ISOSPECTRAL POTENTIALS AND CONFORMALLY

EQUIVALENT ISOSPECTRAL METRICS

ON SPHERES, BALLS AND LIE GROUPS

Carolyn S. Gordon and Dorothee Schueth

To the memory of our colleague and friend Robert Brooks (1952–2002), whose influence is felt throughout our work.

Abstract. We construct pairs of conformally equivalent isospectral Riemannian metrics φ_{1}g and φ_{2}g on spheres S^{n} and balls B^{n+1} for certain dimensions n, the smallest of which is n = 7, and on certain compact simple Lie groups. In the case of Lie groups, the metric g is left-invariant. In the case of spheres and balls, the metric g is not the standard metric but may be chosen arbitrarily close to the standard one. For the same manifolds (M, g) we also show that the functions φ_{1} and φ_{2} are isospectral potentials for the Schrödinger operator ℏ^{2}Δ+φ. To our knowledge, these are the first examples of isospectral potentials and of isospectral conformally equivalent metrics on simply connected closed manifolds.

Introduction

Let M be a compact Riemannian manifold. Two fundamental questions in spectral geometry are:

• To what extent can one recover the geometry of M from the eigenvalue spectrum of the Laplace-Beltrami operator Δ acting on smooth functions on M? (If the manifold has boundary, one may impose either Dirichlet or Neumann boundary conditions on the functions.)

• To what extent does the eigenvalue spectrum of the Schrödinger operator ℏ^{2}Δ+φ determine the potential φ ∈ C^{∞}(M)?

Various geometric invariants of Riemannian manifolds such as dimension, volume and total scalar curvature are known to be spectrally determined. The primary methods for identifying spectral invariants is through the study of the heat and wave equations. If two Schrödinger operators ℏ^{2}Δ + φ and ℏ^{2}Δ + ψ on a fixed

Key words and phrases. Laplace operator, spectrum, Schrödinger operator, conformally equivalent metrics, isospectral potentials.

2000 Mathematics Subject Classification. 58J53, 58J50. The first author is partially supported by National Science Foundation grant DMS-0072534.

The second author is partially supported by SFB 611, Bonn.

Riemannian manifold M are isospectral, then the heat invariants show that φ and ψ have the same average and the same L^{2}-norm (see, e.g., [Gi]).

We will say two compact Riemannian manifolds are isospectral if the associated Laplacians have the same eigenvalue spectrum. We will say that two potentials φ_{1} and φ_{2} are isospectral if the Schrödinger operators ℏ^{2}Δ + φ_{1} and ℏ^{2}Δ + φ_{2} are isospectral for every choice of the Planck’s constant ℏ. The only way of identifying specific geometric invariants, respectively properties of potentials, which are not spectrally determined is through construction of isospectral metrics or potentials. The primary results of this article are the following constructions:

(A) Isospectral conformally equivalent Riemannian metrics on simply-connected manifolds, including spheres of dimension seven and higher, balls of dimension at least eight, and on the Lie groups Spin(2m + 4), m ≥ 5 and SU(2m + 3), m ≥ 3. The metrics on the spheres, respectively balls, may be chosen arbitrarily close to the round, respectively flat metrics. The metrics on the Lie groups are conformally equivalent to left-invariant metrics.

(B) Isospectral potentials for the Schrödinger operators associated with (noncanonical) Riemannian metrics on the same underlying simply-connected manifolds as in (A). Again the Riemannian metrics on the balls and spheres may be chosen arbitrarily close to the standard ones, and the Riemannian metrics on the Lie groups are left-invariant.

To our knowledge, these constructions give the first examples of isospectral conformally equivalent metrics and of isospectral potentials on simply-connected closed manifolds.

We briefly review the literature on isospectral metrics and potentials. (We do not give a complete list of all known examples.)

ISOSPECTRAL POTENTIALS AND CONFORMALLY EQUIVALENT METRICS 3

The known methods for constructing isospectral manifolds roughly fall into three classes: “Explicit” computations, representation theoretic or trace formula methods, and the use of torus actions. Among the examples constructed by more or less explicit computations are 16-dimensional tori constructed by J. Milnor [Mi] in 1964 (these are the first known examples of isospectral manifolds), spherical space forms [Ik1,2], domains in ℝ^{4} [Ur], flat manifolds exhibiting various interesting properties [MR1-3], and the first examples of isospectral manifolds with different local geometry [Sz1] and of pairs of isospectral metrics on spheres [Sz2]. Representation methods were first used to construct isospectral Riemann surfaces [Vi] and continuous isospectral deformations [GW1]. In 1985, T. Sunada [Su] gave an elegant trace formula method (which may also be viewed as a representation theoretic method) for constructing isospectral manifolds with a common finite cover. This method and various generalizations [Be1,2], [DG], [Pe1,2] led to a vast literature of isospectral manifolds, including isospectral Riemann surfaces of every genus greater than three [BT], [Bu], [BGG], isospectral plane domains [GWW], and many other examples. The third method, that of torus actions, was developed in the series of papers [Go1,3], [GW2], [GGSWW], [GSz], [Sch1-3]; see also [Ba]. This method generally produces manifolds with different local geometry, including among others the first examples of isospectral simply-connected manifolds (in fact, continuous deformations) [Sch1], isospectral deformations of left-invariant metrics on simplyconnected compact Lie groups [Sch2], and isospectral deformations of metrics on spheres [Go3], [Sch3]. Recently, C. Sutton [St] modified Sunada’s method (the second method above) to construct new and very different examples of isospectral simply connected manifolds.

R. Brooks, P. Perry and P. Yang [BPY] gave a modification of Sunada’s method which yields isospectral conformally equivalent Riemannian metrics with the same local but different global geometry. (This method is closely related to an earlier modification of Sunada’s method by R. Brooks [Br] which produces isospectral potentials for the Schrödinger operator.) The second author [Sch2] analogously extended the method of torus actions to construct the first examples of isospectral conformally equivalent metrics that do not lift to isometric metrics on the universal covering. These were conformally equivalent to left-invariant metrics on products G × T, where G was a simple compact Lie group (Spin(n), SO(n) or SU(n) for various n) and T was a torus. These examples, together with the constructions of isospectral metrics on Lie groups and spheres referred to above, motivated our construction (A).

For the Schrödinger operator, there is a vast literature concerning isospectral potentials on the circle and on the interval. See, for example, [PT]. An especially interesting connection exists with the Korteveg de Vries equation. Letting an arbitrary function evolve according to the Korteveg de Vries equation, one obtains a continuous family of potentials φ_{t} on the circle such that the operators Δ + φ_{t} are mutually isospectral [GGKM]. In higher dimensions, Brooks’ modification of Sunada’s method, cited above, has yielded isospectral potentials on negatively curved manifolds [Br] and isospectral deformations of potentials [BG]. Although it was not mentioned explicitly in those two papers, these potentials are isospectral in the strong sense defined above; i.e., they are isospectral for the Schrödinger operators with arbitrary Planck constant ℏ. In contrast to the potentials that we will construct, Brooks’ method yields potentials that lift to congruent potentials on the universal covering manifold.

The paper is organized as follows: In Section 1 we present the general principles which will be used for constructing conformally equivalent isospectral metrics and isospectral potentials. In Section 2 we outline a strategy for proving nonisometry. Sections 3 and 4 contain the actual examples: Lie groups in Section 3, balls and spheres in Section 4. In Section 5 we prove nonisometry results for the isospectral metrics and potentials constructed in the previous two sections. The stategy outlined in Section 2 is applied in the case of the ball and sphere examples; a different approach is used in the Lie groups case. Finally, in Section 6 we give a short survey of how our technique may be applied in other settings.

It is a pleasure to thank Steven Zelditch for helpful comments.

As we were completing this paper, we learned with shock and sorrow of the untimely passing of our close friend and colleague, Robert Brooks. Brooks brought to inverse spectral geometry ideas from such diverse areas as complex analysis, probability and combinatorics. His generosity and enthusiasm in sharing his insights has profoundly affected our research. He is deeply missed.

§1 Principles for constructing isospectral potentials and conformally equivalent isospectral metrics

1.1 Definition.

(i) The spectrum of a closed Riemannian manifold (M, g) is the spectrum of eigenvalues, counted with multiplicities, of the associated Laplace operator Δ_{g} acting on functions. The Dirichlet spectrum of a compact Riemannian manifold (M, g) with boundary is the spectrum of eigenvalues corresponding to eigenfunctions which satisfy the Dirichlet boundary condition f_{|_{∂M}} = 0. The Neumann spectrum of such a manifold is defined analogously with respect to the Neumann boundary condition Nf = 0, where N is the inward-pointing unit normal field on the boundary. Given any potential function φ ∈ C^{∞}(M) and a Planck constant ℏ ≠ 0, one analogously defines the (Dirichlet, Neumann) spectrum of the operator ℏ^{2}Δ_{g} + φ : f ↦ ℏ^{2}Δ_{g}f + φf.

(ii) Two closed Riemannian manifolds are called isospectral if they have the same spectrum (including multiplicities). Two compact Riemannian manifolds with boundary are called Dirichlet isospectral, resp. Neumann isospectral, if they have the same Dirichlet spectrum, resp. the same Neumann spectrum. Isospectrality of two operators ℏ^{2}Δ_{g} + φ and ℏ^{2}Δ_{g′} + ψ is defined analogously. If (M, g) = (M′, g′) and if ℏ^{2}Δ_{g} + φ and ℏ^{2}Δ_{g} + ψ are (Dirichlet, Neumann) isospectral for each ℏ ≠ 0, then φ and ψ are called (Dirichlet, Neumann) isospectral potentials on (M, g).

1.2 Notation. By a torus, we always mean a nontrivial, compact, connected abelian Lie group. If a torus T acts smoothly and effectively by isometries on a compact connected Riemannian manifold (M, g), then we denote by M̂ the union of those orbits on which T acts freely. Note that M̂ is an open dense submanifold of M. The action of T gives M̂ the structure of a principal T -bundle. By g_{0}^{T} we denote the unique Riemannian metric on the quotient manifold M̂/T such that the canonical projection π : (M̂, g_{0}) → (M̂/T, g_{0}^{T}) is a Riemannian submersion.

1.3 Theorem. Let T be a torus which acts effectively by isometries on two compact connected Riemannian manifolds (M, g) and (M′, g′). Let φ ∈ C^{∞}(M) and ψ ∈ C^{∞}(M′) be T -invariant potentials. For each subtorus W ⊂ T of codimension one, suppose that there exists a T -equivariant diffeomorphism F_{W} : M → M′ which satisfies:

(a) F_{W}^{∗} dvol_{g′} = dvol_{g},

(b) F_{W} induces an isometry F̄_{W} : (M̂/W, g^{W}) → (M̂′/W, g^{′ W}),

(c) F_{W}^{∗} ψ = φ. Then for each ℏ ≠ 0, the operators ℏ^{2}Δ_{g} + φ on M and ℏ^{2}Δ_{g′} + ψ on M′ are isospectral; if the manifolds have boundary then these operators are Dirichlet and Neumann isospectral.

In particular, if moreover there exists an isometry τ : (M, g) → (M′, g′) such that τ^{∗}ψ ≠ φ, then φ and τ^{∗}ψ are (Dirichlet and Neumann) isospectral potentials on (M, g).

Note that for φ = ψ = 0, this is just Theorem 1.4 from [Sch3], which in turn was a slightly improved version of Theorem 1.2 from [Go3].

Proof. Let H denote the Sobolev space H^{1,2}(M, g) in the case of closed M or in the case of Neumann boundary conditions, and let H denote the closure of C_{0}^{∞}(M) in H^{1,2}(M, g) in the case of Dirichlet boundary conditions. For any subtorus W of T let H_{W} ⊂ H denote the subspace of W -invariant functions. Analogously, we define the function spaces H′ and H′_{W} on M′.

The key point in the proof of Theorem 1.4 from [Sch3] (corresponding to the case φ = ψ = 0) was deducing from (a) and (b) that for each W of codimension 1, the pullback

F_{W}^{∗} : H′_{W} → H_{W}

preserves both the L^{2}- and the H^{1,2}-norm. Combining these maps according to the Fourier decomposition

H = H_{T} ⊕ ⨁_{W} (H_{W} ⊖ H_{T}),

one thus obtained an L^{2}-norm preserving Hilbert space isometry I : H′ → H. The statement (for φ = ψ = 0) then followed from the variational characterization of eigenvalues via the Rayleigh quotient.

We need only slightly extend this proof in order to obtain Theorem 1.3: By condition (c), together with (a), the pullbacks F_{W}^{∗} now also satisfy ∫_{M} φ|F_{W}^{∗} f|^{2}dvol_{g} = ∫_{M′} ψ|f|^{2}dvol_{g′} for all f ∈ H′_{W} . Since φ is T -invariant, multiplication by φ preserves each of the summands H_{T} , H_{W} ⊖ H_{T} in the above orthogonal decomposition of H, and similarly for ψ. One easily concludes that the L^{2}-norm preserving isometry I : H′ → H defined above satisfies ∫_{M} φ|If|^{2}dvol_{g} = ∫_{M′} ψ|f|^{2}dvol_{g′} for all ψ ∈ H′. In particular,

R_{φ}(If) = R_{ψ}(f)

for all f ∈ H′ \ {0}, where

R_{ψ}(f) : = ∫_{M′}(ℏ^{2}∥df∥_{g′}^{2} + ψ|f|^{2})dvol_{g′}/ ∫_{M′} |f|^{2}dvol_{g′}
= ((ℏ^{2}∥f∥_{H^{1,2}(M′,g′)}^{2} + ∫_{M′} ψ|f|^{2}dvol_{g′}) / ∥f∥_{L^{2}(M′,g′)}^{2}) − ℏ^{2}.

is the modified Rayleigh quotient occurring in the variational characterization of the eigenvalues of ℏ^{2}Δ_{g′} + ψ, and where R_{φ} is defined analogously.

Thus the theorem follows again from the variational characterization of eigenvalues. □

1.4 Notation. We fix a torus T with Lie algebra 𝔷. Let L be the cocompact lattice in 𝔷 such that exp : 𝔷 → T induces an isomorphism from 𝔷/L to T, and denote by L^{∗} ⊂ 𝔷^{∗} the dual lattice. We also fix a compact connected Riemannian manifold (M, g_{0}), with or without boundary, and a smooth effective action of T on (M, g_{0}) by isometries.

(i) For Z ∈ 𝔷 we denote by Z^{∗} the vector field x ↦ \frac{d}{dt}_{|_{t=0}} exp(tZ)x on M. For each x ∈ M, we let 𝔷_{x} := {Z_{x}^{∗} | Z ∈ 𝔷} ⊂ T_{x}M.

(ii) We call a smooth 1-form on M admissible if it is T -invariant and horizontal

(i.e., vanishes on the vertical spaces 𝔷_{x}). (iii) For any admissible 𝔷-valued 1-form λ on M we denote by g_{λ} the Riemannian

metric on M given by

g_{λ}(X, Y) = g_{0}(X + λ(X)^{∗}, Y + λ(Y)^{∗}).

Note that g_{λ} is the pullback of g_{0} by a field of unipotent endomorphisms; in particular, dvol_{g_{λ}} = dvol_{g_{0}} . Moreover, g_{λ} is again T -invariant.

(iv) We say that a diffeomorphism F : M → M is T -preserving if conjugation by F preserves T ⊂ Diffeo(M). In that case, we denote by Ψ_{F} the automorphism of 𝔷 induced by conjugation by F. Note that if F is T -equivariant then Ψ_{F} = Id. Obviously, each T -preserving diffeomorphism F of M maps T -orbits to T -orbits and satisfies F_{∗}(Z^{∗}) = Ψ_{F} (Z)^{∗} for all Z ∈ 𝔷, where the vector fields Z^{∗} on M are defined as in (i).

The following theorem gives a useful specialization of Theorem 1.3 and is the generalized analog of Theorem 1.6 from [Sch3], which corresponds to statement (i) below in the special case φ = 0.

1.5 Theorem. In the context of Notation 1.4, let λ, λ′ be two admissible 𝔷-valued 1-forms on M. Assume:

(∗) For every μ ∈ L^{∗} there exists a T -equivariant F_{μ} ∈ Isom(M, g_{0}) such that

μ ∘ λ = F_{μ}^{∗}(μ ∘ λ′). Let φ ∈ C^{∞}(M) be a T -invariant potential function with the property that the isometries F_{μ} occurring in (∗) can be chosen such that φ is invariant under each of them. Then we have:

(i) ℏ^{2}Δ_{g_{λ}} +φ and ℏ^{2}Δ_{g_{λ′}} +φ are isospectral for each ℏ ≠ 0; if M has boundary then these operators are Dirichlet and Neumann isospectral.

(ii) If coincidentally there exists a T -preserving τ ∈ Isom(M, g_{0}) such that, in the notation of 1.4(iv), Ψ_{τ} ∘ λ = τ^{∗}λ′, but φ ≠ τ^{∗}φ, then φ and τ^{∗}φ are isospectral potentials on (M, g_{λ}); if M has boundary then the two potentials are Dirichlet and Neumann isospectral.

Proof. (i) We claim that (M, g_{λ}), (M, g_{λ′}) and φ, ψ := φ satisfy the hypotheses of Theorem 1.3. Given a subtorus W of codimension 1 in T we choose μ ∈ L^{∗} such that ker μ = T_{e}W, and we choose a corresponding F_{μ} as in (∗). In the proof of Theorem 1.6 from [Sch3] (corresponding to the case φ = 0 here) it was shown that F_{W} := F_{μ} satisfies conditions (a) and (b) of Theorem 1.3; more precisely, (a) is immediate from dvol_{g_{λ}} = dvol_{g_{0}} = dvol_{g_{λ′}} , and (b) was deduced from condition (∗). It only remains to check that F_{μ} satisfies condition (c) of Theorem 1.3 (with ψ = φ); but this is already guaranteed by the assumptions on φ.

(ii) This follows immediately from (i) by noting that τ is an isometry from (M, g_{λ}) to (M, g_{λ′}). □

1.6 Corollary. Let λ, λ′ be two admissible 𝔷-valued 1-forms on M satisfying condition (∗) of Theorem 1.5. Moreover, let φ ∈ C^{∞}(M, ℝ_{+}) be invariant under the action of T, and suppose that the isometries F_{μ} occurring in (∗) can be chosen such that φ is invariant under each of them. Then we have:

(i) (M, φg_{λ}) and (M, φg_{λ′}) are isospectral.

(ii) If coincidentally there exists a T -preserving τ ∈ Isom(M, g_{0}) such that Ψ_{τ} ∘ λ = τ^{∗}λ′, but τ^{∗}φ ≠ φ, then the conformally equivalent metrics φg_{λ} and (τ^{∗}φ)g_{λ} are isospectral.

Proof. (i) Note that by definition, φg_{λ} = (φg)_{λ} (and similarly for λ′). Thus the statement follows by applying Theorem 1.5(i) (with trivial potential) to λ, λ′ and to φg_{0} instead of g_{0} .

(ii) This follows immediately from (i) by noting that τ is an isometry from (M, (τ^{∗}φ)g_{λ}) to (M, φg_{λ′}). □

§2 A nonisometry principle

In the following we let (M, g_{0}), T, 𝔷 be as in Notation 1.4, and we consider two admissible 𝔷-valued 1-forms λ, λ′ on M in the sense of Notation 1.4(ii). We define M̂ as in Notation 1.2.

2.1 Notation and Remarks. ([Sch3])

(i) We denote by Aut_{g_{0}}^{T}(M) the group of all T -preserving diffeomorphisms F of M which, in addition, preserve the g_{0}-norm of vectors tangent to the T -orbits and induce an isometry, denoted F̄, of (M̂/T, g_{0}^{T}). We denote the corresponding group of induced isometries by \overline{Aut}_{g_{0}}^{T} (M) ⊂ Isom(M̂/T, g_{0}^{T}).

(ii) We define D := {Ψ_{F} | F ∈ Aut_{g_{0}}^{T}(M)} ⊂ Aut(𝔷) (see Notation 1.4(iv)). Note that D is discrete because it is a subgroup of the discrete group {Ψ ∈ Aut(𝔷) | Ψ(L) = L}, where L is the lattice ker (exp : 𝔷 → T).

(iii) Let ω_{0} denote the connection form on M̂ associated with g_{0} ; i.e., for each x ∈ M̂ the horizontal space ker (ω_{0}_{|_{T_{x}M̂}}) is the g_{0}-orthogonal complement of 𝔷_{x} in T_{x}M̂ and ω_{0}(Z_{x}^{∗}) = Z for Z ∈ 𝔷. Then the connection form on M̂ associated with g_{λ} is given by ω_{λ} := ω_{0} + λ. Let Ω_{λ} denote the curvature form on M̂/T associated with the connection form ω_{λ} on M̂. We have π^{∗}Ω_{λ} = dω_{λ} because T is abelian.

(iv) Since λ is T -invariant and horizontal it induces a 𝔷-valued 1-form λ̄ on M̂/T. We conclude from π^{∗}Ω_{λ} = dω_{λ} = dω_{0}+dλ that Ω_{λ} = Ω_{0}+dλ̄. In particular, Ω_{λ} and Ω_{0} differ by an exact 𝔷-valued 2-form.

2.2 Proposition ([Sch3]). Let λ, λ′ be admissible 𝔷-valued 1-forms on M satisfying the following two conditions:

Ω_{λ} ∉ D ∘ \overline{Aut}_{g_{0}}^{T} (M)^{∗}Ω_{λ′} (N)

(G) No nontrivial 1-parameter group in \overline{Aut}_{g_{0}}^{T} (M) preserves Ω_{λ′} .

Then (M, g_{λ}) and (M, g_{λ′}) are not isometric.

Remark 2.3. (i) For our examples of conformally equivalent isospectral metrics on balls and spheres that we will construct in Sections 4 by applying Proposition 1.6, we will use the nonisometry result above with φg_{0} playing the role of g_{0} . We will not be able to apply this result to the Lie group examples in Section 3, however, and thus will prove the non-isometry of those examples by a different technique.

(ii) To construct isospectral potentials via Theorem 1.5, we may choose to use the same functions φ as for the examples of conformally equivalent isospectral metrics arising from Corollary 1.6. (We do lose a little generality this way, since Theorem 1.5 does not require that the potentials be positive.) Once we show that a pair of conformally equivalent isospectral metrics φg_{λ} and (τ^{∗}φ)g_{λ} constructed as in Corollary 1.6(ii) is nonisometric, it will follow immediately that the isospectral (by Theorem 1.5) potentials φ and τ^{∗}φ for the Schrödinger operator on (M, g_{λ}) are not congruent under any isometry of (M, g_{λ}). Indeed, if the potentials were congruent, then the metrics φg_{λ} and (τ^{∗}φ)g_{λ} would be isometric.

§3 Examples I: Lie groups

The examples in this section are modelled after the construction of isospectral left invariant Riemannian metrics on compact Lie groups given in [Sch2].

Let 𝔤 be a Lie algebra. Recall that ad_{𝔤} is a subalgebra of Der(𝔤), the algebra of derivations of 𝔤. The connected subgroup of Aut(𝔤) with Lie algebra ad(𝔤) is denoted Inn(𝔤) and is called the group of inner automorphisms of 𝔤. If G is any connected Lie group with Lie algebra 𝔤, then Inn(𝔤) = Ad_{G}. In particular, if G is a matrix group and thus 𝔤 is a matrix algebra, then Inn(𝔤) consists of all maps of 𝔤 given by conjugation by matrices in G.

Recall also that a Lie algebra is said to be compact if it is the Lie algebra of a compact Lie group.

3.1 Definition. Let 𝔤 be a compact Lie algebra, let 𝔷 be a real vector space with an inner product, and let j, j′ : 𝔷 → 𝔤 be linear maps.

(i) j and j′ will be said to be isospectral if for each Z ∈ 𝔷, there exists α_{Z} ∈ Inn(𝔤) such that j′(Z) = α_{Z}(j(Z)). (We will focus on matrix algebras, in which case this condition may be rewritten as j′(Z) = Ad(A_{Z})(j(Z)) = A_{Z}j(Z)A_{Z}^{−1} for some A_{Z} in the associated connected matrix group.)

(ii) j and j′ will be said to be equivalent if there exists an automorphism Φ of 𝔤 (not necessarily inner) and an orthogonal linear map C of 𝔷 such that j′(Z) = Φ(j(C(Z))) for all Z ∈ 𝔷. In this case we will say that the pair (Φ, C) is an equivalence between j and j′.

Remarks. The notion of isospectrality and of equivalence of maps j, j′ : 𝔷 → 𝔤 was first introduced in [GW2] in case 𝔤 = 𝔰𝔬(m) and in [Sch2,3] in case 𝔤 = 𝔰𝔲(m). Our notions of isospectrality and equivalence in case 𝔤 = 𝔰𝔬(m) differ slightly from those in [GW2]: the conjugating elements A_{Z} ∈ SO(m) in Definition 3.1(i) were allowed to lie instead in O(m) in [GW2]. In the notion of equivalence used in [GW2], the automorphism Φ was required to lie in Ad_{O(m)} . Since Aut(𝔰𝔬(m)) = Ad_{O(m)} except when m = 4 or m = 8, the two notions of equivalence actually agree except in these special dimensions.

Isospectral maps j and j′ will be used in various constructions of isospectral Riemannian manifolds. In general, equivalence of the maps j and j′ will imply isometry of the resulting manifolds; the converse will hold under generic conditions.

3.2 Proposition. (i) [GW2] Let m be any positive integer other than 1, 2, 3, 4, or 6. Let J be the real vector space consisting of all linear maps j : ℝ^{2} → 𝔰𝔬(m). Then there is a Zariski open subset O of J (i.e., O is the complement of the zero locus of some nonzero polynomial function on J) such that each j ∈ O belongs to a d-parameter family of isospectral, inequivalent elements of J. Here d ≥ m(m − 1)/2 − [m/2]([m/2] + 2) > 1. In particular, d is of order at least O(m^{2}).

(ii) [Sch2] Let m ≥ 3. Let J be the real vector space consisting of all linear maps j : ℝ^{2} → 𝔰𝔲(m). Then there is a Zariski open subset O of J such that each j ∈ O belongs to a continuous family of isospectral, inequivalent elements of J.

Remarks. (i) The article [GW2] also gives an example of a continuous family of isospectral, inequivalent maps from ℝ^{2} to 𝔰𝔬(6).

(ii) The two parts of the proposition were proven in [GW2] and [Sch2,3]. In the case of 𝔰𝔬(m) in [GW2], the notions of isospectrality and equivalence defined in the remark following Definition 3.1 were used. However, the proposition remains true if we instead define isospectrality as in Definition 3.1. Indeed, as pointed out in [Sch2], isospectrality (in the sense of [GW2]) of a continuous family j_{t} as in Proposition 3.2(i) means that on each Z ∈ ℝ^{2}, the path t → j_{t}(Z) lies in a single Ad_{O(m)}-orbit of j_{0}(Z) and thus, by continuity, the path in fact lives in a single Ad_{SO(m)}-orbit. Thus isospectrality also holds in the sense of Definition 3.1. A similar argument may be used to show that inequivalence in the sense of [GW2] of the continuous family {j_{t}} implies inequivalence in the sense of Definition 3.1.

(iii) In Section 5, we will place various genericity conditions on the maps j. The assertions of Proposition 3.2 remain true with these genericity conditions, as one can see from the original proofs.

The following lemma is immediate.

3.3 Lemma. Let 𝔨 be a compact Lie algebra, and let j_{1}, j_{2} : 𝔷 → 𝔨 be isospectral linear maps as in Definition 3.1. Let 𝔤 be a compact Lie algebra containing 𝔨 ⊕ 𝔨 as a subalgebra. Define j, j′ : 𝔷 → 𝔨 ⊕ 𝔨 ⊂ 𝔤 by j(Z) = (j_{1}(Z), j_{2}(Z)) and j′(Z) = (j_{2}(Z), j_{1}(Z)). Then

j′(Z) = (α_{Z} , α_{Z}^{−1})(j(Z))

for each Z ∈ 𝔷, where α_{Z} is given as in Definition 3.1 with respect to the pair of isospectral maps j_{1} , j_{2} . Thus j and j′ are isospectral.

3.4 Notation and Remarks. Let G be a compact Lie group with Lie algebra 𝔤, and let g_{0} be a bi-invariant Riemannian metric on G. Let H < G be a closed connected subgroup of G with Lie algebra 𝔥, and let T < G be a torus with Lie algebra 𝔷. Suppose that [𝔷, 𝔥] = 0, and that 𝔷 is g_{0}-orthogonal to 𝔥 in 𝔷.

(i) For any linear map j : 𝔷 → 𝔥 we define an associated left invariant 𝔷-valued 1-form λ^{j} on G by letting g_{0}(λ^{j}(X), Z) = g_{0}(X, j(Z)) for all Z ∈ 𝔷 and X ∈ 𝔤 (in other words, λ^{j} : 𝔤 → 𝔷 is the g_{0}-transpose of j : 𝔷 → 𝔥 ⊂ 𝔤).

(ii) We note that λ^{j} is admissible in the sense of 1.4 with respect to the action of T on G from the right. In fact, λ^{j} vanishes on the g_{0}-orthogonal complement of 𝔥 and thus, in particular, on 𝔷. The left invariant vector fields Z with Z ∈ 𝔷 are just the vector fields Z^{∗} (as in 1.4) induced by the right action of T. Moreover, λ^{j} is invariant under this action: Since [𝔷, 𝔥] = 0, the Lie algebra 𝔥 is g_{0}-orthogonal to [𝔷, 𝔤]; therefore, λ^{j} vanishes on [𝔷, 𝔤] and is hence invariant under Ad_{T} . Thus λ^{j} is not only left invariant but also right invariant under T.

(iii) Since λ^{j} is admissible with respect to the right action of T, we can define the associated metric g_{λ^{j}} on G as in Notation 1.4. Note that g_{λ^{j}} is left invariant (and right invariant under T).

3.5 Lemma [Sch2]. In the notation of 3.1 and 3.4, suppose that j, j′ : 𝔷 → 𝔥 are isospectral. Let λ, λ′ : 𝔤 → 𝔷 be the associated left-invariant 1-forms. Given a linear functional μ on 𝔷, let Z ∈ 𝔷 satisfy μ(W) = g_{0}(W, Z) for all W ∈ 𝔷. Define F_{μ} := L_{A_{Z}} ∘ R_{A_{Z}}^{−1} ∈ Aut(G), where A_{Z} ∈ H is such that j′(Z) = Ad(A_{Z})(j(Z)) (the existence of A_{Z} follows from Definition 3.1). Then

μ ∘ λ = F_{μ}^{∗}(μ ∘ λ′);

moreover, F_{μ} is T -equivariant because H and T commute, and is a g_{0}-isometry because g_{0} is bi-invariant. In particular, the associated Riemannian metrics g_{λ} and g_{λ′} on G are isospectral by Theorem 1.5(i) (with φ = 0).

3.6 Theorem. Let G be a compact Lie group with Lie algebra 𝔤, let H be a compact Lie subgroup with Lie algebra of the form 𝔥 = 𝔨 ⊕ 𝔨 for some Lie algebra 𝔨, and let T < G be a torus with Lie algebra 𝔷. Suppose that [𝔷, 𝔥] = 0, and that 𝔷 is orthogonal to 𝔥 with respect to a bi-invariant Riemannian metric g_{0} on G. Let j_{1} , j_{2} : 𝔷 → 𝔨 be isospectral linear maps as in 3.1. Define j : 𝔷 → 𝔨 ⊕ 𝔨 = 𝔥 by j(Z) = (j_{1}(Z), j_{2}(Z)). Denote by g_{λ} the associated left invariant metric on G defined as in 3.4, where λ := λ^{j} . Let φ be a smooth function on G which is invariant under the right action of T and under conjugation by elements of H. Suppose that there exists an isometric automorphism τ of (G, g_{0}) such that τ_{|_{T}} = Id and such that τ_{∗} restricts to the map (X, Y) ↦ (Y, X) on 𝔨 ⊕ 𝔨 = 𝔥 ⊂ 𝔤. Then:

(i) φ and τ^{∗}φ are isospectral potentials on (G, g_{λ}).

(ii) If φ is positive then φg_{λ} and (τ^{∗}φ)g_{λ} are conformally equivalent isospectral metrics on G.

Proof. Define j′ : 𝔷 → 𝔨 ⊕ 𝔨 by j′(Z) := (j_{2}(Z), j_{1}(Z)). By Lemma 3.3, j and j′ are isospectral. Let λ′ := λ^{j′} . By Lemma 3.5, λ and λ′ satisfy condition (∗) of Theorem 1.5 with respect to the right action of T on G, with each F_{μ} given as conjugation by an element of H. Since τ is a T -equivariant g_{0}-isometry and obviously satisfies τ_{∗}j(Z) = j′(Z) for all Z ∈ 𝔷, it follows immediately from the definition of λ and λ′ that τ^{∗}λ′ = λ. Thus statements (i) and (ii) follow from Theorem 1.5(ii) and Corollary 1.6, respectively. □

3.7 Examples. (i) Given a vector space 𝔷, say of dimension r, we may realize 𝔷 as the Lie algebra of a maximal torus T in SO(2r). Given isospectral maps j_{1} , j_{2} : 𝔷 → 𝔰𝔬(m) =: 𝔨, let G = SO(2m + 2r) with a bi-invariant Riemannian metric g_{0} . Then G contains SO(m) × SO(m) × SO(2r) as the subgroup formed by three diagonal blocks, which in turn contains SO(m)×SO(m)×T. Set H = SO(m)×SO(m). The map (p, q) ↦ (q, p) of H extends to an inner automorphism — hence an isometry — τ of G (namely, conjugation by the orthogonal map of ℝ^{2m+2r} interchanging the first m coordinates with the next m coordinates). Thus Theorem 3.6 yields an associated left invariant metric g_{λ} on G and, for any smooth function φ on G which is invariant under the right action of T and under conjugation by elements of H, a pair of isospectral potentials for the Schrödinger operator on (G, g_{λ}). Choosing φ positive, we further obtain a pair of isospectral metrics conformally equivalent to g_{λ}.

In particular, Proposition 3.2 and the subsequent remark allow us to construct isospectral potentials and conformally equivalent isospectral metrics on SO(2m+4) for all m ≥ 5. (Although Proposition 3.2 yields continuous families of isospectral j-maps, the construction in Theorem 3.6 requires us to work with pairs of isospectral maps. Thus we do not obtain continuous isospectral deformations of potentials or of conformally equivalent metrics by this construction.)

(ii) Consider the two-fold covering G̃ = Spin(2m + 2r) of SO(2m + 2r). Letting H̃ be the connected subgroup of G̃ with Lie algebra 𝔥 = 𝔰𝔬(m) ⊕ 𝔰𝔬(m) and T̃ the toral subgroup with Lie algebra 𝔷 ⊂ 𝔰𝔬(2r), and letting j_{1} , j_{2} : 𝔷 → 𝔨 = 𝔰𝔬(m) be isospectral linear maps, we then obtain, exactly as in Example (i), an associated metric g_{λ} on G̃ and, for any function φ on G̃ which is invariant under the right action of T and under conjugation by elements of H̃, a pair of isospectral potentials for the Schrödinger operator on (G̃, g_{λ}). Note for that purpose that the automorphism τ of SO(2m + 2r) lifts to an automorphism τ̃ of G̃. If φ is positive then we further obtain a pair of isospectral metrics conformally equivalent to g_{λ}. Thus we get pairs of isospectral potentials and conformally equivalent isospectral metrics on Spin(2m + 4) for all m ≥ 5.

(iii) In the notation of Example (i), we may also realize 𝔷 as a maximal torus in SU(r + 1). The analogous construction with 𝔨 = 𝔰𝔲(m) then yields, for each pair of isospectral maps j_{1} , j_{2} : 𝔷 → 𝔰𝔲(m) =: 𝔨, an associated left invariant Riemannian metric g_{λ} on G := SU(2m + r + 1) and, for appropriately chosen φ, isospectral potentials on (G, g_{λ}) and isospectral metrics on G conformally equivalent to g_{λ}. In particular, applying Proposition 3.2(ii), we obtain such isospectral potentials and conformally equivalent metrics on SU(2m + 3) for all m ≥ 3.

3.8 Remark. In 5.1 we will prove nontriviality of the above examples from 3.7 for a particular choice of φ under the hypothesis that j_{1} is not equivalent to j_{2} and under certain genericity conditions on j.

§4 Examples II: Spheres and Balls

Let r ≥ 2, and let T be the r-dimensional torus ℝ^{r}/L with L = (2πℤ)^{r}. Let {Z_{1} , …, Z_{r}} be the standard basis of its Lie algebra 𝔷 = ℝ^{r}.

4.1 Notation.

(i) For any given n, we consider the action of T = (S^{1})^{r} on ℝ^{n} ⊕ ℂ^{r} which is induced by the canonical action of S^{1}×…×S^{1} on ℂ×…×ℂ given by multiplication in each factor.

(ii) Given any 𝔷-valued 1-form λ on ℝ^{n}, its trivial extension to ℝ^{n} ⊕ ℂ^{r} (again denoted λ) is admissible in the sense of Notation 1.4 with respect to the action of T defined above. Let (B, g_{0}), respectively (S, g_{0}), be the standard unit ball, respectively sphere, in ℝ^{n}⊕ℂ^{r} ≅ ℝ^{n+2r}. We continue to denote by λ the restriction, respectively pullback, of λ to B, respectively S. Then λ is admissible with respect to the induced action of T and thus defines a metric g_{λ} on B and on S.

4.2 Definition. Two 𝔷-valued 1-forms ν, ν′ on ℝ^{m} are called isospectral if for each μ ∈ L^{∗} there exists A_{μ} ∈ O(m) such that μ ∘ ν = A_{μ}^{∗}(μ ∘ ν′).

4.3 Theorem. Let n = 2m, and let M denote either the ball B or the sphere S in ℝ^{n+2r} ≅ ℝ^{n} ⊕ ℂ^{r}. Let ν, ν′ be isospectral 𝔷-valued 1-forms on ℝ^{m}. Define a 𝔷-valued 1-form λ on ℝ^{n} = ℝ^{m} ⊕ ℝ^{m} by

λ_{(p,q)}(X, Y) := ν_{p}(X) + ν′_{q}(Y).

Denote by g_{λ} the associated metric on M as in Notation 4.1(ii). Let ψ : ℝ^{m} → ℝ be a smooth radial function and define φ_{1} , φ_{2} : ℝ^{m} ⊕ℝ^{m} ⊕ℂ^{r} → ℝ by φ_{i}(p_{1} , p_{2} , u) := ψ(p_{i}) for i = 1, 2. Then

(i) φ_{1} and φ_{2} are isospectral potentials on (M, g_{λ}).

(ii) If ψ is strictly positive, then φ_{1}g_{λ} and φ_{2}g_{λ} are isospectral conformally equivalent Riemannian metrics on M.

Proof. We apply the criteria of Theorem 1.5 and Corollary 1.6. Define a second 𝔷-valued 1-form λ′ on ℝ^{n} = ℝ^{m} ⊕ ℝ^{m} by

λ′_{(p,q)}(X, Y) := ν′_{p}(X) + ν_{q}(Y).

Let g_{λ′} be the associated Riemannian metric on M. Given any μ ∈ L^{∗}, choose A_{μ} ∈ O(m) as in Definition 4.2. Then the map

F_{μ} : ℝ^{m} ⊕ ℝ^{m} ⊕ ℂ^{r} ∋ (p, q, u) ↦ (A_{μ}p, A_{μ}^{−1}q, u) ∈ ℝ^{m} ⊕ ℝ^{m} ⊕ ℂ^{r}

restricts to a T -equivariant isometry of (M, g_{0}) satisfying μ ∘ λ = F_{μ}^{∗}(μ ∘ λ′). Moreover, the map

τ : ℝ^{m} ⊕ ℝ^{m} ⊕ ℂ^{r} ∋ (p, q, u) ↦ (q, p, u) ∈ ℝ^{m} ⊕ ℝ^{m} ⊕ ℂ^{r}

also restricts to a T -equivariant isometry of (M, g_{0}) and satisfies λ = τ^{∗}λ′.

We let φ_{1} play the role of φ in the notation of 1.5 and 1.6, and observe that φ_{2} = τ^{∗}φ_{1} . Since ψ is radial, φ_{1} is invariant under each F_{μ}. Thus Theorem 4.3 follows from Theorem 1.5 and Corollary 1.6. □

4.4 Example. Our first family of examples is modelled after the construction of isospectral balls and spheres in [Go3] (see Remark 4.5 below).

Let j, j′ : 𝔷 → 𝔰𝔬(m) be isospectral linear maps, where we allow the weaker notion of isospectrality given in the remark following Definition 3.1. Define two 𝔷-valued 1-forms ν, ν′ on ℝ^{m} by

⟨ν^{(′)}(X_{p}), Z⟩ = ⟨j^{(′)}(Z)p, X⟩

for X_{p} ∈ T_{p}ℝ^{m} and Z ∈ 𝔷, where ⟨ , ⟩ denotes the standard inner products both on ℝ^{m} and on 𝔷 = ℝ^{k}. Then ν and ν′ are isospectral in the sense of Definition 4.2. In fact, let Z ∈ 𝔷 be the vector dual to μ with respect to the inner product ⟨ , ⟩ on 𝔷. Choose A_{Z} ∈ O(m) as in the remark following Definition 3.1; then A_{μ} := A_{Z} will satisfy μ ∘ ν = A_{μ}^{∗}(μ ∘ ν′).

Recall from Proposition 3.2 that for r = 2, i.e. 𝔷 = ℝ^{2}, and each m ≥ 5 there do exist pairs j, j′ : 𝔷 → 𝔰𝔬(m) as above. Using the corresponding forms ν, ν′ for constructing λ in Theorem 4.3, we obtain a metric g_{λ}, pairs of isospectral metrics conformally equivalent to g_{λ}, and pairs of potentials which are isospectral with respect to g_{λ} on each of the spheres S^{2m+3≥13} and the balls B^{2m+4≥14}.

4.5 Remark. In 5.2 we will show that the examples from 4.4 with 𝔷 = ℝ^{2} are nontrivial provided that

(i) the maps j, j′ are not equivalent in the sense of Definition 3.1, (ii) j and j′ are generic in the sense that no nonzero element of 𝔰𝔬(m) commutes

with both j(Z_{1}), j(Z_{2}), or with both j′(Z_{1}), j′(Z_{2}), and (iii) the restriction to \overline{B_{1}(0)} ⊂ ℝ^{m} of the smooth radial function ψ : ℝ^{m} → ℝ_{+}

attains its maximum precisely in p = 0.

Remark. In the notation of Theorem 4.3 and Example 4.4 above, the isospectral spheres and balls from [Go3] were just the unit spheres, respectively balls, in ℝ^{m} ⊕ ℂ^{r}, endowed with the associated isospectral metrics g_{ν} and g_{ν′} .

Note that in Example 4.4, we had r ≥ 2 and m ≥ 5, where r and m are the dimensions used in Theorem 4.3. We now give an example with r = 2 and m = 3, yielding conformally equivalent isospectral metrics and isospectral potentials on S^{9} ⊂ ℝ^{3} ⊕ ℝ^{3} ⊕ ℂ^{2} and on B^{10}. The approach is related to certain constructions from [Sch2,3].

4.6 Example. Let r = 2, thus 𝔷 = ℝ^{2}. Let Sym_{0}(ℝ^{3}) denote the space of symmetric traceless real 3 × 3 -matrices. For each linear map c : 𝔷 ≅ ℝ^{2} → Sym_{0}(ℝ^{3}) we define a 𝔷-valued 1-form ν on ℝ^{3} by letting

⟨ν(X_{p}), Z⟩ = ⟨c(Z)p × p, X⟩

for all X ∈ T_{p}ℝ^{3} and Z ∈ 𝔷, where ⟨ , ⟩ denotes the standard euclidean inner products on both ℝ^{3} and on 𝔷, and × denotes the vector product in ℝ^{3}.

The key observation is that there exist pairs of linear maps c, c′ : 𝔷 → Sym_{0}(ℝ^{3}) such that

1.) c and c′ are isospectral, that is, for each Z ∈ 𝔷 there exists E_{Z} ∈ SO(3) such

that c′(Z) = E_{Z}c(Z)E_{Z}^{−1} ; 2.) c and c′ are not equivalent, where being equivalent means: There exists E ∈ O(3) and C ∈ O(2) such that c′(Z) = Ec(C(Z))E^{−1} for all Z ∈ 𝔷; 3.) c and c′ are generic, that is, no nonzero element of 𝔰𝔬(3) commutes with

both c(Z_{1}) and c(Z_{2}), and the analogous statement holds for c′.

An explicit example of a pair c, c′ satisfying the conditions above is given by

c(Z_{1}) = c′(Z_{1}) = ( \begin{matrix}−1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) , c(Z_{2}) = ( \begin{matrix}0 & 1/\sqrt{2} & 0 \\ 1/\sqrt{2} & 0 & 1/\sqrt{2} \\ 0 & 1/\sqrt{2} & 0 \end{matrix} ) , c′(Z_{2}) = ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} )

(compare [Sch3], Proposition 3.3.6; actually c(Z_{2}) and c′(Z_{2}) were rescaled by \sqrt{2} there, but this does not affect any of the three properties above).

It is easy to check that condition 1.) implies isospectrality (in the sense of 4.2) of the associated 𝔷-valued 1-forms ν, ν′ on ℝ^{3}. Using these in Theorem 4.3, we obtain pairs of isospectral potentials and pairs of conformally equivalent isospectral metrics on S = S^{9} ⊂ ℝ^{3} ⊕ ℝ^{3} ⊕ ℂ^{2}, resp. on B = B^{10}.

In 5.2 we will prove the corresponding nontriviality statements, provided that the restriction to \overline{B_{1}(0)} ⊂ ℝ^{3} of the smooth radial function ψ : ℝ^{3} → ℝ_{+} attains its maximum precisely in p = 0.

Having thus obtained examples of conformally equivalent isospectral metrics and isospectral potentials on S^{9} and B^{10}, we do a final twist in order to obtain such examples even on S^{7} and B^{8}. The idea is to lift the forms ν, ν′ used in Example 4.6 from ℝ^{3} to ℂ^{2} via the Hopf projection. The new λ = ν⊕ν′ will live on ℂ^{2}⊕ℂ^{2} ≅ ℝ^{8} instead of ℝ^{3} ⊕ ℝ^{3}. In return, the third component of our previous ambient space ℝ^{3} ⊕ ℝ^{3} ⊕ ℂ^{2} will become unnecessary because we will use a different torus action, living on ℂ^{2} ⊕ ℂ^{2} itself. This construction is related to a construction from [Sch3] of isospectral metrics on S^{5} ⊂ ℂ^{2} ⊕ ℂ and on B^{6}.

First we need some preparations.

4.7 Notation. (i) Let r = 2, thus 𝔷 = ℝ^{2} and T = ℝ^{2}/(2πℤ)^{2}. For any m ∈ ℕ we identify the real vector spaces ℂ^{m} and ℝ^{2m} via the linear isomorphism which sends {e_{1} , ie_{1} , …, e_{m} , ie_{m}} (in this order) to the standard basis of ℝ^{2m}, where {e_{1} , …, e_{m}} denotes the standard basis of ℂ^{m}. We let the torus T act on ℂ^{m}⊕ℂ^{m} by

exp(aZ_{1} + bZ_{2}) : (p, q) ↦ (e^{ia}p, e^{ib}q)

for all a, b ∈ ℝ, p, q ∈ ℂ^{m}.

(ii) Let (B, g_{0}), resp. (S, g_{0}), be the standard unit ball of dimension 4m, resp. the standard unit sphere of dimension 4m − 1, in ℝ^{4m} ≅ ℂ^{m} ⊕ ℂ^{m}. Given any 𝔷-valued 1-form λ on ℂ^{m} ⊕ ℂ^{m} which is admissible with respect to the action of T defined above, we also denote by λ the restriction, resp. the pullback, to B, resp. S. Then λ is admissible with respect to the induced action of T on B and S, and thus defines a corresponding metric g_{λ} on B, resp. S.

4.8 Definition.

(i) A 𝔷-valued 1-form ν on ℂ^{m} ≅ ℝ^{2m} is called Hopf admissible if it is admissible, in the sense of 1.4, with respect to the Hopf action of the 1-dimensional Torus S^{1} on ℂ^{m}.

(ii) Two Hopf admissible 𝔷-valued 1-forms ν, ν′ on ℂ^{m} are called isospectral if for each μ ∈ L^{∗} there exists A_{μ} ∈ SU(m) such that μ ∘ ν = A_{μ}^{∗}(μ ∘ ν′).

4.9 Theorem. Let M denote either the ball B or the sphere S in ℂ^{m} ⊕ℂ^{m} ≅ ℝ^{4m}. Let ν and ν′ be Hopf admissible 𝔷-valued 1-forms on ℂ^{m}, isospectral in the sense of Definition 4.8. Write ν = (ν_{1} , ν_{2}) and ν′ = (ν′_{1} , ν′_{2}) with respect to the coordinates on 𝔷 = ℝ^{2} defined by the basis {Z_{1}, Z_{2}}. Assume that there exist A, A′ ∈ SU(m) such that A^{∗}ν_{1} = ν_{2} , A^{∗}ν_{2} = ν_{1} , and similarly for A′ with respect to ν′. Define a 𝔷-valued 1-form λ on ℂ^{m} ⊕ ℂ^{m} by

λ_{(p,q)}(X, Y) := ν_{p}(X) + ν′_{q}(Y).

for all (p, q) ∈ ℂ^{m} ⊕ ℂ^{m}. Let ψ : ℂ^{m} ≅ ℝ^{2m} → ℝ be a smooth radial function and define φ_{1} , φ_{2} : ℂ^{m} ⊕ ℂ^{m} → ℝ by φ_{i}(p_{1} , p_{2}) := ψ(p_{i}) for i = 1, 2. Then

(i) φ_{1} and φ_{2} are isospectral potentials on (M, g_{λ}).

(ii) If ψ is strictly positive, then φ_{1}g_{λ} and φ_{2}g_{λ} are isospectral conformally equivalent Riemannian metrics on M.

Proof. The proof is very similar to that of Theorem 4.3, slightly complicated by the fact that switching the components of ℂ^{m} ⊕ ℂ^{m} is no longer a T -equivariant map here. Define a second 𝔷-valued 1-form λ′ on ℂ^{m} ⊕ ℂ^{m} by

λ′_{(p,q)}(X, Y) := ν′_{p}(X) + ν_{q}(Y).

Let g_{λ′} be the associated Riemannian metric on M. Given any μ ∈ L^{∗} and the associated A_{μ} as in Definition 4.8(ii), the map

F_{μ} : ℂ^{m} ⊕ ℂ^{m} ∋ (p, q) ↦ (A_{μ}p, A_{μ}^{−1}q) ∈ ℂ^{m} ⊕ ℂ^{m}

restricts to an isometry of (M, g_{0}) satisfying μ ∘ λ = F_{μ}^{∗}(μ ∘ λ′); note that F_{μ} is T -equivariant since each A_{μ} ∈ SU(m) commutes with multiplication by complex scalars. Moreover, the map

τ : ℂ^{m} ⊕ ℂ^{m} ∋ (p, q) ↦ (A′q, Ap) ∈ ℂ^{m} ⊕ ℂ^{m}

restricts to a T -preserving isometry of (M, g_{0}). The corresponding linear automorphism Ψ_{τ} of 𝔷 interchanges the basis vectors Z_{1} and Z_{2} . By the conditions on A and A′, it is easy to check that τ satisfies Ψ_{τ} ∘ λ = τ^{∗}λ′.

We let φ_{1} play the role of φ in the notation of 1.5 and 1.6, and observe that φ_{2} = τ^{∗}φ_{1} because ψ is radial. By the same reason, φ_{1} is invariant under each F_{μ}. Thus Theorem 4.9 follows from Theorem 1.5 and Corollary 1.6. □

4.10 Example. We fix a realization of the Hopf projection P : S^{3} → S_{1/2}^{2} ⊂ ℝ^{3} in coordinates, say

P : (α, β, γ, δ) ↦ (\frac{1}{2}(α^{2} + β^{2} − γ^{2} − δ^{2}), αγ + βδ, αδ − βγ).

We extend P to a smooth map from ℝ^{4} ≅ ℂ^{2} to ℝ^{3}, defined by the same formula (note that P will map S_{a}^{3} to S_{a^{2}/2}^{2} for each radius a ≥ 0).

With each linear map c : 𝔷 ≅ ℝ^{2} → Sym_{0}(ℝ^{3}) we associate a 𝔷-valued 1-form ν = (ν_{1} , ν_{2}) on ℂ^{2} by letting

ν_{k}(X_{p}) = ⟨c(Z_{k})P (p) × P (p), P_{∗p}(X)⟩

for k = 1, 2 and all X ∈ T_{p}ℂ^{2}.

Consider the specific pair of linear maps c, c′ : ℝ^{2} → Sym_{0}(ℝ^{3}) from Example 4.6. In addition to the three properties (isospectrality, inequivalence, genericity) stated there, we observe that

(I) There exist E, E′ ∈ SO(3) satisfying Ec(Z_{1})E^{−1} = c(Z_{2}) , Ec(Z_{2})E^{−1} =

c(Z_{1}) , and similarly for E′ with respect to c′. In fact, the elements E, E′ of SO(3) given by

E = ( \begin{matrix}1/2 & −1/\sqrt{2} & 1/2 \\ −1/\sqrt{2} & 0 & 1/\sqrt{2} \\ 1/2 & 1/\sqrt{2} & 1/2 \end{matrix} ) , E′ = ( \begin{matrix}−1/\sqrt{2} & 0 & 1/\sqrt{2} \\ 0 & −1 & 0 \\ 1/\sqrt{2} & 0 & 1/\sqrt{2} \end{matrix} )

satisfy E^{2} = E^{′ 2} = Id and conjugate c(Z_{1}) to c(Z_{2}) , respectively c′(Z_{1}) to c′(Z_{2}) .

Again, condition 1.) on c, c′ from Example 4.6 implies isospectrality of the associated Hopf admissible ℝ^{2}-valued 1-forms ν, ν′ on ℂ^{2}. In fact, for any given μ ∈ L^{∗} ⊂ 𝔷^{∗} let Z ∈ 𝔷 be the dual vector (via the chosen basis). Choose E_{Z} as in condition 1.), and choose A_{Z} ∈ SU(2) ⊂ SO(4) such that P ∘ A_{Z} = E_{Z} ∘ P. Then it is straightforward to check that A_{μ} := A_{Z} satisfies μ ∘ ν′ = A_{μ}^{∗}(μ ∘ ν).

Moreover, condition (I) implies existence of elements A, A′ ∈ SU(2) as required in the conditions of Theorem 4.9: Just choose A, A′ such that P ∘ A = E ∘ P, P ∘ A′ = E′ ∘ P. (Note that A, A′ are both unique up to sign and will satisfy A^{2} = A^{′ 2} = −Id = e^{iπ}Id.)

Thus we are able to apply Theorem 4.9 and obtain pairs of isospectral potentials and pairs of conformally equivalent isospectral metrics on S = S^{7} ⊂ ℂ^{2} ⊕ ℂ^{2} and on B = B^{8}. In 5.3 we will show nontriviality of these examples for every choice of a positive smooth radial function ψ : ℝ^{4} ≅ ℂ^{2} → ℝ_{+} which is strictly monotonically increasing with the radius.

4.11 Remark. The choice of the metrics g_{λ} to which our isospectral metrics are conformal, resp. which admit the pairs of isospectral potentials constructed above, allows an interesting modification. In each of our examples of this section (4.4, 4.6, 4.10) one can actually modify g_{λ} in such a way that it becomes equal to the standard metric g_{0} outside certain subsets of arbitrarily small volume. The idea, first used in [Sch3] for a similar aim, is to exploit the following obvious fact:

If λ, λ′ are 𝔷-valued admissible forms on M which satisfy condition (∗) of Theorem 1.5, then so do fλ, fλ′, where f ∈ C^{∞}(M) is any function which is invariant under T and under each F_{μ} occurring in (∗).

In particular, the statements 1.5(i) and 1.6(i) will still hold with λ, λ′ replaced by fλ, fλ′, and so will 1.5(ii) and 1.6(ii), provided that f is invariant under τ, too. In Theorems 4.3 and 4.9, which were specializations of 1.5/1.6, we can therefore, without affecting any of the isospectrality statements, replace λ by fλ, where f is the restriction to M of

– (for Theorem 4.3, hence the Examples 4.4 and 4.6:) any smooth function on ℝ^{m} ⊕ℝ^{m} ⊕ℂ^{r} which is of the form (p, q, u) ↦ α(|p|, |q|, |u|) with α(r, s, t) = α(s, r, t) for all (r, s, t);

– (for Theorem 4.9, hence Example 4.10:) any smooth function on ℂ^{m} ⊕ ℂ^{m} which is of the form (p, q) ↦ α(|p|, |q|) with α(a, b) = α(b, a) for all (a, b).

In particular, we may in each case choose α to have as small support (in the parameter set relevant for M) as we like. Then the support of f in M will have small volume with respect to the standard metric g_{0} and hence also with respect to g_{fλ} (recall from 1.4(iii) that dvol_{g_{0}} = dvol_{g_{fλ}}).

In 5.4, we will show that the corresponding nonisometry statements will remain true, provided that f does not vanish identically on M; only in the case of Example 4.10 and M = B^{8}, we will need an additional technical assumption: Namely, that the support of α nontrivially intersects a certain open set U ⊂ {(a, b) ∈ B_{1}(0) | a > b > 0} (whose definition will actually depend on the radial function ψ used for constructing the potentials).

§5 Non-triviality of the examples

5.1 Nonisometry proof for the examples from 3.7. We consider the construction of metrics and potentials on Lie groups given in Example 3.7(i–iii). We first prove that for a special choice of the function φ and with a very mild genericity condition on j (condition 5.1.2(a) below), the isospectral potentials φ and ψ := τ^{∗}φ on (G, g_{λ}) are not congruent if j_{1} and j_{2} are inequivalent (Theorem 5.1.8). We then show under an additional genericity hypothesis on j = (j_{1}, j_{2}) (condition 5.1.2(b)) that the isospectral conformally equivalent metrics φg_{λ} and ψg_{λ} are not isometric (Theorem 5.1.13). For notational simplicity, we will focus throughout on the two cases G = SO(2m + 2r) and G = SU(2m + r + 1). With minor adjustments to the proofs in the case of SO(2m + 2r), one obtains the same results for G = Spin(2m + 2r).

5.1.1 The function φ. For any matrix A ∈ M(m, m, ℂ) and any column vector V ∈ ℂ^{m} we write

d(A, V) := det(V, AV, A^{2}V, …, A^{m−1}V).

Note that d is invariant under the action of SL_{ℂ}(m) on M(m, m, ℂ) × ℂ^{m} given by B(A, V) = (BAB^{−1}, BV) for B ∈ SL_{ℂ}(m). It is easy to see that if A has pairwise different eigenvalues λ_{1} , …, λ_{m}, then d(A, V) is the Vandermonde determinant |( λ_{k}^{ℓ} )|_{{1≤k≤m \atop 0≤ℓ≤m−1}} = ∏_{i<j}(λ_{j} −λ_{i}), multiplied by the product of the coefficients of V

with respect to an eigenbasis of determinant one of A. If A is diagonizable and has at least one multiple eigenvalue, then d(A, V) = 0 for all V.

For A ∈ M(m, m, ℂ) and C ∈ M(m, ℓ, ℂ) we let

d(A, C) := ∏_{k=1,... ,ℓ} d(A, C_{k}),

where C_{1} , …, C_{ℓ} denote the column vectors of C.

We write matrices in G = SO(2m + 2r) or G = SU(2m + r + 1) as

X = ( \begin{matrix}A & B & C \\ D & E & F \\ H & J & L \end{matrix} ) ,

where A, B, D, E are m × m -matrices and L is a 2r × 2r -matrix or an (r + 1) × (r + 1)-matrix, respectively. Let c_{1} > c_{2} > 0. In case G = SO(2m + 2r), we define φ : G → ℝ_{+} by

φ(X) := exp(c_{1} det A + c_{2} det E).

In case G = SU(2m + r + 1), we let

φ(X) := exp(c_{1} Re det A + c_{2} Re det E − (Re d(A, C))^{2} − (Re d(E, F))^{2}),

where Re denotes the real part.

5.1.2 Genericity conditions.

(a) The kernel of the map j = (j_{1}, j_{2}) : 𝔷 → 𝔥 is trivial, where 𝔥 = 𝔨 ⊕ 𝔨.

(b) The image of j has trivial centralizer in 𝔥.

5.1.3 Notation. For any subgroup U of G, denote by U_{L}, respectively U_{R}, the group of all left, respectively right, translations of G by elements of U. Moreover, denote by Inn_{G}(U) the subgroup of the group of inner automorphisms of G consisting of conjugations by elements of U.

5.1.4 Lemma [OT]. Let G be a connected simple compact Lie group with a left invariant Riemannian metric g. Then:

(i) The full group of isometries of (G, g) is the semi-direct product of G_{L} with the group of all automorphims of G that preserve g.

(ii) Let W = {a ∈ G | R_{a} preserves g} and let Z(G) be the (finite) center of G. Then the identity component Isom_{0}(G, g) of the full isometry group is given by (G_{L} × W_{R})/Z(G), where Z(G) is embedded by a ↦ L_{a} ∘ R_{a}^{−1}.

Note that a ∈ W if and only if the associated inner automorphism I_{a} of G is an isometry of (G, g).

5.1.5 Notation. Let P be the subgroup of G given by those matrices of the form

( \begin{matrix}Id_{m} & 0 & 0 \\ 0 & Id_{m} & 0 \\ 0 & 0 & ∗ \end{matrix} ) .

Thus P is isomorphic to SO(2r), respectively SU(r + 1), in case G = SO(2m + 2r), respectively G = SU(2m + r + 1). Let Q = H × P. Recall that H is the subgroup SO(m) × SO(m), respectively SU(m) × SU(m), of G, embedded in the upper left corner, and that T is the standard maximal torus in P.

5.1.6 Lemma. Let φ be defined as in 5.1.1. Let G denote the subgroup of G_{L} ×G_{R} consisting of those elements which leave φ invariant, and let G_{0} denote the connected component of the identity in G. Then:

(i) If L_{a}R_{b} ∈ G then a·b ∈ Q and a, b are elements of S(O(m)×O(m)×O(2r)), respectively of S(U(m) × U(m) × U(r + 1)). In particular, I_{a} and I_{b} leave the subgroups H and P of G invariant.

(ii) G_{0} = Q_{L} × Q_{R} in case G = SO(2m + 2r), and G_{0} = Inn_{G}(H) × P_{L} × T_{R} in case G = SU(2m + r + 1).

Proof. (i) It is obvious from the definition of φ that it assumes its maximum value precisely on Q ⊂ G. By determining first which elements of G_{L} × G_{R} preserve Q, and then ruling out those of these candidates which obviously do not preserve φ (taking into particular account that c_{1} > c_{2} > 0) it is not hard to see that a and b must be of the asserted form. Moreover, aQb = Q implies a · b ∈ Q.

(ii) If G = SO(2m + 2r) then (i) implies G_{0} ⊂ Q_{L} × Q_{R}. Since Q_{L} × Q_{R} does leave φ (as defined for this G) invariant, the statement follows.

Now let G = SU(2m + r + 1). In this case, (i) says that G, and hence G_{0} , is contained in

Q_{L} × Q_{R} × Inn_{G}(D) = H_{L} × H_{R} × P_{L} × P_{R} × Inn_{G}(D),

where D is the subgroup of G consisting of the matrices ( \begin{matrix}αId_{m} \\ & βId_{m} \\ & & γId_{r+1} \end{matrix} ) ∈ G

with α, β, γ ∈ e^{iℝ}. It is clear from the definition of φ and the comments in 5.1.1 that Inn_{G}(H)×P_{L}×T_{R} does preserve φ, hence is contained in G_{0} . We have to show that it is in fact equal to G_{0} . We claim that any 1-parameter family in H_{L} × P_{R} × Inn_{G}(D) which leaves φ invariant is actually contained in T_{R}; this will obviously imply the statement. Let h(t), p(t), d(t) be 1-parameter families in H, P, and D, respectively, with the property that L_{h(t)}R_{p(t)}I_{d(t)} leaves φ invariant. Write h(t) =

( \begin{matrix}k_{1}(t) \\ & k_{2}(t) \\ & & Id \end{matrix} ), where k_{1}(t), k_{2}(t) are 1-parameter families in SU(m), and write

d(t) as above with 1-parameter families α(t), β(t), γ(t) in e^{iℝ}. Identify p(t) with the corresponding 1-parameter family in SU(r + 1). Note that the first two summands in the exponent of φ are invariant under L_{h(t)}R_{p(t)}I_{d(t)} ; consequently, this family must also leave the function ψ : G ∋ X ↦ (Re d(A, C))^{2} + (Re d(E, F))^{2} ∈ ℝ invariant, where X is written as in 5.1.1.

We first show that the family h(t) is trivial. Assume the contrary. Then k_{1}(t) or k_{2}(t) is nontrivial; without loss of generality we assume k_{1}(t) is nontrivial. Choose t_{0} such that k_{1}(t_{0}) has at least two different eigenvalues. Choose A′ ∈ SU(m) such that A′ has less than m different eigenvalues, but k_{1}(t_{0})^{−1}A′ has m pairwise different eigenvalues. Choose 0 < c < 1. It is easy to see that there exist elements X of G, written as in 5.1.1, with A = ck_{1}(t_{0})^{−1}A′ and F = 0. It is also not hard to show that the condition that no column vector of C be orthogonal to any of the eigenvectors of A — equivalently, d(A, C) ≠ 0 — is a generic condition on this subset of G, and that so is the condition Re d(A, C) ≠ 0. Choose X ∈ G with these properties. Then ψ(X) = (Re d(A, C))^{2} + 0 ≠ 0, but ψ(L_{h(t_{0})}R_{p(t_{0})}I_{d(t_{0})}X) = (Re d(cA′, k_{1}(t_{0})Cp(t_{0})α(t_{0})γ(t_{0})^{−1})^{2} + 0 = 0 since A′ has less than m different eigenvalues (recall the comments in 5.1.1). This is a contradiction; so the family h(t) must indeed be trivial.

Next we attack p(t). Assume that the family p(t) were not contained in the maximal torus T consisting of the diagonal elements in P. Let k be a diagonal matrix in SU(m) with m different eigenvalues, and set A := ck for some c with 0 < c < 1. Choose t_{0} such that some column vector p_{i}(t_{0}) of p(t_{0}) has at least two nonzero entries. It is easy to see that there exist elements X of G, written as in 5.1.1, with the given A = ck and with F = 0, such that some row vector of C is orthogonal to p_{i}(t_{0}). It is also not hard to show that the condition that C have only nonvanishing entries — equivalently: d(A, C) ≠ 0 — is a generic condition on this subset of G (recall the choice of p_{i}(t_{0})), and that so is the condition Re d(A, C) ≠ 0. Choose X ∈ G with these properties. Then ψ(X) = (Re d(A, C))^{2} ≠ 0, but ψ(L_{h(t_{0})}R_{p(t_{0})}I_{d(t_{0})}X) = (Re d(A, Cp(t_{0})α(t_{0})γ(t_{0})^{−1})^{2} = 0 since Cp_{i}(t_{0}) has a zero entry and thus d(A, Cp_{i}(t_{0})) = 0 (recall that A is diagonal). This is a contradiction, so the family p(t) must be contained in T.

Finally, for every X ∈ G with F = 0 we have that ψ(R_{p(t)}I_{d(t)}X) is equal to (Re d(A, Cp(t)α(t)γ(t)^{−1})^{2} = (Re((α(t)γ(t)^{−1})^{m(r+1)}d(A, C))^{2} since p(t) ∈ T. The fact that for every given such X this expression must be constant in t implies α(t) = γ(t) for all t. Analogously we obtain β(t) = γ(t) for all t, hence d(t) = α(t)Id_{2m+r+1} . But d(t) ∈ SU(2m + r + 1); hence the family d(t) is trivial. □

5.1.7 Lemma. Assume that j satisfies the genericity condition 5.1.2(a). Let Φ be any automorphism of Q which preserves the subgroups H and P and which is an isometry of the metric g_{λ}. Then Φ preserves T, and (Φ_{∗}^{−1}_{|_{𝔥}} , Φ_{∗}_{|_{𝔷}}) is a selfequivalence of j in the sense of Definition 3.1.

Proof. For X ∈ 𝔥 and U ∈ 𝔭 we have g_{λ}(X, U) = g_{0}(λ(X), U) since λ_{|_{𝔭}} = 0 and since 𝔥 is g_{0}-orthogonal to 𝔭. From the facts that g_{λ}(X, U) = g_{λ}(Φ_{∗}(X), Φ_{∗}(U)) and that Φ_{∗} preserves 𝔥 and 𝔭, it follows that

g_{0}(λ(X), U) = g_{0}(λ(Φ_{∗}X), Φ_{∗}U).

Using the genericity condition 5.1.2(a), we may conclude that Φ_{∗}(𝔭 ∩ 𝔷^{⊥}) ⊂ 𝔭 ∩ 𝔷^{⊥} and thus Φ_{∗}(𝔷) = 𝔷. Letting U ∈ 𝔷 in the displayed equation above, we then have

g_{0}(X, j(U)) = g_{0}(Φ_{∗}X, j(Φ_{∗}U)) = g_{0}(X, Φ_{∗}^{−1}(j(Φ_{∗}U))),

and the lemma follows. □

5.1.8 Theorem. We use the notation of Example 3.7. Let φ be the function given in 5.1.1 and let ψ = τ^{∗}φ. Assume that j_{1} is not equivalent to j_{2} and that j satisfies the genericity condition 5.1.2(a). Then ψ is not congruent to φ under an isometry of (G, g_{λ}).

Thus by Example 3.7, φ and ψ are non-congruent isospectral potentials on (G, g_{λ}).

Proof. Suppose F is an isometry of (G, g_{λ}) such that F^{∗}ψ = φ. By Lemma 5.1.4, F ∈ G_{L} × Aut(G). Recall that τ = I_{c}, where

c = ( \begin{matrix}0 & Id & 0 \\ Id & 0 & 0 \\ 0 & 0 & Id \end{matrix} ) .

Thus I_{c}^{∗}ψ = φ, so φ is invariant under I_{c^{−1}} ∘ F. Let σ be the outer automorphism of G given (in case G = SO(2m+2r)) by conjugation by a diagonal matrix of determinant −1 all of whose diagonal entries lie in {±1}, or (in case G = SU(2m + r + 1)) by the map A ↦ Ā. Observe that σ leaves φ invariant. Since for both choices of G, Inn(G) has index two in Aut(G), I_{c}^{−1} ∘ F may be written either as σ ∘ I_{a} ∘ L_{b} or else as I_{a}∘L_{b} for some a, b ∈ G. In either case (I_{a}∘L_{b})^{∗}φ = φ, so Lemma 5.1.6(i) implies that I_{a} and I_{b} preserve H and P. Consequently, the automorphism Φ := F ∘ L_{b}^{−1} (which is either I_{ca} or I_{c} ∘ σ ∘ I_{a}) is an isometry of (G, g_{λ}) which preserves both H and P and interchanges the two factors of H. By Lemma 5.1.7, we see that (Φ_{∗}^{−1}_{|_{𝔥}} , Φ_{∗}_{|_{𝔷}}) is a self-equivalence of j which intertwines j_{1} and j_{2} , contradicting the fact that j_{1} and j_{2} are not equivalent. □

We now prepare for the proof that the conformally equivalent metrics in Example 3.7 are not isometric for our given choice of φ and with the genericity hypotheses 5.1.2.

5.1.9 Proposition. Suppose that (M, g) is a compact Riemannian manifold admitting a free action by isometries by a torus T and that the orbits all have the same volume (i.e., they are totally geodesic). Let φ and ψ be positive smooth T -invariant functions on M. Let S be a non-trivial subtorus of T and let F : (M, φg) → (M, ψg) be an isometry that carries S-orbits to S-orbits. Then F^{∗}ψ = φ and F ∈ Isom(M, g). In particular (letting ψ = φ), the normalizer of S in Isom(M, φg) is contained in {F ∈ Isom(M, g) | F^{∗}φ = φ}.

Proof. The second statement is immediate from the first. To prove that F^{∗}ψ = φ, first note that all S-orbits have the same volume relative to the metric g; in fact, they are totally geodesic. Moreover, since φ and ψ are S-invariant, they are constant on each orbit. Since the isometry F preserves volumes, it must intertwine the values of φ and ψ on each orbit. Hence ψ ∘ F = φ. Consequently, F is an isometry of (M, g). □

5.1.10 Lemma. Let 5.1.2(a,b) be satisfied. Denote by C the connected component of the identity in {F ∈ Isom(G, g_{λ}) | F^{∗}φ = φ}. Then C = Q_{L} × T_{R} in case G = SO(2m + 2r), and C = P_{L} × T_{R} in case G = SU(2m + r + 1), where Q and P are as in Notation 5.1.5.

Proof. If G = SO(2m + 2r) then by Lemmas 5.1.4 and 5.1.6(ii) and the fact that φ and g_{λ} are invariant under Q_{L} × T_{R}, we have Q_{L} × T_{R} ⊂ C ⊂ Q_{L} × Q_{R}. If G = SU(2m + r + 1) then the same lemmas and the fact that φ is invariant under P_{L} × T_{R} imply P_{L} × T_{R} ⊂ C ⊂ Inn_{G}(H) × P_{L} × T_{R}. By Lemma 5.1.7 and the fact that the group of automorphisms of the torus T is discrete, the only oneparameter subgroups of Inn_{G}(Q) which lie in Isom(g_{λ}) (consequently also: the only such subgroups of Inn_{G}(H)) are those which restrict to the identity on 𝔷 and also on j(𝔷). Since the centralizer of j(𝔷) in 𝔥 is trivial by condition 5.1.2(b), and since the centralizer of 𝔷 in 𝔭 is 𝔷, these one-parameter subgroups must lie in Inn_{G}(T). The corollary follows. □

5.1.11 Lemma. Let U be a compact Lie group and T a torus in U of dimension greater than one. Suppose that every subtorus of T of positive dimension has the same centralizer in U. Then T lies in the center of U (and thus its centralizer is U).

Proof. Let 𝔲 denote the Lie algebra of U and 𝔱 ⊂ 𝔲 the Lie algebra of T. Consider the simultaneous root space decomposition of the complexified space 𝔲^{ℂ} with respect to the operators ad(X), X ∈ 𝔱. If Y = Y_{1}+iY_{2} (with Y_{1}, Y_{2} ∈ 𝔲) is a root vector, say ad(X)(Y) = iα(X)Y for all X ∈ 𝔲, where α ∈ 𝔱^{∗}, then either α ≡ 0 or else ker(α) is a subspace 𝔰 of 𝔱 of codimension one. In the latter case the centralizer of the associated subtorus S of T contains Y and thus properly contains the centralizer of T, contradicting our hypothesis. Thus α ≡ 0 for every root α. Hence 𝔱 is central in 𝔲 and T is central in U. □

5.1.12 Proposition. If the genericity conditions 5.1.2(a,b) are satisfied, then the identity component of the center of Isom_{0}(G, φg_{λ}) is equal to T_{R}.

Proof. By Proposition 5.1.9 and Lemma 5.1.10, the centralizer in Isom_{0}(G, φg_{λ}) of any nontrivial subtorus of T_{R} is given in case G = SO(2m + 2r) by Q_{L} × T_{R} and in case G = SU(2m + r + 1) by P_{L} ×T_{R}. Thus by Lemma 5.1.11, Isom_{0}(G, g_{λ}) equals Q_{L} × T_{R}, respectively P_{L} × T_{R}. Since in either case, T_{R} is the identity component of the center of this group, the proposition follows. □

Of course, the proposition is also valid with φ replaced by ψ.

5.1.13 Theorem. Let φ be the function given in 5.1.1 and let ψ = τ^{∗}φ. Assume that j_{1} is not equivalent to j_{2} and that j satisfies the genericity conditions 5.1.2(a,b). Then the metrics φg_{λ} and ψg_{λ} on G are not isometric.

Proof. Suppose that F : (G, φg_{λ}) → (G, ψg_{λ}) were an isometry. By Proposition 5.1.12, the resulting isomorphism from Isom(G, φg_{λ}) to Isom(G, ψg_{λ}) given by μ ↦ F ∘ μ ∘ F^{−1} carries T_{R} to T_{R}. Hence F carries T_{R}-orbits in G to T_{R}-orbits. By Proposition 5.1.9, F is an isometry of g_{λ} and satisfies F^{∗}ψ = φ, contradicting Theorem 5.1.8. □

5.2 Nonisometry proof for the examples from 4.4/4.6. We prove here the nonisometry of the conformally equivalent metrics on both the balls and the spheres in Examples 4.4/4.6. As an aside, we observe that the nonisometry of the metrics on the balls would follow immediately from that on the boundary spheres. However, we will not use this observation below but instead prove the nonisometry of the metrics on the balls directly, so that we will be able later (in 5.4) to adapt the proof to address the nonisometry of the metrics in Remark 4.11, where the boundary spheres may in fact be isometric.

In the context of Example 4.4 (resp. 4.6), suppose that the linear maps j, j′ : 𝔷 = ℝ^{2} → 𝔰𝔬(m) (resp. c, c′ : ℝ^{2} → Sym_{0}(ℝ^{3})) satisfy the conditions from Remark 4.5 (resp. those specified in 4.6); let ν, ν′ be the associated ℝ^{2}-valued 1-forms on ℝ^{m} (where m = 3 in the case of 4.6). Furthermore, assume that the positive smooth radial function ψ on ℝ^{m} satisfies the condition from 4.5 (resp. 4.6); writing ψ(p) = ρ(|p|), this means that ρ_{|_{[0,1]}} attains its maximum precisely in 0.

For M = B ⊂ ℝ^{m} ⊕ ℝ^{m} ⊕ ℂ^{2}, or M = S ⊂ ℝ^{m} ⊕ ℝ^{m} ⊕ ℂ^{2}, we are to show that the metrics φ_{1}g_{λ} and φ_{2}g_{λ} on M are nonisometric, where λ is the ℝ^{2}-valued 1-form associated with ν ⊕ ν′ on ℝ^{m} ⊕ ℝ^{m} as in 4.1(ii), and where the φ_{i} (i = 1, 2) are defined as in Theorem 4.3. By the proof of Theorem 4.3, the metric φ_{2}g_{λ} is isometric to φ_{1}g_{λ′}, where λ′ is the ℝ^{2}-valued 1-form associated with ν′ ⊕ ν. Following the strategy described in Proposition 2.2 and Remark 2.3, it thus suffices to show that the 1-forms λ and λ′ satisfy the conditions (N) and (G) in Proposition 2.2 with respect to the metric φg_{0} on M, where φ := φ_{1} . Recall that φ_{1}(p, q, u) = ψ(p) = ρ(|p|) ∈ ℝ_{+} .

We canonically identify the set M̂/T (see Notation 1.2) with

B^{2m+2} ∩ Q = {(p, q, a, b) ∈ ℝ^{m} ⊕ ℝ^{m} ⊕ ℝ ⊕ ℝ | a, b > 0, |p|^{2} + |q|^{2} + a^{2} + b^{2} < 1}

in case M = B, resp. with

S^{2m+1} ∩ Q = {(p, q, a, b) ∈ ℝ^{m} ⊕ ℝ^{m} ⊕ ℝ ⊕ ℝ | a, b > 0, |p|^{2} + |q|^{2} + a^{2} + b^{2} = 1}

in case M = S, where Q denotes the open “quadrant” of ℝ^{2m+2} defined by a, b > 0. The metric φg_{0}^{T} on M̂/T is in each case given as φ̄g_{0} , where φ̄(p, q, a, b) := ρ(|p|), and where g_{0} now denotes the standard metric on ℝ^{2m+2} as well.

Let F̄ ∈ \overline{Aut}_{φg_{0}}^{T}(M).

If M = B then F̄, being an isometry of φ̄g_{0}, extends to a continuous self-map, again denoted F̄, of the closure of B^{2m+2} ∩ Q. Note that F̄ must preserve the 2m-dimensional “edge” B^{2m} × {0} ⊂ ℝ^{2m} ⊕ ℝ^{2} because this is the image under the projection M → M/T of those points whose T -orbit is just a point. Hence F̄ must also preserve the opposite “face” S^{2m+1} ∩ Q. So F̄ preserves S^{2m+1} ∩ Q regardless of whether M = B or M = S.

Note that the T -orbit of a point (p, q, v, w) ∈ M with |v| = a and |w| = b, endowed with the metric induced by φg_{0} , is a rectangular torus with side lengths \sqrt{ρ(|p|)}·2πa and \sqrt{ρ(|p|)}·2πb. Considering the ratio of the side lengths, we see that F̄ must either preserve or switch the functions (p, q, a, b) ↦ a/b and (p, q, a, b) ↦ b/a on M̂/T and thus on S^{2m+1} ∩ Q (by continuity in case M = B).

The length of the diagonal of the rectangular torus which is the T -orbit of a point (p, q, a, b) ∈ M̂/T is equal to ℓ(p, q, a, b), where ℓ : ℝ^{2m+2} ∋ (p, q, a, b) ↦ \sqrt{ρ(|p|)}·2π\sqrt{a^{2} + b^{2}} ∈ ℝ. Consequently F̄ must preserve ℓ, too. By our assumption on ψ (hence on ρ), the restriction of ℓ to S^{2m+1} ∩ Q attains its maximum precisely in the points (0, 0, a, b); hence the set of these points, too, is invariant under F̄. Summarizing, for each vector (a, b) ∈ S^{1} with a, b > 0, the map F̄ either preserves or switches the two 2m-dimensional hemispheres S_{(a,b)} and S_{(b,a)} where S_{(a,b)} := S^{2m+1} ∩ {(p, q, ta, tb) | p, q ∈ ℝ^{m}, t ∈ ℝ_{+}}, and accordingly F̄ fixes, resp. switches, their poles (0, 0, a, b) and (0, 0, b, a).

Recall that F̄ is, at the same time, an isometry of φ̄g_{0} . In particular, it is a conformal self-map of the open domain B^{2m+2} ∩ Q ⊂ ℝ^{2m+2} (in case M = B), resp. of the open domain S^{2m+1} ∩ Q ⊂ S^{2m+1} (in case M = S). By Liouville’s theorem (see, e.g., [Sp], pp. 302–313), and since the dimensions of these domains are certainly greater than two, it follows that F̄ is the restriction to M̂/T of a Möbius transformation Φ of \overline{ℝ^{2m+2}} := ℝ^{2m+2} ∪ \{∞\}; that is, a composition of reflections in hyperspheres and hyperplanes. (To see this for S^{2m+1} ∩ Q, one has to conjugate F̄ by the stereographical projection, which is itself the restriction of a Möbius transformation, and to use the fact that each Möbius transformation of \overline{ℝ^{2m+1}} extends to one of \overline{ℝ^{2m+2}}.)

From our previous arguments we now conclude that Φ, combined with the reflection in the (a = b)-hyperplane if necessary, preserves each of the 2m-dimensional hemispheres S_{(a,b)} considered above and, moreover, fixes their poles (0, 0, a, b). By well-known properties of Möbius transformations, this implies that on each of these hemispheres, Φ is uniquely determined by its restriction to the boundary (2m − 1)-sphere, on which it acts as an orthogonal transformation. Since all the hemispheres S_{(a,b)} have the same boundary, namely, S^{2m−1} × {0} ⊂ ℝ^{2m} ⊕ ℝ^{2}, it

follows that Φ|_{S^{2m+1}} is of the form (U, σ) with U ∈ O(2m) and σ ∈ \{Id, ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )\} ⊂ O(2).

Recall once more that F̄ — and hence Φ — has to preserve the function ℓ : (p, q, a, b) ↦ \sqrt{ρ(|p|)}·2π\sqrt{a^{2} + b^{2}} on S^{2m+1}∩Q. Since our assumption on ρ implies, in particular, that ρ is nonconstant on [0, 1], it follows that U is of the form (A, A′) ∈ O(m) × O(m). Thus Φ|_{S^{2m+1}} = (A, A′, σ). If M = B then Φ must moreover preserve B^{2m+2} because it extends F̄. Summarizing, we have shown so far:

(1)

Each F̄ ∈ \overline{Aut}_{φg_{0}}^{T}(M) is the restriction to M̂/T ⊂ ℝ^{2m+2} of
some map (A, A′, σ) ∈ O(m) × O(m) × O(2).

Moreover, another look at the isometry classes of the T -orbits in (M, φg_{0}) shows that each Ψ ∈ D (see 2.1(ii)) will preserve the set {±Z_{1} , ±Z_{2}} ⊂ 𝔷 = ℝ^{2}; in particular, Ψ ∈ O(2).

Proof of condition (N).

Suppose condition (N) were violated. Let Ψ ∈ D and F̄ ∈ \overline{Aut}_{φg_{0}}^{T}(M) be such that Ω_{λ} = Ψ ∘ F̄^{∗}Ω_{λ′} . Note that we have dω_{0} = 0 here and hence Ω_{0} = 0; thus Ω_{λ} = dλ̄, Ω_{λ′} = dλ̄′ with λ̄, λ̄′ as in 2.1(iv). Let pr : ℝ^{m} ⊕ ℝ^{m} ⊕ ℝ^{2} → ℝ^{m} ⊕ ℝ^{m} denote the canonical projection and also its restriction to M̂/T = B^{2m+2} ∩ Q, resp. M̂/T = S^{2m+1} ∩ Q. Then

dλ̄ = pr^{∗}(dν ⊕ dν′), dλ̄′ = pr^{∗}(dν′ ⊕ dν).

For X, Y ∈ T_{p}ℝ^{m} and Z ∈ 𝔷, we have

⟨dν^{(′)}(X, Y), Z⟩ = 2⟨j^{(′)}(Z)X, Y ⟩ (in 4.4), resp.
⟨dν^{(′)}(X, Y), Z⟩ = 3⟨c^{(′)}(Z)p × X, Y ⟩, (in 4.6)

(compare the proof of [Sch3], 4.3, for the second equation). By (1), F̄ is the restriction to M̂/T of some map (A, A′, σ) ∈ O(m) × O(m) × O(2). Moreover, the interior of pr(M̂/T) ∈ ℝ^{2m} is the whole open unit ball B^{2m}, regardless of whether M = B or M = S. Therefore, one elementarily derives from the equation dλ̄ = Ψ ∘ F̄^{∗}dλ̄′ and the formulas above:

(2)

j′(^{T} Ψ(Z)) = Aj(Z)A^{−1}, j(^{T} Ψ(Z)) = A′j′(Z)A^{′ −1}, resp.
c′(det(A) ·^{T} Ψ(Z)) = Ac(Z)A^{−1}, c(det(A′) ·^{T} Ψ(Z)) = A′c′(Z)A^{′ −1}

for all Z ∈ 𝔷, where ^{T} Ψ denotes the transpose of Ψ with respect to the standard inner product on 𝔷 = ℝ^{2}. Since Ψ ∈ O(𝔷) (see above) and det(A), det(A′) ∈ {±1}, the equations in (2) contradict the nonequivalence assumption on j, j′ from 4.5, resp. on c, c′ from 4.6.

Proof of condition (G).

Suppose to the contrary that λ′ did not have Property (G); i.e., there is a nontrivial 1-parameter family F̄_{t} ∈ \overline{Aut}_{φg_{0}}^{T}(M) such that F̄_{t}^{∗}Ω_{λ′} ≡ Ω_{λ′} . Proceeding exactly as in the proof of condition (N), replacing Ψ by Id and λ by λ′, we now obtain 1-parameter families A_{t} , A′_{t} ∈ SO(m) such that

(3)  \begin{matrix}j′(Z) ≡ A_{t}j′(Z)A_{t}^{−1}, & j(Z) ≡ A′_{t}j(Z)A_{t}^{′ −1}, resp. \\ c′(Z) ≡ A_{t}c′(Z)A_{t}^{−1}, & c(Z) ≡ A′_{t}c(Z)A_{t}^{′ −1} \end{matrix}

for all Z ∈ 𝔷 and t ∈ ℝ (compare (2)). The families A_{t} , A′_{t} are those occurring in F̄_{t} = (A_{t} , A′_{t} , Id) ∈ SO(m)×SO(m)×SO(2). Note that the SO(2)-component of F̄_{t} is trivial because F̄_{t} must preserve the open quadrant Q of ℝ^{2m+2} considered earlier. Since the family F̄_{t} is assumed to be nontrivial, at least one of the 1-parameter families A_{t} , A′_{t} in SO(m) is nontrivial. The corresponding equations in (3) thus contradict the genericity assumptions from 4.5 on j and j′, resp. from 4.6 on c and c′. □

5.3 Nonisometry proof for Example 4.10. The proof will be similar to that of [Sch3], Proposition 4.3. Let c, c′ : 𝔷 = ℝ^{2} → Sym_{0}(ℝ^{3}) be the pair of linear maps from Example 4.10, and let ν, ν′ be the associated 𝔷-valued 1-forms on ℂ^{2}. We are going to show that the 𝔷-valued 1-forms λ, λ′ on M = B^{8} ⊂ ℂ^{2} ⊕ ℂ^{2} or M = S^{7} ⊂ ℂ^{2} ⊕ ℂ^{2}, induced by ν ⊕ ν′, resp. ν′ ⊕ ν, satisfy conditions (N) and (G) from Proposition 2.2 with respect to the metric φg_{0} on M, where φ := φ_{1} : ℂ^{2} ⊕ ℂ^{2} ∋ (p, q) ↦ ψ(p) ∈ ℝ_{+} . Since the metrics φ_{1}g_{λ′} and φ_{2}g_{λ} on M are isometric, as seen in the proof of Theorem 4.9, we may then conclude from Proposition 2.2 that the conformally equivalent metrics φ_{1}g_{λ} and φ_{2}g_{λ} are not isometric.

For a, b ≥ 0 let M_{a,b} := {(p, q) ∈ ℂ^{2} ⊕ ℂ^{2} | |p|^{2} = a^{2}, |q|^{2} = b^{2}} = S_{a}^{3} × S_{b}^{3}. Then M̂ is the union of those M_{a,b} ⊂ M with a, b > 0 and a^{2} + b^{2} < 1 (in case M = B^{8}), resp. a^{2} + b^{2} = 1 (in case M = S^{7}).

Write ψ(p) = ρ(|p|). The T -orbit of (p, q) ∈ M_{a,b}, endowed with the metric induced by φg_{0} , is isometric to the rectangular torus with sides of length \sqrt{ρ(a)}·2πa and \sqrt{ρ(a)}·2πb. Since ρ is strictly increasing by assumption, so is the function a ↦ \sqrt{ρ(a)} · a; note that this function maps (0, ∞) bijectively to itself. Consequently, for every (a, b) ∈ (0, ∞)^{2} there is exactly one (a′, b′) ∈ (0, ∞)^{2} such that

\sqrt{ρ(a′)} · a′ = \sqrt{ρ(a)} · b and \sqrt{ρ(a′)} · b′ = \sqrt{ρ(a)} · a

(the first condition determines a′, the second then determines b′). The map σ : (a, b) ↦ (a′, b′) on (0, ∞)^{2} is continuous and obviously has the following property: Whenever a > b, then a′ < a (since a ↦ \sqrt{ρ(a)} · a is strictly increasing); hence \sqrt{ρ(a′)} < \sqrt{ρ(a)} and thus ∥(a′, b′)∥^{2} > ∥(a, b)∥^{2}. In particular, σ(a, b) lies outside the closed unit disc for all (a, b) ∈ C := {(a, b) ∈ S^{1} | a > b > 0}, and the same is true for some open neighborhood A ⊂ {(a, b) | a > b > 0} of this arc C of S^{1}. Choose (a, b) ∈ C (in case M = S^{7}), resp. (a, b) ∈ U := A∩B_{1}(0) (in case M = B^{8}). Then M_{a,b} is precisely the set of points in (M̂, φg_{0}) whose T -orbit is isometric to the rectangular torus with sides of length \sqrt{ρ(a)} · 2πa and \sqrt{ρ(a)} · 2πb.

Each F ∈ Aut_{φg_{0}}^{T}(M) (see Definition 2.1) must therefore preserve M_{a,b} =: L. Note that (L/T, φg_{0}^{T}) is isometric to (S_{a/2}^{2} × S_{b/2}^{2} , ρ(a)^{2}g_{stdd}), where g_{stdd} denotes the restriction of the standard metric on ℝ^{3} ⊕ ℝ^{3}. Since a > b, every isometry of (L/T, φg_{0}^{T}) — and thus the restriction to L/T of every F̄ ∈ \overline{Aut}_{φg_{0}}^{T}(M) — must preserve the two factors. In particular, every F ∈ Aut_{φg_{0}}^{T}(M) must preserve the factors of M_{a,b}; we conclude from this that every Ψ ∈ D is of the form Z_{1} ↦ ε_{1}Z_{1}, Z_{2} ↦ ε_{2}Z_{2} with ε_{1} , ε_{2} ∈ {±1}; in particular, Ψ ∈ O(𝔷).

Proof of condition (N).

Suppose condition (N) were violated. Let Ψ ∈ D and F̄ ∈ \overline{Aut}_{φg_{0}}^{T}(M) be such that Ω_{λ} = Ψ ∘ F̄^{∗}Ω_{λ′} . Let L = M_{a,b} ⊂ M̂ be chosen as above. By the arguments above, F̄ induces an isometry of (L/T, φg_{0}^{T}). Denote by Ω_{λ}^{L}, Ω_{λ′}^{L} the ℝ^{2}-valued 2-forms induced by Ω_{λ}, Ω_{λ′} on L/T ⊂ M̂/T; then Ω_{λ}^{L} = Ψ ∘ F̄^{∗}Ω_{λ′}^{L} because F̄ preserves L/T.

Observe that the first, resp. second, component of Ω_{0}^{L} is a scalar multiple of the standard volume form of the first, resp. second, factor of L/T (compare [Sch3], 4.1(iii)). Recall from 2.1(iv) that Ω_{λ} = Ω_{0}+dλ̄, Ω_{λ′} = Ω_{0}+dλ̄′, and similarly for the forms induced on L/T. Note that each of Ω_{0}^{L}, dλ̄^{L} and dλ̄^{′ L} is a sum of pullbacks of forms defined on one of the factors of L/T. Since each 2-form on S^{2} is uniquely decomposable into an exact component and a multiple of the volume form, and since F̄ preserves the two factors of L/T, we therefore obtain dλ̄^{L} = Ψ ∘ F̄^{∗}dλ̄^{′ L}.

The restriction of F̄ to L/T is given as (A, A′) with A, A′ ∈ O(3). Evaluating the previous equation on 2-vectors tangent to the first factor of L/T, we get dν̄ = Ψ ∘ A^{∗}dν̄′, where

⟨dν̄^{(′)}(X, Y), Z⟩ = 3a^{3}⟨c^{(′)}(Z)x × X, Y ⟩

for all X, Y ∈ T_{x}S_{a/2}^{2} and Z ∈ 𝔷; compare the proof of [Sch3], 4.3. As in [Sch3] one now derives

(4)  c′(Φ(Z)) = Ac(Z)A^{−1}

for all Z ∈ 𝔷, with Φ := det A · Ψ ∈ O(𝔷). But this contradicts the nonequivalence of c and c′ (see condition 2.) in 4.6).

Proof of condition (G).

Suppose to the contrary that λ′ did not have Property (G); i.e., there is a nontrivial 1-parameter family F̄_{t} ∈ \overline{Aut}_{φg_{0}}^{T}(M) such that F̄_{t}^{∗}Ω_{λ′} ≡ Ω_{λ′} . Note that F̄_{t}, being an isometry of M̂/T, is determined by its restriction to any nonempty open subset. The submanifolds L/T = M_{a,b}/T do foliate an open subset of M̂/T if we vary the tuple (a, b) in the set from which we had chosen it. It follows that F̄_{t} induces a nontrivial 1-parameter family of isometries on at least one of these M_{a,b}/T; again, we denote the corresponding M_{a,b} by L.

Proceeding exactly as in the proof of condition (N), replacing Ψ by Id and λ by λ′, we now obtain 1-parameter families A_{t} , A′_{t} in SO(3), such that dλ̄^{′ L} ≡ (A_{t} , A′_{t})^{∗}dλ̄^{′ L} on L/T. Evaluating this on the first, resp. second, factor of L/T, we see that dν̄′ is invariant under the A_{t}, resp. that dν̄ is invariant under the A′_{t}. We derive that

c′(Z) ≡ A_{t}c′(Z)A_{t}^{−1} and c(Z) ≡ A′_{t}c(Z)A_{t}^{′ −1}

for all Z ∈ 𝔷 (compare (4)). However, since at least one of the families A_{t} , A′_{t} is nontrivial, this contradicts the genericity assumption on c and c′ (see condition 3.) in 4.6). □

5.4 Nonisometry proof for the modified metrics from Remark 4.11.

The case of Example 4.4/4.6.

We have to suitably adapt the proof given in 5.2. Note that λ, λ′ did not play any role in the derivation of statement (1) of 5.2. In order to prove the conditions (N) and (G) for fλ, fλ′, we proceed as follows:

Choose any tuple (r, s, t) such that α(r, s, t) ≠ 0; we can assume r, s, t ≠ 0. Let L := {(p, q, u) ∈ M̂ | |p| = r, |q| = s, |u| = t}. Then f has the constant value C := α(r, s, t) ≠ 0 on L. For all differential forms occurring in the proof of condition (N) in 5.2, we now restrict attention to the forms they induce on L, resp. on L/T. These induced forms are, up to multiplication by the constant C, the same as those induced by fλ, fλ′ etc. The only problem in the course of the arguments from 5.2 is the fact that pr(L/T) = S_{r}^{m−1} × S_{s}^{m−1} has empty interior in ℝ^{2m}. But the equations in (2) can be derived nevertheless, even when using the equation dν ⊕ dν′ = Ψ ∘ (A, A′)^{∗}(dν′ ⊕ dν) only restricted to the tangent bundle of S_{r}^{m−1} × S_{s}^{m−1}:

In the case of Example 4.4, this is obvious from the formula for dν^{(′)}, because for all X, Y ∈ ℝ^{m} there is certainly some p ∈ S_{r}^{m−1} such that X, Y ∈ T_{p}S_{r}^{m−1} (and similarly for S_{s}^{m−1}). In the case of Example 4.6, we first obtain only

⟨(c(Z) − A^{−1}c′(det(A) ·^{T} Ψ(Z))A)p × X, Y ⟩ = 0

for all Z ∈ 𝔷 and all X, Y ∈ T_{p}S_{r}^{2} (and similarly for c, c′ reversed, with A′ instead of A). But this implies (c(Z) − A^{−1}c′(det(A) ·^{T} Ψ(Z))A)p ⊥ p for all p ∈ S_{r}^{2} . Since on the left hand side a symmetric map is applied to p, this map must be zero. Thus the equations in (2) follow nevertheless.

For both examples, the proof of condition (G) now goes through verbatim.

The case of Example 4.10.

In case M = B^{8}, we make the assumption that the support of α nontrivially intersects U, where U ⊂ B_{1}(0) ∩ {(a, b) | a > b > 0} is the nonempty open set occurring in the proof given in 5.3. In case M = S^{7}, since α does not identically vanish on the relevant parameter set {(a, b) ∈ S^{1} | a, b ≥ 0}, and because of α(a, b) = α(b, a), we know that α does not identically vanish on the arc C = {(a, b) ∈ S^{1} | a > b > 0} either.

Thus we can choose (a, b) ∈ U, resp. (a, b) ∈ C, such that C := α(a, b) ≠ 0. In 5.3 we saw that every F̄ ∈ \overline{Aut}_{φg_{0}}^{T}(M) must then preserve L := M_{a,b} = S_{a}^{3} × S_{b}^{3} ⊂ M̂. Moreover, f_{|_{L}} ≡ C. In order to prove the conditions (N) and (G) for fλ, fλ′, we now work with Cλ^{L}, Cλ^{′ L} instead of λ^{L}, λ^{′ L} in 5.3, which does not affect any of the arguments. □

§6 Other types of examples

The techniques used in Sections 3 and 4 apply to the construction of isospectral conformally equivalent metrics and isospectral potentials in various settings. We first observe the common properties of the constructions, particularly the constructions of Section 3 and of Example 4.4. For simplicity, we will consider here linear maps j from a vector space 𝔷 into 𝔰𝔬(m) rather than into a more general compact Lie algebra 𝔥.

(i) Linear maps j : 𝔷 → 𝔰𝔬(m) where 𝔷 is a k-dimensional vector space, viewed as the Lie algebra of a torus T, were used to construct Riemannian metrics g_{j} on a fixed manifold M = M_{k,m}; the manifold M depended only on k and m. (The metrics were denoted by g_{λ^{j}} rather than g_{j} . In Section 3, M was a Lie group, while in Section 4, M was either a ball or sphere.) The torus T acted by isometries on each (M, g_{j}).

(ii) In each setting, we knew from previous work that g_{j} and g_{j′} were isospectral whenever the linear maps j and j′ were isospectral in the sense of Definition 3.1. Moreover, the isospectrality of the metrics g_{j} and g_{j′} had been proven using Theorem 1.3 with φ = ψ = 0. Subtori W ⊂ T of codimension one correspond to linear functionals μ ∈ L^{∗} or, equivalently (given an inner product on 𝔷) to elements Z of 𝔷. The diffeomorphisms F_{W} of Theorem 1.3 (referred to in Sections 3 and 4 as F_{μ}) arose from the linear maps A_{Z} in Defintion 3.1.

(iii) Given isospectral linear maps j_{1}, j_{2} : 𝔷 → 𝔰𝔬(m), we defined isospectral maps j = (j_{1}, j_{2}) and j′ = (j_{2}, j_{1}) : 𝔷 → 𝔰𝔬(2m). The associated metrics g_{j} and g_{j′} on M = M_{k,2m} were isometric. Thus we had diffeomorphisms F_{W} satisfying (ii), arising from the isospectrality of j and j′, and separately an isometry τ between g_{j} and g_{j′}. We then chose a conformal factor or potential φ invariant under all the F_{W} but not under τ. The functions φ and τ^{∗}φ then gave isospectral potentials or conformal factors on (M, g_{j}).

We now briefly describe how this procedure can be used to obtain isospectral potentials and conformal factors in other settings.

6.1 Example. Let T be a torus with Lie algebra 𝔷. Let N = N_{m} denote either the standard unit ball or the standard unit sphere in ℝ^{m}. Given j : 𝔷 → 𝔰𝔬(m), define a 𝔷-valued 1-form λ = λ_{j} on ℝ^{m} by λ_{p}(X) = ⟨j(p), X⟩, where ⟨ , ⟩ is the standard inner product on ℝ^{m}, and where T_{p}ℝ^{m} is identified with ℝ^{m}. Then λ pulls back, first by the inclusion, to a 1-form on N and then, by the canonical projection, to an admissible form on N × T. Let g_{j} denote the associated Riemannian metric on N × T as in Notation 1.4. As seen in [GW2] and [GGSWW], isospectral linear maps j, j′ : 𝔷 → 𝔰𝔬(m) result in isospectral metrics g_{j} and g_{j′} ; in the notation of (ii) above, the role of the diffeomorphisms F_{W} is played by the maps A_{Z} × Id : N × T → N × T. In the notation of (iii), the role of τ is played by τ_{0} × Id where τ_{0} is the diffeomorphism of the ball or sphere N_{2m} obtained by restriction of the map (x, y) ↦ (y, x) of ℝ^{2m}. Choose a function ψ on N which is invariant under the action of O(m)×O(m) but which is not τ_{0}-invariant. Pull ψ back by the projection to N × T to obtain a function φ satisfying the conditions of (iii).

6.2 Example. Let T be a torus with Lie algebra 𝔷, and let Q be a domain with boundary in the hyperbolic space H^{m+1} invariant under the isometric action of O(m). With respect to the upper half space model of H^{m+1} with coordinates (a, x), a ∈ ℝ^{+}, x ∈ ℝ^{m}, the action of α ∈ O(m) on H^{m+1} is given by α(a, x) = (a, α(x)).

The article [GSz] constructed from each linear map j : 𝔷 → 𝔰𝔬(m) a Riemannian metric g_{j} on Q × T. After multiplying the linear map j by a small constant if necessary, the metric g_{j} was shown to have negative curvature. Isospectral linear maps resulted in isospectral metrics, with the role of the diffeomorphisms in (ii) being played by the restriction to Q × T of the maps A_{Z} × Id on H^{m+1} × T. The procedure outlined in (iii) results in isospectral potentials or isospectral conformally equivalent metrics on Q×T where now Q is an O(2m)-invariant domain in H^{2m+1}. The role of τ in (iii) is played by the restriction to Q×T of τ_{0}×Id, where τ_{0} ∈ O(2m) is given as in Example 6.1. To obtain a function φ as in (iii), begin with a function ψ on H^{2m+1} invariant under the action of O(m) × O(m) but not τ_{0}-invariant. The function φ can then be taken to be the pullback to Q × T of ψ_{|_{Q}}. As in (iii), φ and φ ∘ τ give isospectral potentials on the negatively curved manifold (Q × T, g_{j}). By choosing φ to be positive, one obtains isospectral conformally equivalent metrics on Q × T. By moreover choosing φ to be bounded by a sufficiently small constant, one can guarantee that the conformally equivalent metrics are negatively curved.

If we choose m = 4ℓ with ℓ ≥ 2, choose 𝔷 to be 3-dimensional and choose j_{1} , j_{2} : 𝔷 → 𝔰𝔬(m) as in Theorem 3.8(i) of [GSz], then the metric g_{j} on Q × T, where Q is a domain in H^{2m+1} and j = (j_{1} , j_{2}), has constant Ricci curvature; indeed (Q × T, g_{j}) is a domain in a harmonic manifold. We thus obtain isospectral potentials on an Einstein manifold with boundary.

References

[Ba] W. Ballmann, On the construction of isospectral manifolds, Preprint (2000).

[Be1] P. Bérard, Transplantation et isospectralité I, Math. Ann. 292 (1992), 547–559.

[Be2] , Transplantation et isospectralité II, J. London Math. Soc. 48 (1993), 565–576.

[Br] R. Brooks, On manifolds of negative curvature with isospectral potentials, Topology 26 (1987), 63–66.

[BG] R. Brooks and C.S. Gordon, Isospectral families of conformally equivalent Riemannian metrics, Bull. Amer. Math. Soc. 23 (1990), no. 3, 433–436.

[BGG] R. Brooks, R. Gornet, and W. H. Gustafson, Mutually isospectral Riemann surfaces, Adv. Math. 138 (1998), 306–322.

[BPY] R. Brooks, P. Perry, and P. Yang, Isospectral sets of conformally equivalent metrics, Duke Math J. 58 (1989), 131–150.

[BT] R. Brooks and R. Tse, Isospectral surfaces of small genus, Nagoya Math. J. 107 (1987), 13–24.

[Bu] P. Buser, Isospectral Riemann surfaces, Ann. Inst. Fourier 36 (1986), 167–192.

[DG] D. DeTurck and C. Gordon, Isospectral Deformations II: trace formulas, metrics, and potentials, Comm. Pure Appl. Math. 42 (1989), 1067–1095.

[GGKM] C.S. Gardner, J.M. Greene, M.D. Kruskal, and R.M. Miura, A method for solving the Korteveg de Vries equation, Phys. Rev. Letters 19 (1967), 1095-1097.

[Gi] P. Gilkey, Recursion relations and the asymptotic behavior of the eigenvalues of the Laplacian, Compositio Math. 38, 201-240.

[Go1] C.S. Gordon, Isospectral closed Riemannian manifolds which are not locally isometric: II, Geometry of the Spectrum (R. Brooks, C. Gordon, P. Perry, eds.), Contemp. Math. 173 (1994), AMS, 121–131.

[Go2] , Survey of isospectral manifolds, Handbook of Differential Geometry, F.J.E. Dillen and L.C.A. Verstraelen, eds., vol. 1, Elsevier, 2000, pp. 747–778.

[Go3] , Isospectral deformations of metrics on spheres, Invent. Math. 145 (2001), no. 2, 317–331.

[GGSWW] C. Gordon, R. Gornet, D. Schueth, D. Webb, E. Wilson, Isospectral deformations of closed Riemannian manifolds with different scalar curvature, Ann. Inst. Fourier 48 (1998), no. 2, 593–607.

[GSz] C.S. Gordon and Z.I. Szabo, Isospectral deformations of negatively curved Riemannian manifolds with boundary which are not locally isometric, Duke Math. J. 113 (2002), no. 2, 355–383.

[GWW] C. Gordon, D. Webb, and S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math. 110 (1992), 1–22.

[GW1] C. Gordon and E.N. Wilson, Isospectral deformations of compact solvmanifolds, J. Diff. Geom. 19 (1984), 241–256.

[GW2] , Continuous families of isospectral Riemannian metrics which are not locally isometric, J. Diff. Geom. 47 (1997), 504–529.

[Gt1] R. Gornet, A new construction of isospectral Riemannian nilmanifolds with examples, Michigan Math. J. 43 (1996), no. 1, 159–188.

[Gt2] , Continuous families of Riemannian manifolds isospectral on functions but not on 1-forms, J. Geom. Anal. 10 (2000), no. 2, 281–298.

[Ik1] A. Ikeda, Isospectral problem for spherical space forms, Spectra of Riemannian Manifolds (M. Berger, S. Murakami, and T. Ochiai, eds.), Kaigai Publications, 1983, pp. 57–63.

[Ik2] , Riemannian manifolds p-isospectral but not (p + 1)-isospectral, Geometry of Manifolds (Matsumoto), Perspect. Math., vol. 8, Academic Press, Boston, MA, 1989, pp. 383–417.

[MR1] R. Miatello and J.P. Rossetti, Flat manifolds isospectral on p-forms, J. Geom. Anal. 11 (2001), 649–667.

[MR2] , Comparison of twisted p-form spectra for flat manifolds with diagonal holonomy, Ann. Global Anal. Geom. 21 (2002), no. 4, 341–376.

[MR3] , Length spectra and p-spectra of compact flat manifolds, preprint (2001), math.DG/0110325.

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

[OT] T. Ochiai and T. Takahashi, The group of isometries of a left invariant Riemannian metric on a Lie group, Math. Ann. 223 (1976), 91–96.

[Pe1] H. Pesce, Représentations relativement équivalentes et variétés Riemanniennes isospectrales, C. R. Acad. Sci. Paris, Série I 3118 (1994), 657–659.

[Pe2] , Quelques applications de la théorie des représentations en géométrie spectrale, Rend. Mat., Série VII 18 (1998), 1–63.

[PT] J. Pöschel and E. Trubowitz, Inverse spectral theory, Pure and Applied Mathematics, vol. 130, Academic Press, 1987.

[Sch1] D. Schueth, Continuous families of isospectral metrics on simply connected manifolds, Ann. of Math. (2) 149 (1999), no. 1, 287–308.

[Sch2] , Isospectral manifolds with different local geometries, J. Reine Angew. Math. 534 (2001), 41–94.

[Sch3] , Isospectral metrics on five-dimensional spheres, J. Diff. Geometry 58 (2001), no. 1, 87–111.

[Sp] M. Spivak, Differential Geometry III, Publish or Perish, 1975.

[Su] T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. (2) 121 (1985), 169–186.

[St] C. Sutton, Isospectral simply-connected homogeneous spaces and the spectral rigidity of group actions, Comment. Math. Helv. 77 (2002), no. 4, 701–717.

[Sz1] Z.I. Szabo, Locally non-isometric yet super isospectral spaces, Geom. Funct. Anal. 9 (1999), no. 1, 185–214.

[Sz2] , Isospectral pairs of metrics constructed on balls, spheres, and other manifolds with different local geometries, Ann. of Math. (2) 154 (2001), no. 2, 437–475.

[Sz3] , Cornucopia of isospectral pairs of metrics on balls and spheres with different local geometries, Preprint (2000), math.DG/0011034.

[Ur] H. Urakawa, Bounded domains which are isospectral but not congruent, Ann. Scient. Éc. Norm. Sup. 15 (1982), 441–456.

[Vi] M.-F. Vignéras, Variétés Riemanniennes isospectrales et non isométriques, Ann. of Math. 112 (1980), 21–32.

Dartmouth College, Hanover, New Hampshire 03755, U.S.A.

   E-mail address: csgordon@dartmouth.edu

Mathematisches Institut, Universität Bonn, Beringstr. 1, D-53115 Bonn, Germany

   E-mail address: schueth@math.uni-bonn.de