection of $\lambda$'s with their finite multiplicities is called the spectrum of $( M , \Delta )$ and is denoted by $\operatorname { spec } ( M , \Delta )$. Two compact Riemannian manifolds $( M , g )$ and $( M ^ { \prime } , g ^ { \prime } )$ are said to be isospectral if $\operatorname { spec } ( M , \Delta ) = \operatorname { spec } ( M ^ { \prime } , \Delta ^ { \prime } )$. Two isometric compact Riemannian manifolds are necessarily isospectral. The inverse problem, namely to what extent does $\operatorname { spec } ( M , \Delta )$ determine $( M , g )$, 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 $16$-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 [a2], [a4], [a9], [a10], [a11], [a13], [a23], and [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 $> 4$, 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 [a22]), and H. Pesce succeeded in giving a major strengthening of this method (see [a20], [a21]).

Since it is difficult to study $\operatorname { spec } ( M , \Delta )$ 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 $\partial M = \emptyset$. S. Minakshisundaram and A. Pleijel have proved that for every closed Riemannian manifold $( M , g )$ there exists an asymptotic expansion

\begin{equation*} \sum _ { k = 0 } ^ { \infty } \operatorname { exp } ( - \lambda _ { j } t ) \sim ( 4 \pi t ) ^ { - \operatorname { dim } ( M ) / 2 } \sum _ { k = 0 } ^ { \infty } a _ { k } t ^ { k } \end{equation*}

as $t \searrow 0$. The numerical invariants $a_k$ are the heat coefficients and they determine and are uniquely determined by $\operatorname { spec } ( M , \Delta )$. The numbers $a_k$ are locally computable from the metric. In fact, they are universal polynomials in the curvature of the Levi-Civita connection associated to $( M , g )$ and their covariant derivatives. In particular, it follows from this expansion that $\operatorname { spec } ( M , \Delta )$ determines $\operatorname{dim}( M )$, $\operatorname{Vol}( M , g )$ and the total scalar curvature of $( M , g )$ (and hence, by the Gauss–Bonnet theorem, the Euler characteristic of $M$ if $M$ is a surface). There are more results along these lines; for instance, the fact that standard spheres and real projective spaces in dimensions $\leq 6$ are characterized by their spectra can be deduced from $a_1$, $a _2$, $a_3$. The $a_k$'s get more and more complicated as $k$ increases, but at least the leading terms (i.e. the terms with a maximal number of derivatives) in the $a_k$ for all $k$ can be described. General references for this area are [a1], [a3], [a7], [a8], and [a14].

The regularized determinant $\operatorname { det } ( \Delta )$ of $\Delta$.

This regularized determinant is defined by

\begin{equation*} \operatorname { det } ( \Delta ) = \operatorname { exp } \left( - \frac { d } { d s } \zeta ( s ) | _ { s = 0 } \right), \end{equation*}

where $\zeta$ is the meromorphic extension to $\mathbf{C}$ of the zeta-function associated to the non-zero eigenvalues $\lambda _ { k }$. 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 $\operatorname { det } ( \Delta )$ as a function of the metric $g$ on a given closed surface. References are [a17], [a16] and [a18].

The regularized characteristic determinant $\operatorname { det } ( \Delta + z )$ of $\Delta + z$.

This regularized characteristic determinant is defined by

\begin{equation*} \operatorname { det } ( \Delta + z ) = \operatorname { exp } \left( - \frac { \partial } { \partial s } \zeta ( s , z ) | _ { s = 0 } \right), \end{equation*}

where $\zeta$ is the analytic continuation to $\mathbf{C} \times ( \mathbf{C} \backslash ( - \infty , 0 ) )$ of the function

\begin{equation*} \sum _ { k } ( z + \lambda _ { k } ) ^ { - s } , \operatorname { Re } ( s ) > \frac { 1 } { 2 } \operatorname { dim } M, \end{equation*}

where the sum ranges over the non-zero eigenvalues $\lambda _ { k }$. This function generalizes the concept of a characteristic polynomial in finite dimension. For a closed surface $M$ of constant negative curvature $- 1$, $\operatorname { det } ( \Delta + z )$ is closely related to the Selberg zeta-function. A reference is [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 $\mathbf{C}$ of the zeta-functions associated respectively to the non-zero eigenvalues of $\operatorname { spec } ( M , \Delta ^ { ( 0 ) } ) , \ldots , \operatorname { spec } ( M , \Delta ^ { ( \dim M ) } )$ gives the Euler characteristic of $M$ and another linear combination of the derivatives at zero of these functions gives the Reidemeister torsion. A general reference for this area as well as for spectral problems on non-compact Riemannian manifolds is [a15].

References

[a1]	P.H. Bérard, "Spectral geometry: Direct and inverse problems" , Lecture Notes Math. , 1207 , Springer (1986)
[a2]	P.H. Bérard, "Variétés Riemanniennes isospectrales non isométriques" Astérisque , 177–178 (1989) pp. 127–154
[a3]	M. Berger, P. Gauduchon, E. Mazet, "Le spectre d'une variété Riemannienne" , Lecture Notes Math. , 194 , Springer (1971)
[a4]	P. Buser, "Geometry and spectra of compact Riemann surfaces" , Progr. Math. , 106 , Birkhäuser (1992)
[a5]	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
[a6]	I. Chavel, "Eigenvalues in Riemannian geometry" , Acad. Press (1984)
[a7]	P.B. Gilkey, "Spectral geometry of Riemannian manifolds" Contemp. Math. , 101 (1989) pp. 147–153
[a8]	P.B. Gilkey, "Invariance theory, the heat equation, and the Atiyah–Singer index theorem" , CRC (1995) (Edition: Second)
[a9]	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
[a10]	C.S. Gordon, "Isospectral closed Riemannian manifolds which are not locally isometric I" J. Diff. Geom. , 37 (1993) pp. 639–649
[a11]	C.S. Gordon, "Isospectral closed Riemannian manifolds which are not locally isometric II" Contemp. Math. , 173 (1994) pp. 121–131
[a12]	M. Kac, "Can one hear the shape of a drum?" Amer. Math. Monthly , 73 (1966) pp. 1–23
[a13]	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
[a14]	H.P. McKean, I.M. Singer, "Curvature and the eigenvalues of the Laplacian" J. Diff. Geom. , 1 (1967) pp. 43–69
[a15]	W. Müller, "Spectral theory and geometry" , First European Congress of Mathematics (Paris, July 6-10, 1992) , I , Birkhäuser (1994) pp. 153–185
[a16]	B. Osgood, R. Phillips, P. Sarnak, "Extremals of determinants of Laplacians" J. Funct. Anal. , 80 (1988) pp. 148–211
[a17]	B. Osgood, R. Phillips, P. Sarnak, "Compact isospectral sets of surfaces" J. Funct. Anal. , 80 (1988) pp. 212–234
[a18]	B. Osgood, R. Phillips, P. Sarnak, "Moduli spaces, heights and isospectral sets of plain domains" Ann. of Math. , 129 (1989) pp. 293–362
[a19]	H. Pesce, "Variétés isospectrales et représentations de groupes" Contemp. Math. , 173 (1994) pp. 231–240
[a20]	H. Pesce, "Représentations relativement équivalentes et variétés Riemanniennes isospectrales" Comment. Math. Helvetici , 71 (1996) pp. 243–268
[a21]	H. Pesce, "Une réciproque générique du théorème de Sunada" Compositio Math. , 109 (1997) pp. 357–365
[a22]	T. Sunada, "Riemannian coverings and isospectral manifolds" Ann. of Math. , 121 (1985) pp. 169–186
[a23]	M.F. Vignéras, "Variétés Riemanniennes isospectrales et non isométriques" Ann. of Math. , 112 (1980) pp. 21–32