SPECTRAL ISOLATION OF BI-INVARIANT METRICS ON COMPACT LIE GROUPS

CAROLYN S. GORDON ^{†}, DOROTHEE SCHUETH ^{‡}, AND CRAIG J. SUTTON ^{♯}

Abstract. We show that a bi-invariant metric on a compact connected Lie group G is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a biinvariant metric g_{0} on G there is a positive integer N such that, within a neighborhood of g_{0} in the class of left-invariant metrics of at most the same volume, g_{0} is uniquely determined by the first N distinct non-zero eigenvalues of its Laplacian (ignoring multiplicities). In the case where G is simple, N can be chosen to be two.

RÉSUMÉ. Soit G un groupe de Lie compact et connexe, et soit g_{0} une métrique bi-invariante sur G. On démontre que g_{0} est isolée spectralement dans la classe des métriques invariantes à gauche: Plus précisément, il existe un entier positif N tel que, dans un voisinage de g_{0} dans la classe des métriques invariantes à gauche et de volume égal ou inférieur à celui de g_{0}, la métrique g_{0} est determinée de manière unique par les N premières valeurs propres strictement positives de son Laplacien (sans multiplicités). Si G est simple, on peut choisir N = 2.

1. Introduction

Given a connected closed Riemannian manifold (M, g) its spectrum, denoted Spec(M, g), is defined to be the sequence of eigenvalues, counted with multiplicities, of the associated Laplacian Δ acting on smooth functions. Two Riemannian manifolds (M_{1}, g_{1}) and (M_{2}, g_{2}) are said to be isospectral if their spectra (counting multiplicities) agree. Inverse spectral geometry is the study of the extent to which geometric properties of a Riemannian manifold (M, g) are determined by its spectrum.

A long standing question is whether very special Riemannian manifolds – e.g., manifolds of constant curvature or symmetric spaces – may be spectrally distinguishable from other Riemannian manifolds. The strongest results are for constant curvature: Tanno showed that a round sphere of dimension at most six is uniquely determined by its spectrum among all orientable Riemannian manifolds [T1], and in arbitrary dimensions round metrics on spheres are at least spectrally isolated among all Riemannian metrics on spheres [T2]. In contrast,

2000 Mathematics Subject Classification. 53C20, 58J50. Key words and phrases. Laplacian, eigenvalue spectrum, Lie group, left-invariant metric, bi-invariant metric. †Research partially supported by National Science Foundation grant DMS 0605247. ‡Research partially supported by DFG Sonderforschungsbereich 647. This work was started during a visit of

the second author to Dartmouth College, which she would like to thank for its great hospitality. ♯Research partially supported by National Science Foundation grant DMS 0605247.

the first and third author have shown that in dimension 7 and higher there are isospectrally deformable metrics on spheres arbitrarily close to the standard metric [G, Sch2]. Hence, for geometries that are in some sense extremely close to being “nice” or “ideal”, spectral uniqueness can fail profoundly.

While many examples exist of isospectral flat manifolds, Kuwabara [K] has proven that flat metrics are at least spectrally isolated within the space of all metrics. However, even the question of whether a flat torus may be isospectral to a non-flat manifold remains open! One cannot resolve this question by appealing to the heat invariants of a Riemannian manifold as there are examples of non-flat manifolds all of whose heat invariants vanish [Pa].

Outside of the setting of constant curvature, we are not aware of any examples of Riemannian metrics that are known to be spectrally isolated among arbitrary Riemannian metrics. Various results show that within certain classes of Riemannian metrics, isospectral sets are finite. Even here, many of the results involve constant curvature. For example, isospectral sets of flat tori are finite (see [W] or unpublished work of Kneser) as are isospectral sets of Riemann surfaces [McK]. As for the class of symmetric spaces, the first and third author have recently shown that any collection of mutually isospectral compact symmetric spaces is finite [GS].

This article is motivated by the question of whether one can tell from the spectrum whether a compact Riemannian manifold is symmetric. Given that this question has resisted solution even in the case of spheres, it does not appear tractable at this time to compare the spectrum of a symmetric space with that of a completely arbitrary Riemannian manifold. Instead, we ask whether symmetric spaces can be spectrally distinguished within a larger class of homogeneous Riemannian manifolds.

The compact symmetric spaces fall into two types; the type we consider are those given by bi-invariant Riemannian metrics on compact (not necessarily semisimple) Lie groups. We compare the spectrum of each such symmetric space with the spectra of arbitrary left-invariant metrics on the Lie group. As a departure point we note that the second author showed that there are no non-trivial continuous isospectral deformations of a bi-invariant metric within the class of left-invariant metrics on a compact Lie group [Sch1]. This prompts one to ask whether a bi-invariant metric on a compact Lie group G is spectrally isolated within the class of leftinvariant metrics. We give an affirmative answer; in fact we obtain a significantly stronger result.

Let M_{left}(G) denote the set of left-invariant metrics on a Lie group G. This set can be canonically identified with the set of Euclidean inner products on the Lie algebra of G. The latter set can in turn be identified, after some choice of basis, with the set of positive definite symmetric (m × m)-matrices, where m is the dimension of G. The canonical topology on this set of matrices gives rise to a topology on M_{left}(G) (independent of the choice of basis), and it is this topology that we consider. We call a left-invariant metric g_{0} on G spectrally isolated in M_{left}(G) if it is locally spectrally determined within M_{left}(G); that is, there is a neighborhood U of g_{0} in M_{left}(G) such that no g ∈ U \ {g_{0}} is isospectral to g_{0}. We prove the following:

Result. Let g_{0} be a bi-invariant metric on a compact Lie group G.

(1) There is a neighborhood U of g_{0} in M_{left}(G) and a positive integer N such that if g is any metric in U with vol(g) ≤ vol(g_{0}) and whose first N distinct eigenvalues (ignoring multiplicities) agree with those of g_{0}, then g is isometric to g_{0}. (See Theorem 2.3.)

(2) The metric g_{0} is spectrally isolated in M_{left}(G). (See Corollary 2.4.)

(3) Let α_{1} < α_{2} < α_{3} be three distinct consecutive eigenvalues (ignoring multiplicities) of the associated Laplacian Δ_{0}. If G is simple, then there exists a neighborhood U of g_{0} in M_{left}(G) such that if g ∈ U satisfies vol(g) ≤ vol(g_{0}) and the condition that three consecutive distinct eigenvalues of Δ_{g} agree with α_{1}, α_{2} and α_{3} (again ignoring multiplicities), then g = g_{0}. In particular, letting α_{1} = 0, then the first two distinct non-zero eigenvalues along with the volume bound distinguish g_{0} withing U. (See Theorem 3.3.)

The second result above is immediate from the first since the spectrum of a compact Riemannian manifold determines its volume. Hence, within the class of left-invariant metrics on a compact Lie group G, any metric g ≠ g_{0} that is isospectral to a bi-invariant metric g_{0} must be sufficiently far away from g_{0}. In contrast, we note that the second author exhibited the first examples of continuous isospectral families of left-invariant metrics on compact simple Lie groups [Sch1]; see also [Pr].

In light of the fact that most examples of isospectral manifolds in the literature exploit metrics with “large” symmetry groups, the spectral isolation results above lend strong support to the conjecture that a bi-invariant metric on a compact Lie group G is spectrally isolated within the class of all metrics on G. In fact, these results lead one to speculate on whether a bi-invariant metric on a semisimple Lie group is uniquely determined by its spectrum.^{1} In Section 3 we present strong evidence that the bi-invariant metric on a compact simple Lie group is uniquely determined by its spectrum within the class of left-invaraint metrics. In particular, we show the following.

Result. Let g_{0} be a bi-invariant metric on a compact simple Lie group G, and let g ≠ g_{0} be a left-invariant metric on G, which is isospectral to g_{0}. Then there is a constant C ≡ C(g) > 1, such that for every subspace V ≤ L^{2}(G) that is invariant under the right regular action of G, we have

\frac{Tr(Δ_{g} ↾ V)}{Tr(Δ_{0} ↾ V)} ≡ C > 1.

(See Proposition 3.1 for a more precise statement.)

This implies that if g ≠ g_{0} ∈ M_{left}(G) is isospectral to the bi-invariant metric g_{0}, then a very special rearrangement of the eigenvalues must occur.

1We must restrict our attention to semisimple Lie groups due to the existence of nontrivial pairs of isospectral flat tori (e.g., [M] and [CS]).

The outline of this paper is as follows. In Section 2 we establish the main results for biinvariant metrics on arbitrary compact Lie groups. In Section 3 we restrict our attention to compact simple Lie groups to obtain the stronger results in this setting.

2. Proof of the main result

Following [LMNR] we introduce the notion of eigenvalue equivalence, which is weaker than that of isospectrality. The same notion was introduced earlier by Z.I. Szabo [Sz, p. 212], who referred to it as isotonality. We also define a notion of partial eigenvalue equivalence.

2.1. Definition. Given a compact Riemannian manifold (M, g), we define the eigenvalue set of (M, g) to be the ordered collection of eigenvalues of the associated Laplace operator Δ_{g} on functions on M, not counting multiplicities. We will say that two compact Riemannian manifolds are eigenvalue equivalent if their eigenvalue sets coincide. For N a positive integer, we will say that two compact Riemannian manifolds are eigenvalue equivalent up to level N if the first N elements of their eigenvalue sets coincide.

2.2. Lemma. Let G be a compact Lie group, and let g_{0} be a bi-invariant metric on G with associated Laplacian Δ_{0}. Let V ≤ L^{2}(G) be a finite dimensional subspace which is invariant under the right-regular representation of G on L^{2}(G). Then there exists a positive integer N and a neighborhood U of g_{0} in M_{left}(G) such that if g ∈ U is eigenvalue equivalent to g_{0} up to level N, then

Δ_{g} ↾ V = Δ_{0} ↾ V.

Proof. First note that V, being a finite dimensional subspace of L^{2}(G) which is invariant under the right-regular representation, contains only smooth functions. Moreover, V is a direct sum of finitely many irreducible representations of G; it is therefore enough to prove the result in the case that V is irreducible. For any g ∈ M_{left}(G) and any g-orthonormal basis {Y_{1}, …, Y_{n}} of the Lie algebra of G, the associated Laplace operator on smooth functions on G is given by

Δ_{g} = − ∑_{j=1}^{n}(ρ_{∗}Y_{j})^{2},

where ρ : G → U(L^{2}(G)) is the right-regular representation of G. Thus, V is invariant under Δ_{g}. Since g_{0} is bi-invariant, right translations in G are g_{0}-isometries; hence Δ_{0} : V → V commutes with the action of G on V. Irreducibility of V implies by Schur’s Lemma that Δ_{0} ↾ V is a multiple of the identity, say Δ_{0} ↾ V = λ Id_{V}. We may choose ϵ > 0 such that (λ − ϵ, λ + ϵ) ∩ Spec(Δ_{0}) = {λ}. Choose N large enough so that the Nth element of the eigenvalue set is greater than λ (and hence greater than λ + ϵ). The hermitian operators Δ_{g} ↾ V on the finite dimensional vector space V depend continuously on g. Therefore, their eigenvalues also depend continuously on g. Consequently, there is a neighborhood U of g_{0} in M_{left}(G) such that for each g ∈ U the eigenvalues of Δ_{g} ↾ V must lie in (λ − ϵ, λ + ϵ). If g ∈ U is eigenvalue equivalent to g_{0} up to level N, it follows that Δ ↾ V = Δ_{0} ↾ V = λ Id_{V}. □

We now establish the spectral isolation of bi-invariant metrics on compact connected Lie groups. In fact, we prove a little more; namely, we replace the isospectrality condition by the much weaker condition of partial eigenvalue equivalence together with an upper volume bound.

2.3. Theorem. Let g_{0} be a bi-invariant metric on a compact connected Lie group G. Then there is a positive integer N, depending only on g_{0}, and a neighborhood U of g_{0} in M_{left}(G) such that if g ∈ U is eigenvalue equivalent to g_{0} up to order N and satisfies vol(g) ≤ vol(g_{0}), then g = g_{0}.

2.4. Corollary. Let g_{0} be a bi-invariant metric on a compact connected Lie group G. Then g_{0} is spectrally isolated in M_{left}(G).

The corollary follows from the theorem by the fact that isospectrality implies eigenvalue equivalence and equality of volumes; in fact, the volume is the first of the classical heat invariants.

Proof of Theorem 2.3. We have G = G_{ss}T where G_{ss} is semisimple, T is a torus and G_{ss} ∩ T is finite. The Lie algebra of G is a direct sum 𝔤 = 𝔤_{ss} ⊕ 𝔱, where 𝔤_{ss} and 𝔱 are the Lie algebras of G_{ss} and T. In particular, G_{ss} and T commute, and 𝔤_{ss} is g_{0}-orthogonal to 𝔱.

We first claim that there exists a positive integer N′ and a neighborhood U′ of g_{0} in M_{left}(G) such that if g ∈ U′ is eigenvalue equivalent to g_{0} up to level N′, then g and g_{0}, viewed as inner products on 𝔤, induce the same inner product on 𝔤/𝔤_{ss}. By the inner product induced by g we mean the one obtained by identifying 𝔤/𝔤_{ss} with the g-orthogonal complement of 𝔤_{ss} in 𝔤. To prove the claim, note that the Lie group \overline{T} := G/G_{ss} ≅ T/(G_{ss} ∩ T) is a torus which is finitely covered by T. In particular, the Lie algebra of \overline{T} is canonically identified with 𝔱. Let p : G → \overline{T} be the homomorphic projection. Given g ∈ M_{left}(G), denote by ḡ the induced (flat) metric on \overline{T} (i.e., the metric for which p : (G, g) → (\overline{T} , ḡ) becomes a Riemannian submersion). Let L be the lattice in 𝔱 which is the kernel of the Lie group exponential map 𝔱 → \overline{T}, and let L^{∗} ⊂ 𝔱^{∗} be the dual lattice. For μ ∈ L^{∗}, denote by ∥μ∥ḡ the norm of μ with respect to the dual inner product on 𝔱^{∗}. Let ν_{1}, …, ν_{k} be a basis of L^{∗}, where k = dim(T). Write L := k + \binom{k}{2}, and let {μ_{1}, …, μ_{L}} be the set containing the vectors ν_{i} as well as the ν_{i} + ν_{j} for i ≠ j. Note that, by polarization, the norm ∥ . ∥ḡ on 𝔱^{∗} – and hence ḡ itself – is uniquely determined by the norms of the vectors μ_{1}, …, μ_{L}. For each s ∈ \{1, …, L\} let f̄_{s} : \overline{T} → U(1) (where U(1) is the unitary group of unit complex numbers) be the associated character of \overline{T}. Then Δḡf̄_{s} = 4π^{2}∥μ_{s}∥2 ; ḡf̄_{s}. Now f_{s} := f̄_{s} ∘p is a character on G. Since the Riemannian submersion p : G → \overline{T} has minimal fibers, f_{s} is an eigenfunction of Δ_{g} with eigenvalue 4π^{2}∥μ_{s}∥2 ; ḡ for each s = 1, …, L. (One can also verify this fact by direct computation.)

The one-dimensional space F_{s} ≤ L^{2}(G) spanned by the character f_{s} is invariant under the right-regular representation. Let N′ be a positive integer and U′ be a neighborhood of g_{0} in M_{left}(G) satisfying the property from Lemma 2.2 with respect to F_{1} ⊕ … ⊕ F_{L}, and let g ∈ U′ be eigenvalue equivalent to g_{0} up to level N′. Then we must have ∥μ_{s}∥ḡ = ∥μ_{s}∥ḡ_{0} for each s = 1, …, L. As remarked above, this implies ḡ = ḡ_{0}. The claim follows.

In the case of the bi-invariant metric g_{0}, the metric ḡ_{0} on 𝔱 coincides with the restriction of the inner product g_{0} to 𝔱 since 𝔤_{ss} and 𝔱 are g_{0}-orthogonal. However, for more general g, one has only that the differential p_{∗} : 𝔤 → 𝔱 of the projection p : G → \overline{T} restricts to an inner product space isometry p_{∗} : (𝔤_{ss}^{⊥_{g}}, g) → (𝔱, ḡ). In particular, if g ∈ U′, then it follows from the claim that

(2.5)  the projection from (𝔤_{ss}^{⊥_{g}}, g) to (𝔱, g_{0}) along 𝔤_{ss} is an isometry.

For the remaining part of the argument, let 𝔤_{ss} = 𝔤_{1} ⊕ … ⊕ 𝔤_{r} be the decomposition of 𝔤_{ss} into simple Lie subalgebras. The adjoint representation of G on 𝔤, restricted to the invariant subspace 𝔤_{ℓ}, is an irreducible representation of G for each ℓ = 1, …, r. Note that for ℓ ≠ ℓ′ these representations of G are inequivalent even in the case when 𝔤_{ℓ} and 𝔤_{ℓ′} happen to be isomorphic as Lie algebras. By the Peter-Weyl Theorem, every irreducible representation of G occurs in the right-regular representation of G on L^{2}(G) (with multiplicity equal to its dimension). Thus, for each ℓ = 1, …, r we can choose a corresponding irreducible subspace V_{ℓ} ≤ L^{2}(G), and the action of G on the subspace V_{1} ⊕ … ⊕ V_{r} of L^{2}(G) will then be equivalent to the adjoint representation of G acting on 𝔤_{ss}.

Let N′′ be a positive integer and U′′ be a neighborhood of g_{0} in M_{left}(G) satisfying the property from Lemma 2.2 with respect to V_{1} ⊕ … ⊕ V_{r}. We are going to show that N := max{N′, N′′} and U := U′ ∩ U′′ satisfy the property stated in the Theorem.

If g is any left-invariant metric on G and {U_{1}, …, U_{m}} is a g-orthonormal basis of 𝔤, then

(2.6)  Tr(Δ_{g} ↾ V_{ℓ}) = − ∑_{j=1}^{m} Tr((ad_{U_{j}} ↾ 𝔤_{ℓ})^{2}).

Since g_{0} is bi-invariant, 𝔤_{ℓ} is g_{0}-orthogonal to 𝔤_{ℓ′} for ℓ ≠ ℓ′. Let n_{ℓ} denote the dimension of 𝔤_{ℓ}, and let n = n_{1} +…+n_{r} be the dimension of 𝔤_{ss}. Choose a g_{0}-orthonormal basis {X_{1}, …, X_{n}} of 𝔤_{ss} such that the first n_{1} elements lie in 𝔤_{1}, the next n_{2} elements lie in 𝔤_{2}, etc. Complete to a g_{0}-orthonormal basis B_{0} = {X_{1}, …, X_{n}, Z_{1}, …, Z_{k}}, where (necessarily) Z_{1}, …, Z_{k} ∈ 𝔱.

Let g be a metric in U which is eigenvalue equivalent to g_{0} up to level N and satisfies vol(g) ≤ vol(g_{0}). Since g ∈ U′ and N ≥ N′, statement (2.5) holds and thus there exist elements W_{i} ∈ 𝔤_{ss} such that {Z_{1} + W_{1}, …, Z_{k} + W_{k}} is a g-orthonormal basis of 𝔤_{ss}^{⊥_{g}}. Complete to a g-orthonormal basis B = {Y_{1}, …, Y_{n}, Z_{1} + W_{1}, …, Z_{k} + W_{k}} of 𝔤 with Y_{1}, …, Y_{k} ∈ 𝔤_{ss}. The change of basis matrix which expresses the elements of B in terms of B_{0} is given by

[\begin{matrix}A & R \\ 0 & I_{k} \end{matrix}] ,

where A = (a_{ij}) ∈ Mat_{n×n}(ℝ), R = (r_{ij}) ∈ Mat_{n×k}(ℝ) and I_{k} is the k × k identity matrix. Hence, Y_{j} = ∑_{i=1}^{n} a_{ij}X_{i} for j = 1, …, n and W_{s} = ∑_{i=1}^{n} r_{is}X_{i} for s = 1, …, k. The condition vol(g) ≤ vol(g_{0}) implies that | det(A)| ≥ 1. Without loss of generality we assume det(A) > 0 and hence det(A) ≥ 1.

Since g_{0} is bi-invariant, there exist numbers c_{ℓ} > 0 for ℓ = 1, …, r such that the restriction of g_{0} to 𝔤_{ℓ} coincides with −c_{ℓ}B_{ℓ}, where B_{ℓ} is the Killing form of 𝔤_{ℓ} (which in turn coincides with the restriction to 𝔤_{ℓ} of the Killing form B : (X, Y) ↦ Tr(ad_{X} ∘ ad_{Y}) of 𝔤). In particular, by equation (2.6), we have

Tr(Δ_{0} ↾ V_{ℓ}) = \frac{n_{ℓ}}{c_{ℓ}} .

Since g ∈ U ⊂ U′′ and N ≥ N′′, we have Δ_{g} ↾ V_{ℓ} = Δ_{0} ↾ V_{ℓ} for each ℓ = 1, …, r. In particular, for ℓ = 1:

\frac{n_{1}}{c_{1}} = Tr(Δ_{0} ↾ V_{1}) = Tr(Δ_{g} ↾ V_{1}) = − ∑_{j=1}^{n} Tr((ad_{Y_{j}} ↾ 𝔤_{1})^{2}) − ∑_{s=1}^{k} Tr((ad_{Z_{s}+W_{s}} ↾ 𝔤_{1})^{2})

= − ∑_{i=1}^{n_{1}} ∑_{j=1}^{n} a_{ij}^{2} Tr((ad_{X_{i}} ↾ 𝔤_{1})^{2}) − ∑_{i=1}^{n_{1}} ∑_{s=1}^{k} r_{is}^{2} Tr((ad_{X_{i}} ↾ 𝔤_{1})^{2})

= \frac{1}{c_{1}} ∑_{i=1}^{n_{1}} ∑_{j=1}^{n} a_{ij}^{2} + \frac{1}{c_{1}} ∑_{i=1}^{n_{1}} ∑_{s=1}^{k} r_{is}^{2},

where the fourth equality holds because ad_{X} ↾ 𝔤_{1} = 0 for X ∈ 𝔤_{ℓ} with ℓ ≠ 1 and because {X_{1}, …, X_{n_{1}}} is orthonormal with respect to −c_{1}B_{1}. We hence obtain n_{1} = ∑_{i=1}^{n_{1}} ∑_{j=1}^{n} a_{ij}^{2} + ∑_{i=1}^{n_{1}} ∑_{s=1}^{k} r_{is}^{2}. Summing over the analogous equations for ℓ = 1, …, r we conclude that

n = ∥A∥^{2} + ∥R∥^{2},

where ∥ . ∥ denotes the standard Euclidean norm of matrices viewed as points in the appropriate ℝ^{N}. However, n is the minimal value (in fact, the only critical value) of the function SL(n, ℝ) ∋ C ↦ ∥C∥^{2} ∈ ℝ and is attained precisely on SO(n). It thus follows from det(A) ≥ 1 that A ∈ SO(n) and R = 0; hence g = g_{0}. □

2.7. Remark. Some of the techniques used in this section are similar to those used by Urakawa in [U].

3. A stronger spectral isolation result for simple groups

In the proof of Lemma 2.2 it was observed that if g is a left-invariant metric on a compact Lie group G, with associated Laplacian Δ_{g}, then any subspace V ≤ L^{2}(G) that is invariant under the right-regular representation of G is also invariant under Δ_{g}. With this in mind we have the following result concerning the trace of the Laplacian on compact simple Lie groups.

3.1. Proposition. Let g_{0} be a bi-invariant metric on a compact simple Lie group G, and let g ≠ g_{0} be a left-invariant metric on G which satisfies vol(g) ≤ vol(g_{0}). Then there exists a constant C = C(g) > 1 such that

Tr(Δ_{g} ↾ V) = C Tr(Δ_{0} ↾ V)

for every finite dimensional subspace V ≤ L^{2}(G) which is invariant under the right-regular representation ρ of G and on which G acts nontrivially.

Proof. By rescaling g and g_{0} we can assume without loss of generality that g_{0} coincides with −B on the Lie algebra 𝔤 of G, where B is the Killing form on 𝔤. Define h : 𝔤 × 𝔤 → ℝ by

h(X, Y) := − Tr(((ρ_{∗}X) ↾ V) ∘ ((ρ_{∗}Y) ↾ V)).

Obviously h is bilinear, symmetric, and Ad_{G}-invariant. The map X ↦ (ρ_{∗}X) ↾ V is nonzero because V is not a trivial representation space of G. Since 𝔤 is simple, this map has trivial kernel, which implies that h is positive-definite; in particular, there exists some c > 0 such that h = −cB. We now proceed similarly as in the last part of the proof of Theorem 2.3: Let {X_{1}, …, X_{n}} be a g_{0}-orthonormal basis and {Y_{1}, …, Y_{n}} be a g-orthonormal basis of 𝔤, and define A = (a_{ij}) by Y_{j} = ∑_{i=1}^{n} a_{ij}X_{i} for j = 1, …, n. By the volume condition, | det(A)| ≥ 1; we can assume det(A) ≥ 1. Moreover, g ≠ g_{0} implies A ∉ SO(n) and therefore ∥A∥^{2} > n. Thus,

Tr(Δ_{g} ↾ V) = ∑_{j=1}^{n} h(Y_{j}, Y_{j}) = −c ∑_{j=1}^{n} B(Y_{j}, Y_{j}) = −c ∑_{i,j=1}^{n} a_{ij}^{2}B(X_{i}, X_{i}) = c∥A∥^{2}

> cn = ∑_{i=1}^{n} h(X_{i}, X_{i}) = Tr(Δ_{0} ↾ V).

The proposition follows with C = \frac{∥A∥^{2}}{n}. □

3.2. Remark. Since volume is a spectral invariant the previous proposition implies the following: Let g_{0} be a bi-invariant metric on a compact simple Lie group G, and suppose there exists a left-invariant metric g ≠ g_{0} on G which is isospectral to g_{0}. Then Tr(Δ_{g} ↾ V) > Tr(Δ_{0} ↾ V) for any finite dimensional invariant subspace V of L^{2}(G) on which G acts nontrivially. Thus Δ_{g}, although isospectral to Δ_{0}, must have greater trace than Δ_{0} on every Δ_{0}-eigenspace (except for the eigenvalue 0), even on each irreducible subspace. This is not a priori a contradiction because some wild reordering of eigenvalues could occur to produce this situation. Nevertheless, this seems a strong indication in support of the conjecture that a bi-invariant metric on a compact simple Lie group is globally spectrally determined among left-invariant metrics.

3.3. Theorem. Let g_{0} be a bi-invariant metric on a compact simple Lie group G. Let α_{1} < α_{2} < α_{3} be three consecutive elements of the eigenvalue set of (G, g_{0}). (See Definition 2.1.) Then there exists a neighborhood U of g_{0} in M_{left}(G) (depending on α_{1}, α_{2}, α_{3}) such that if g ∈ U satisfies vol(g) ≤ vol(g_{0}) and if α_{1}, α_{2} and α_{3} are also consecutive elements of the eigenvalue set of (G, g), then g = g_{0}.

3.4. Corollary. A bi-invariant metric on a compact simple Lie group is locally determined within the set of left-invariant metrics of at most the same volume by its first two distinct non-zero eigenvalues 0 < λ_{1} < λ_{2} (ignoring multiplicities).

The corollary follows from the theorem since 0, λ_{1}, λ_{2} are three consecutive elements of the eigenvalue set.

Proof of Theorem 3.3. Since Δ_{0} commutes with right translations in G, the α_{2}-eigenspace V of Δ_{0} is invariant under the right-regular representation. Note that V is finite dimensional. As remarked in the proof of Lemma 2.2, the eigenvalues of Δ_{g} ↾ V depend continuously on g. Thus there exists a neighborhood U of g_{0} in M_{left}(G) such that for any g ∈ U the eigenvalues of Δ_{g} on V are contained in the interval (α_{1}, α_{3}). Let g ∈ U be a metric which satisfies vol(g) ≤ vol(g_{0}) and the condition that α_{1}, α_{2} and α_{3} are also consecutive eigenvalues of Δ_{g}. Then necessarily Δ_{g} ↾ V = α_{2} Id_{V} = Δ_{0} ↾ V. Finally, note that G acts nontrivially on V since α_{2} ≠ 0. Proposition 3.1 now implies g = g_{0}. □

References

[CS] J. H. Conway and N.J.A. Sloane, Four-dimensional lattices with the same theta series, Int. Math. Res. Not. 1992, No. 4, 93–96.

[G] C.S. Gordon, Isospectral deformations of metrics on spheres, Invent. Math. 145 (2001), 317–331.

[GS] Carolyn S. Gordon and Craig J. Sutton, Spectral isolation of naturally reductive metrics on simple Lie groups, Math. Z., to appear.

[K] R. Kuwabara, On the characterization of flat metrics by the spectrum, Comment. Math. Helv. 55 (1980), No. 3, 427–444.

[LMNR] C.J. Leininger, D.B. McReynolds, W.D. Neumann and A.W. Reid, Length and eigenvalue equivalence, Int. Math. Res. Not. IMRN 2007, no. 24, Art. ID rnm135, 24 pp.

[McK] H.P. McKean, Selberg’s trace formula as applied to a compact Riemann surface, Comm. Pure Appl. Math. 25 (1972), 225–246.

[M] J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Natl. Acad. Sci. USA 51 (1964), 542.

[Pa] V. Patodi, Curvature and the fundamental solution of the heat equation, J. Indian Math. Soc. 34 (1970), 269–285.

[Pr] E. Proctor, Isospectral metrics and potentials on classical compact simple Lie groups, Michigan Math. J. 53 (2005), 305–318.

[Sch1] D. Schueth, Isospectral manifolds with different local geometries, J. Reine Angew. Math., 534 (2001), 41–94.

[Sch2] D. Schueth, Isospectral metrics on five-dimensional spheres, J. Differential Geom., 58 (2001), 87–111.

[Sz] Z. I. Szabo, Locally non-isometric yet super isospectral spaces, Geom. Funct. Anal. 9 (1999), 185–214.

[T1] S. Tanno, Eigenvalues of the Laplacian of Riemannian manifolds, Tôhoku Math. Journ. 25 (1973), 391–403.

[T2] S. Tanno, A characterization of the canonical spheres by the spectrum, Math. Z. 175 (1980), 267–274.

[U] H. Urakawa, On the least positive eigenvalue of the Laplacian for compact group manifolds, J. Math. Soc. Japan 31 (1979), 209–226.

[W] S. Wolpert, The eigenvalue spectrum as moduli for flat tori, Trans. Amer. Math. Soc. 244 (1978), 313–321.

Dartmouth College, Department of Mathematics, Hanover, NH 03755

E-mail address: carolyn.s.gordon@dartmouth.edu

Humboldt-Universität, Institut für Mathematik, Berlin, Germany

E-mail address: schueth@math.hu-berlin.de

Dartmouth College, Department of Mathematics, Hanover, NH 03755

E-mail address: craig.j.sutton@dartmouth.edu