Using elementary techniques from Geometric Analysis, Partial Differential Equations, and Abelian Algebras, we uncover a novel, yet familiar, global geometric invariant — namely the indexed set of integrals of triple products of eigenfunctions of the Laplace-Beltrami operator, to precisely characterize which isospectral closed Riemannian manifolds are isometric.
For a closed Riemannian manifold , characterizing its class of non-isometric, isospectral manifolds is a type of Inverse Problem [DH11] in Spectral Geometry. Naïvely one might speculate that this class would always be empty. However, the academic literature is rich with decades-old constructions of specific pairings of counterexamples: beginning in 1964 with John Milnor’s 16-dimensional pair of non-isometric, isospectral flat tori [JM64], and continuing [CS92] towards the generic dimensional characterization of flat tori in Alexander Schiemann’s 1993 doctoral thesis [AS94] — replete with a computer aided search for the critical case. A modern survey of the full flat tori history appears in [NRR22].
Along the way were insightful offshoots into more sophisticated, non-Euclidean symmetric covering spaces; constructing such isospectral, non-isometric “duets” involving nontrivial curvature tensors (and their spectrum-determined Euler characteristics in dimension 2 [MS67].) A prime example of this effort was Toshikazu Sunada’s 1985 [TS85] invention of a general-purpose covering space framework, which he then deployed in the same work to construct hyperbolic duets in dimensions 2 and 3.
For inhomogeneous Riemannian metrics, Carolyn Gordon discovered duets that are not even locally isometric [CG93].
Work continues in many related areas [DH11], such as determining topological characteristics of the class of isospectral, non-isometric manifolds in general (empty [ST80], finite [AS93], rigid [GK80], and compact [GZ97]) as a subset of different moduli spaces of Riemannian metrics.
What we offer in this article is a new perspective on a familiar tool: indexed Fourier coefficients of pairwise products of eigenfunctions as a discrete “algebraic/topological invariant” to complement the existing, discrete “analytic invariant” — the non-negative spectrum of the Laplace-Beltrami operator (herein referred to as the Laplacian) on . Combined, we observe the pair provides a “discrete global geometric representation” of the isometry classes of isospectral, closed Riemannian manifolds.
Given a (non-decreasing on the eigenvalues) orthonormal basis of eigenfunctions for the (non-negative) Laplacian on associated with a closed Riemannian manifold , define
To be isometric to , it is a necessary and sufficient condition for another iosospectral closed Riemannian manifold to have an orthonormal basis of eigenfunctions (for its Laplacian) that both preserves the associated eigenvalues and possesses an invariant under each basis.
Symmetry plays an important role in computationally tractable cases [TF17] [LS18] [PS94], which is aptly illustrated in our flat tori Example below. However, the strength of our approach is perhaps best made apparent in the case of manifolds with the fewest number of Riemannian symmetries, which is the generic case often coinciding with the eigenvalues being unique (i.e., without nontrivial multiplicity.) In this circumstance we offer an unproven conjecture: if the elements of the set , for a pair of real-valued eigenfunction bases, agree up to changes of sign, then there must be a change of basis map that exactly aligns them (via changes of sign) and thus proves the underlying closed Riemannian manifolds are isometric by application of the Theorem above.
The motivation for the study of is loosely derived from the study of the role of the linear multiplication operator in the definition of a Vertex Operator Algebra [FBZ04] associated with a Chiral Conformal Field Theory. Here is the Vector Space of States and is the space of formal Laurent series in with coefficients in . Since often comes equipped as a Hilbert Space with a traditional Fourier series orthonormal basis, indexing using the Fourier basis elements of is only slightly more involved than the case studied here, but quite similar in spirit. However a detailed comparison is out of scope for this article.
These results were first demonstrated during a similarly titled talk by the author at MSRI in 1997, but they appear here in published form for the first time.
Now with as above, for and note that the Fourier coefficients
since is uniquely representable as its rapidly converging Fourier Series (-specific Sobolev Embeddings [MT13] [RS75], together with Weyl’s Asymptotic Law [HW11], imply the terms in the sum are uniformly in [LH68], .) Then we see that for , the Fourier coefficients of the pointwise product are
and so, critically, any multivariate polynomial (on smooth functions) commutes with any spectrum-preserving -eigenfunction orthonormal basis map that preserves .
Moreover if is Borel-measurable, then the results above hold pointwise for the characteristic function of everywhere except along the boundary of : if and ,
and by uniqueness, we have the following identity
This implies any such basis map as above carries characteristic functions (as members of ) to characteristic functions in a measure-preserving fashion.
The point of these computations is to emphasize the fact that characterizes the Harmonic Analysis of the pointwise multiplication operator on , which is a dense subalgebra of the Abelian algebra , by the Stone-Weierstrass theorem.
For the rapid convergence of these above sums involving , note that products of eigenfunctions are smooth, so these Fourier coefficients decay as above (in each index). For more details, see Emmett Wyman’s work in 2022 with these coefficients as it relates to the triangle inequality on the eigenvalues [EW22].
Note: we may always assume
where is the Kronecker delta. Since is a spectral invariant [HW11], this information is already available from isospectrality considerations.
For necessity, let be an isometry between closed Riemannian manifolds, and let the target orthonormal basis of eigenfunctions on be the pull-back via of the orthonormal basis on above. Since
we are done with the necessity argument.
For sufficiency, we now consider the linear, bijective orthonormal eigenfunction basis map from to and note that from the calculations in the Preliminaries above, preserves pointwise products for smooth functions (and preserves characteristic functions when extended to ) by the premise that is invariant under this map.
preserves the uniform norm.
Let be a smooth partition of unity on .
Thus (Kronecker delta).
By the dominated convergence theorem,
which is a characteristic function of positive measure on each disjoint subset . This means the Lemma is proven for each , since the limiting characteristic function of a set with positive measure is preserved (and hence has uniform norm 1, as do all ).
Without loss of generality, we may apply the special case result shown for the smooth partition of unity , where has positive measure, and the Lemma is proven in full.
This means that on a dense set of (and ), we have established as an isomorphism of Abelian algebras, and thus can be extended to an isomorphism of and in the same category.
Now we apply the Gelfand-Naimark Representation Theorem (in contravariant functor form) for Abelian algebras [JC19] to represent this isomorphism by a homeomorphism between and . Since it is bijective on smooth functions, it too must be smooth.
As this now diffeomorphism preserves eigenvalues and eigenfunctions (by hypothesis on ), it must preserve the Laplacian on smooth functions. Hence it also must preserve the principal symbols of these same elliptic operators [MT13]. The principal symbols of the Laplacian are simply another means of expressing the Riemannian metric on the manifolds in question.
This completes the proof of the Theorem.
Let be an indexed, rank lattice of Lie Algebra weights for the quotient space representation of as translation invariant (i.e., constant) vector fields on itself, when is also viewed as associated Lie Group over a torus defined by . These weights define integrable lifts of 1-forms over the torus that integrate to linear functionals as its Lie Group (covering the torus). These linear functionals can then be uniformly rescaled (by ) and exponentiated to form multiplicative characters that descend to form an orthonormal basis of , with Lebesgue (Haar) measure .
Moreover, this basis simultaneously diagonalizes the flat torus’s Laplacian because the Laplacian is the image of a symmetric, negative-definite quadratic Casimir element under this (constant coefficient linear differential operator) quotient space representation of the universal enveloping algebra. Hence, its eigenvalues are in constant proportion (of ) to the Casimir-element-determined-length-squared of each character’s weight in the lattice.
We presently view the above basis
to be our Theorem-applicable Fourier basis of orthonormal (multiplicative character) eigenfunctions (of this quotient representation of the (negative) Euclidean Casimir element) directly corresponding to . By our Theorem’s hypotheses, we must have (with the Euclidean norm on the weights).
Now we can compute
As this condition is linear on the weight lattice , only an orthonormal eigenfunction basis map which is induced from a volume-preserving invertible linear map between two such indexed, rank weight lattices will keep the “algebraic/topological” indexed data set invariant.
However, in order to apply our Theorem, it is essential that such a linear map be on the weight lattice, because the induced eigenfunction basis map
must also preserve the “analytic” invariants — the Casimir-element induced figure for each indexed weight, i.e. the individual eigenvalues of the flat-tori’s Laplacian.
This representation-theoretical account [AK01] is exactly equivalent to the prior development of lattice congruence [NRR22] traditonally used to delineate isometry classes of flat tori. In fact, the matrix transpose of such a linear map , as described in the prior paragraph, is the contravariant Riemannian isometry between the tori, as provided by application of the Gelfand-Naimark Representation Theorem during the Proof of our Theorem.
If every eigenvalue has multiplicity , given a pair of eigenvalue preserving orthormal bases as described in the hypothesis of the Theorem, the manifolds are isometric if and only if the for one basis agrees, up to absolute value in the individual terms, with the other basis.
The original research was funded in part by a gracious James Simons Research Award in 1995-1996, and the generous support of an Alfred P. Sloan Dissertation Fellowship in 1996-1997.
The author would also like to thank Tanya Christiansen, Carolyn Gordon, Hamid Hezari, Harish Seshadri, and especially Leon Takhtajan for their technical assistance and review in the preparation of this manuscript for publication.
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