Isospectrality I: Introduction

 In this post and the next few, $M$ will denote a Riemannian manifold, and often will be compact and oriented. We will mostly be following a couple of survey papers of Gordon on isospectrality: 'Survey of isospectral manifolds' and 'Sunada's isospectrality technique: two decades later'. Both are excellently written, highly interesting, and informative. I can wholeheartedly recommend them to anyone who's interested in learning more about this topic.

Spectral geometry studies the relationship between the geometry of a Riemannian manifold $M$ and the spectra of various differential operators defined on the manifold. We will see below that the spectrum carries quite a lot of information, which has given rise to a large number of lovely results in the sea of literature. However, this isn't the focus of this series of posts: in the next couple of posts I hope to discuss some of the limits of what the spectrum can see by presenting previous work of many people on finding isospectral but not isometric Riemannian manifolds (I will sometimes be sloppy and omit the 'not isometric' part in stating results when this is implied). One of the most famous questions asked in this vein is Kac's 'Can one hear the shape of a drum?', i.e. whether knowing the spectrum of a 2-dimensional planar region, which if one were to make a drum in that shape is equivalent to knowing what sort of notes can be played, determines the isometry type of the region. We will discuss this in due course. First, we recall some facts about operators.

The Laplace-Beltrami operator

When \( M \) is an oriented Riemannian manifold, the orientation allows one to specify a definite volume form on \( M \), given in an oriented coordinate system \( x^i \) by

\[\text{vol}_n := \sqrt{\left|g\right|} \, dx^1 \wedge \cdots \wedge dx^n\]

where \(\left|g\right| :=\left|\det(g_{ij})\right|\) is the absolute value of the determinant of the metric tensor, and the \( dx^i \) are the 1-forms forming the dual frame to the frame

\[\partial_i := \frac{\partial}{\partial x^i}\]

of the tangent bundle \( TM \) and \( \wedge \) is the wedge product.

The divergence of a vector field \( X \) on the manifold is then defined as the scalar function \(\nabla \cdot X\) with the property

\[(\nabla \cdot X) \, \text{vol}_n := L_X \, \text{vol}_n\]

where \( L_X \) is the Lie derivative along the vector field \( X \). 

The gradient of a scalar function \( f \) is the vector field grad \( f \) that may be defined through the inner product \(\langle\cdot,\cdot\rangle\) on the manifold, as

\[\langle \text{grad } f(x), v_x \rangle = df(x)(v_x)\]

for all vectors \( v_x \) anchored at point \( x \) in the tangent space \( T_xM \) of the manifold at point \( x \). Here, \( df \) is the exterior derivative of the function \( f \); it is a 1-form taking argument \( v_x \). 


The Laplace--Beltrami operator is the (Riemannian) divergence of the (Riemannian) gradient:

\[\Delta f = \text{div}(\nabla f).\]

(There is also a description in local coordinates which is omitted.) This is one of the most important objects in Riemannian geometry, because so much information is contained in the set of eigenvalues. One can show that it is self-adjoint, hence all its eigenvalues are real.

Let \( M \) denote a compact Riemannian manifold without boundary. We want to consider the eigenvalue equation,

\[-\Delta u = \lambda u,\]

where \( u \) is the eigenfunction associated with the eigenvalue \(\lambda\). The compactness of the manifold \(M\) allows one to show that the eigenvalues are discrete and furthermore, the vector space of eigenfunctions associated with a given eigenvalue \(\lambda\), i.e. the eigenspaces are all finite-dimensional. Notice by taking the constant function as an eigenfunction, we get \(\lambda = 0\) is an eigenvalue. 

In fact, all the eigenvalues have to be non-negative. If we multiply the eigenvalue equation through by the eigenfunction \(u\) and integrate the resulting equation on \(M\) we get (using the notation \(dV = \text{vol}_n\)):

\[-\int_M \Delta u \, u \, dV = \lambda \int_M u^2 \, dV\]

Performing an integration by parts or what is the same thing as using the divergence theorem on the term on the left, and since \(M\) has no boundary we get

\[-\int_M \Delta u \, u \, dV = \int_M |\nabla u|^2 \, dV\]

Putting the last two equations together we arrive at

\[\int_M |\nabla u|^2 \, dV = \lambda \int_M u^2 \, dV\]

We conclude from the last equation that \(\lambda \geq 0\).

Here is an example of what one can conclude given information about the spectrum.

Theorem 1: Let \( M \) be an \( n \)-dimensional, compact Riemannian manifold with Ricci curvature satisfying, for some given constant \( \kappa > 0 \), 
\[ \rm{Ric}(\xi, \xi) \geq \kappa(n - 1)|\xi|^2 \] 
for all \( \xi \in TM \). 

1. (Lichnerowicz) The largest eigenvalue $\lambda_1(M)$ satisfies \[ \lambda_1(M) \geq n\kappa, \] 
2.  (Obata) The equality case is precisely when $M$ is isometric to the $n$-sphere of constant sectional curvature $\kappa$
3. (Generalised Topogonov Theorem, proved by Cheng) If \( M \) furthermore has maximal diameter allowed by the Bonnet--Myers theorem, i.e. \[d(M) = \pi / \sqrt{\kappa},\], then \( M \) is isometric to the \( n \)-sphere of constant sectional curvature \( \kappa \).

We show Lichnerowicz's result and leave the interested reader to look up the proofs of the other results in Chavel's book 'Eigenvalues in Riemannian geometry'. 

Proof: Recall that if \( V \) is an \( n \)-dimensional, real inner product space, and \( T: V \to V \) a linear transformation, then the norm of \( T, |T| \), is defined by 
\[ |T|^2 = \sum_{j,k=1}^n \langle Te_j, e_k \rangle^2, \] 
where \( \{e_1, \ldots, e_n\} \) is any orthonormal basis of \( V \). One easily checks that \( |T| \) is well defined, and that 
\[ |T|^2 \geq (\text{tr } T)^2 / n, \] 
with equality if and only if \( T \) is a scalar multiple of the identity. 

If \( M \) is our given Riemannian manifold, with Levi-Civita connection \( \nabla \), and \( f \) is a \( C^2 \) function on \( M \), then the Hessian of \( f \), \( \text{Hess } f: TM \to TM \), is defined by 
\[ (\text{Hess } f)(\xi) = \nabla_\xi \text{ grad } f, \] 
for \( \xi \in TM \). Of course, \( (\text{Hess } f)(M_p) \subseteq M_p \) for all \( p \in M \). Note that 
\[ \Delta f = \text{tr } \text{Hess } f. \]

Recall also the famous Bochner--Lichnerowicz formula: \[\frac{1}{2} \Delta |\text{grad } f|^2 = |\text{Hess } f|^2 + \langle \text{grad } f, \text{ grad } \Delta f \rangle + \text{Ric}(\text{grad } f, \text{grad } f)\] for any \( f \in C^\infty(M) \).

If \( f \) is an eigenfunction on \( M \) with eigenvalue \( \lambda= \lambda_1 \), then the results above combine to imply \[\frac{1}{2} \Delta |\text{grad } f|^2 \geq \lambda^2 f^2 / n + \{ \kappa(n - 1) - \lambda \} |\text{grad } f|^2\] on all of \( M \). Integrating this over \( M \) gives

\[0 \geq \{ \lambda^2 / n + [\kappa(n - 1) - \lambda] \lambda \} \| f \|^2 = \lambda (n - 1) / n (\kappa - \lambda) \| f \|^2 / n\] and the inequality follows.    $\blacksquare$

The heat equation

The heat equation on a Riemannian manifold is given by

\[ u_t + \Delta (u) = 0 \]

for functions \( u: [0, \infty] \times M \to \mathbb{R} \), where \(\Delta(u)\) denotes the Laplacian in the space variable. A function \( K : \mathbb{R}^+ \times M \times M \to M \) is called a heat kernel, or fundamental solution of the heat equation, if it satisfies the following properties:

• [(K1)] \( K(t, x, y) \) is \( C^1 \) in t and \( C^2 \) in \((x, y)\);
• [(K2)] \[ \left( \frac{\partial}{\partial t} + \Delta_x \right) K(t, x, y) = 0 \] where \(\Delta_x\) denotes the Laplacian acting on the second variable;
• [(K3)] \[ \lim_{t \to 0^+} \int_M K(t, x, y) f(y) \, dy = f(x) \] for any smooth function \( f \) with compact support on \( M \).

These conditions imply that the solution of the heat equation with initial condition \( u(0, x) = f(x) \) is given by \( u(t, x) = \int_M K(t, x, y) f(y) \, dy \).

For compact Riemannian manifolds, the heat kernel exists uniquely and may be expressed as

\[ K(t, x, y) = \sum_j e^{-\lambda_j t} \phi_j(x)\phi_j(y) \]

where the \(\lambda_j\) are the eigenvalues of the Laplace-Beltrami operator, and the \(\phi_j\) are the associated eigenfunctions, normalized so as to form an orthonormal basis of \( L^2(M) \).

The trace of the heat kernel, defined by \( Z(t) = \int_M K(t, x, x) \, dx \), satisfies

\[ Z(t) = \sum_{j=0}^\infty e^{-\lambda_j t} \]

where \(\lambda_0 < \lambda_1 < \lambda_2 < \cdots\) is the Laplace spectrum of \( M \). Thus knowledge of the spectrum is equivalent to knowledge of the heat trace.

Minakshisundaram and Pleijel showed that the heat trace of a closed manifold has an asymptotic expansion as \( t \to 0^+ \) of the form

\[ Z(t) = (4\pi t)^{-\frac{n}{2}} \sum_{k=0}^\infty a_k t^k \]

where \( n \) is the dimension of \( M \). The coefficients \( a_k \) are integrals over \( M \) of universal homogeneous polynomials in the curvature and its covariant derivatives. These coefficients are spectral invariants. The first few are given by:

\[ a_0 = \text{vol}(M), \quad a_1 = \frac{1}{6} \int_M \tau, \quad a_2 = \frac{1}{360} \int_M (5\tau^2 - 2|\text{Ric}|^2 - 10|R|^2) \]

where \( \tau \) is the scalar curvature and \( R \) is the curvature tensor. See

From the heat trace, we see that the spectrum determines the dimension, volume, and total scalar curvature of \( M \). In particular, for surfaces, the Gauss–Bonnet theorem implies that the spectrum determines the Euler characteristic of \( M \). Using \( a_1 \) and \( a_2 \), one can tell from the spectrum whether a surface has constant curvature. There is a vast literature concerning geometric information which may be inferred from the heat invariants and thus from the spectrum. For example, Patodi showed that from the spectrum of the Laplacian on functions, one-forms and two-forms, one can tell whether a manifold has constant scalar curvature, whether it is Einstein, and whether it has constant sectional curvature. Brüning and Lesch used heat asymptotics to prove that from the spectrum of an algebraic curve, equipped via projective embedding with the Fubini–Study metric, one can tell whether the curve has singularities, other than multiple points.

Some compact Riemannian manifolds are known to be uniquely determined, or to be uniquely determined within some class of manifolds, by their spectra. Results in this flavour include:

• By a more careful use of the heat invariants, Tanno showed that the round sphere is spectrally determined in all dimensions less than six. 
• Croke and Sharafutdinov proved that there are no nontrivial continuous isospectral deformations of negatively curved manifolds in any dimension.
• Although examples are known of isospectral flat tori and isospectral Riemann surfaces, S. Wolpert nonetheless proved that almost all Riemann surfaces and almost all flat tori are uniquely determined by their spectra; the exceptions lie in a proper real subvariety of the moduli space of all compact Riemann surfaces, respectively, flat tori.  Moreover, Kneser proved that there can be at most finitely many tori with a given spectrum and McKean proved the analogous result for Riemann surfaces. 

In the next post in this series we will discuss the isospectral examples in the last item in some detail.