ISOSPECTRAL METRICS ON FIVE-DIMENSIONAL SPHERES

Dorothee Schueth

Abstract. We construct isospectral pairs of Riemannian metrics on S^{5} and on B^{6}, thus lowering by three the minimal dimension of spheres and balls on which such metrics have been constructed previously (S^{n≥8} and B^{n≥9}). We also construct continuous families of isospectral Riemannian metrics on S^{7} and on B^{8}. In each of these examples, the metrics can be chosen equal to the standard metric outside certain subsets of arbitrarily small volume.

Introduction

During the past two decades, research on isospectral manifolds – that is, Riemannian manifolds sharing the same spectrum (including multiplicities) of the Laplace operator acting on functions – has been very active; see, for example, the survey article [Go2]. However, it was only recently that Zoltan I. Szabó and Carolyn Gordon independently discovered the first examples of isospectral metrics on spheres: Pairs of such metrics on S^{n≥10} [Sz2,3], and continuous families on S^{n≥8} [Go3].

In spite of the wealth of other isospectral manifolds obtained before, the construction of isospectral spheres had seemed beyond reach for a long time. The reason was that both of the main methods of construction which were known and used until 2000 had excluded spheres:

• The so-called Sunada method [Su] and its various generalizations (see, e.g., [DG]) produces isospectral quotients (M/Γ_{1} , g), (M/Γ_{2} , g) of a common Riemannian covering manifold (M, g); in particular, these isospectral manifolds were always nonsimply connected. Many interesting examples of locally isometric isospectral manifolds of the form (M/Γ_{1} , g), (M/Γ_{2} , g) – some of Sunada type, some using other special constructions – can be found, for example, in [Vi], [Ik], [GW1], [DG], [Gt1,2], [Sch1]; also the famous examples of isospectral plane domains [GWW] arise as orbifolds in the Sunada type setting.

• The method of principal torus bundles (see [Go1], [GW2], [GGSWW], [Sz1], [GSz], [Sch2,3]) produces certain pairs of isospectral principal bundles for which the structural group is a torus of dimension at least two; the metrics are invariant under the torus action. Since no sphere is a principal T^{k≥2}-bundle, spheres cannot be obtained by using this method either (although products of spheres could [Sch2]; these were the first examples of simply connected isospectral manifolds). See [Sch3] for a detailed treatment of the method of principal torus bundles, a systematical approach for applying it, and many examples (among others, isospectral left invariant metrics on compact Lie groups).

Key words and phrases. Laplace operator, spectrum, spheres, torus actions

2000 Mathematics Subject Classification. 58J53, 58J50.

The author is partially supported by SFB 256, Bonn.

The key to Gordon’s construction of isospectral metrics on the spheres S^{n≥8} (and on the balls B^{n≥9}) was a new approach which still involves T^{k≥2}-actions, but does not require them to be free anymore. On the other hand, Z.I. Szabó’s pairs of isospectral metrics on S^{n≥10} (and on B^{n≥11}) do not arise by such a construction and seem to be of a completely different type. A spectacular feature of his examples is that they include pairs of isospectral metrics on S^{11} in which one of the metrics is homogeneous while the other is not.

The present paper serves several purposes.

First, we reformulate Gordon’s new theorem [Go3, Theorem 1.2] in a somewhat more elegant way – see Theorem 1.4 below –, and in Theorem 1.6 we establish a special version of it which turns to be a useful tool for finding new applications. It also accounts for all applications of Theorem 1.4 which are known so far.

Second, we derive a general sufficient nonisometry condition for the type of Riemannian metrics occurring in Theorem 1.6; see Proposition 2.4 below. Although we will apply this criterion only to certain new examples constructed in this paper, we wish to point out that it could also be used to unify most of the various nonisometry proofs from [GW2], [GGSWW], [GSz], [Go3] (however, some of the locally homogeneous manifolds with boundary from [GW2] and [GSz] cannot be proven nonisometric using this approach because they violate a certain genericity condition; see (G) in Proposition 2.3). Similarly, it would be possible to apply our method to the isospectral examples from [Sch2,3] where we had instead performed explicit curvature computations in order to show that objects like ∫ scal^{2}, ∫ ∥R∥^{2}, the critical values of the scalar curvature, or the dimension of their loci are not spectrally determined.

Our third (and main) purpose is to use Theorem 1.6 for constructing isospectral pairs of metrics on S^{5} (and on B^{6}) and thus decreasing the minimal dimensions of Gordon’s examples by three; see Example 3.3. We also obtain continuous isospectral families of metrics on S^{7} (and on B^{8}); see Example 3.2. For the nonisometry proofs we use the general criterion mentioned above.

Fourth, we show that in these new examples – in particular, on S^{5} and B^{6} (pairs), and on S^{7} and B^{8} (continuous families) – it is possible to choose the isospectral metrics in such a way that they are equal to the round (resp. flat) metric outside certain subsets of arbitrarily small volume; see Theorm 5.3. This gives a nice contrast to the fact that in dimensions up to six, the round spheres themselves are completely determined by their spectra [Ta1] (the corresponding problem in higher dimensions has not yet been solved), and to the fact that no round sphere in any dimension admits a continuous isospectral deformation [Ta2].

The paper is organized as follows:

In Section 1 we present our reformulation (Theorem 1.4) of Gordon’s theorem and the afore-mentioned specialization (Theorem 1.6). Section 2 contains our general sufficient nonisometry criterion for the metrics occurring in Theorem 1.6. In Section 3 we construct our new examples: Continuous families of isospectral metrics on S^{7} and on B^{8} (Example 3.2), and pairs of such metrics on S^{5} and on B^{6} (Examle 3.3). Moreover, we give a survey of a number of related examples in 3.4; each of them can be obtained using Theorem 1.6. Section 4 gives the nonisometry proof for Examples 3.2 / 3.3. Finally, we show in Section 5 how to make the isospectral metrics round (resp. flat) on large subsets (Theorem 5.3).

The author would like to thank Carolyn Gordon and Werner Ballmann for interesting discussions concerning these and related topics.

§1 Isospectrality via effective torus actions

1.1 Definition. (i) The spectrum of a closed Riemannian manifold is the spectrum of eigenvalues, counted with multiplicities, of the associated Laplace operator acting on functions. The Dirichlet spectrum of a compact Riemannian manifold M 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.

(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.

1.2 Remark. Let (M, g) be a compact Riemannian manifold with or without

boundary. Consider the Hilbert spaces H^{N} := H^{1,2}(M, g) and H^{D} := H^{∘}^{1,2}(M, g) ⊆ H^{N}. Note that H^{N}, resp. H^{D}, is the completion of C^{∞}(M), resp. {f ∈ C^{∞}(M) | f_{|_{∂M}} = 0}, with respect to the H^{1,2}-norm associated to (M, g). For each f ∈ H^{N} \ {0} the Rayleigh quotient is defined as

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

Let 0 ≤ λ_{1}^{N} ≤ λ_{2}^{N} ≤ … → ∞, resp. 0 ≤ λ_{1}^{D} ≤ λ_{2}^{D} ≤ … → ∞, denote the Neumann spectrum, resp. the Dirichlet spectrum, of (M, g); if ∂M = ∅ then both of these sequences coincide with the spectrum of (M, g). Finally, denote by L_{k}^{N}, resp. L_{k}^{D}, the set of all k-dimensional subspaces of H^{N}, resp. H^{D}. Then we have the following variational characterization of eigenvalues (see, e.g., [Be]):

(1)  λ_{k}^{N} = inf_{U∈L_{k}^{N}} sup_{f∈U\{0}} R(f) and λ_{k}^{D} = inf_{U∈L_{k}^{D}} sup_{f∈U\{0}} R(f).

1.3 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^{T} we denote the unique Riemannian metric on the quotient manifold M̂/T such that the canonical projection π : (M̂, g) → (M̂/T, g^{T}) is a Riemannian submersion.

1.4 Theorem. Let T be a torus which acts effectively on two compact connected Riemannian manifolds (M, g) and (M′, g′) by isometries. For each subtorus W ⊂ T of codimension one, suppose that there exists a T -equivariant diffeomorphism F_{W} : M → M′ which satisfies F_{W}^{∗} dvol_{g′} = dvol_{g} and induces an isometry F̄_{W} between the quotient manifolds (M̂/W, g^{W}) and (M̂′/W, g^{′ W}). Then (M, g) and (M′, g′) are isospectral; if the manifolds have boundary then they are Dirichlet and Neumann isospectral.

Remark. Theorem 1.4 above is a slight variation of Carolyn Gordon’s Theorem 1.2 in [Go3]. Instead of our condition F_{W}^{∗} dvol_{g′} = dvol_{g}, Gordon assumes the condition that F̄_{W ∗} maps the projected mean curvature vector field H̄_{W} of the submersion (M̂, g) → (M̂/W, g^{W}) to the corresponding vector field H̄′_{W} . While the two conditions actually turn out to be equivalent in this context, our volume preserving condition is not only easier to formulate but also more convenient to check in applications. More than that, it is inherent and automatically satisfied in the specialization described below in 1.5 / 1.6 which covers all applications of the above theorem which are known so far. Our different formulation of the theorem has also led to a different proof (construction of a certain isometry between the H^{1,2}-spaces instead of intertwining the Laplacians). An advantage of our proof is that it does not require a certain additional condition which Gordon assumes in the case of manifolds with boundary (namely, that M̂ ∩ ∂M be dense in ∂M).

Proof of Theorem 1.4. Consider the Hilbert space H := H^{1,2}(M, g) in the case of manifolds without boundary or in case of Neumann boundary conditions, resp. H := H^{∘}^{1,2}(M, g) in the case of Dirichlet boundary conditions. Let H′ be defined analogously with respect to (M′, g′). We claim that there is a Hilbert space isometry from H to H′ which moreover preserves L^{2}-norms; the theorem will thus follow from the variational characterization of eigenvalues (1).

Consider the unitary representation of T on H defined by (zf)(x) = f(zx) for all f ∈ H, z ∈ T, x ∈ M. Write T = 𝔷/L and let L^{∗} be the dual lattice. Since T is abelian, H decomposes as the orthogonal sum ⨁_{μ∈L^{∗}} H_{μ} with H_{μ} = {f ∈ H | zf = e^{2πiμ(Z)}f for all z ∈ T }, where Z denotes any representative for z in 𝔷. In particular, this implies the coarser decomposition

(2)  H = H_{0} ⊕ ⨁_{W} (H_{W} ⊖ H_{0}),

where W runs though the set of all subtori of codimension 1 in T, and H_{W} is the sum of all H_{μ} such that μ ∈ L^{∗} and T_{e}W ⊆ ker μ. In other words, H_{W} is just the space of W -invariant functions in H. Let H′_{W} and H′_{0} be the analogously defined subspaces of H′. Now let W be any subtorus of codimension 1 in T and choose a diffeomorphism F_{W} as in the assumption. Since F_{W} intertwines the T -actions, F_{W}^{∗} sends H′_{W} to H_{W} and H′_{0} ⊂ H′_{W} to H_{0} ⊂ H_{W} . We will now show that F_{W}^{∗} : H′_{W} → H_{W} is a Hilbert space isometry which also preserves L^{2}-norms; in view of the decomposition (2) this will prove our above claim. Preservation of L^{2}-norms is trivial by the assumption F_{W}^{∗} dvol_{g′} = dvol_{g}. Moreover, this assumption implies that it suffices to show that for each ψ ∈ C^{∞}(M′) which is invariant under the W -action and for all y ∈ M′ we have ∥dψ_{|_{y}}∥_{g′} = ∥dφ_{|_{x}}∥_{g}, where φ := F_{W}^{∗} ψ and x := F_{W}^{−1}(y). We can assume x ∈ M̂; let φ̄ and ψ̄ be the functions induced on M̂/W and M̂′/W. Then φ̄ = F̄_{W}^{∗} ψ̄; since g^{W} and g^{′ W} are the submersion metrics and F̄_{W} is an isometry we obtain indeed, at the appropriate points: ∥dφ∥_{g} = ∥dφ̄∥_{g^{W}} = ∥dψ̄∥_{g^{′ W}} = ∥dψ∥_{g′} . □

1.5 Notation and Remarks. In the following we fix a torus T with Lie algebra 𝔷 = T_{e}T. 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 and each subspace 𝔴 of 𝔷 we let 𝔴_{x} := {Z_{x}^{∗} | Z ∈ 𝔴}.

(ii) We call a smooth 𝔷-valued 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)^{∗}).

In other words, g_{λ} = (Φ_{λ}^{−1})^{∗}g_{0} , where Φ_{λ} is the smooth endomorphism field on M given by X ↦ X −λ(X)^{∗} for all X ∈ T M. Note that Φ_{λ} is unipotent on each tangent space; in particular, dvol_{g_{λ}} = dvol_{g_{0}} .

(iv) Finally, note that g_{λ} is again invariant under the action of T, that g_{λ} restricts to the same metric as g_{0} on the vertical subspaces 𝔷_{x}, and that the submersion metric g_{λ}^{T} on M̂/T is equal to g_{0}^{T} .

1.6 Theorem. In the context of Notation 1.5, 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}) which

satisfies μ ∘ λ = F_{μ}^{∗}(μ ∘ λ′). Then (M, g_{λ}) and (M, g_{λ′}) are isospectral; if M has boundary then the two manifolds are Dirichlet and Neumann isospectral.

Proof. We show that (M, g_{λ}) and (M, g_{λ′}) satisfy the hypotheses of Theorem 1.4. Let W be a subtorus of codimension 1 in T. Choose μ ∈ L^{∗} such that 𝔴 := T_{e}W equals ker μ, and choose a corresponding F_{μ} as in (∗). We claim that F_{W} := F_{μ} satisfies the conditions required in Theorem 1.4. First of all note that F_{μ}, being an isometry of g_{0} , trivially satisfies F_{μ}^{∗}dvol_{g_{λ′}} = dvol_{g_{λ}} because of dvol_{g_{λ}} = dvol_{g_{0}} = dvol_{g_{λ′}} (see 1.5(iii)). Thus it remains to prove that F_{μ} induces an isometry from (M̂/W, g_{λ}^{W}) to (M̂/W, g_{λ′}^{W}).

Let V ∈ T_{x}M be any vector which is g_{λ}-orthogonal to 𝔴_{x}; then V = Φ_{λ}(X) for some X which is g_{0}-orthogonal to 𝔴_{x}. By condition (∗) we know that λ(X) equals λ′(F_{μ∗}X) modulo 𝔴. Keeping in mind that F_{μ} commutes with the T -action, we conclude that the vector F_{μ∗}V = F_{μ∗}(Φ_{λ}X) equals Y := Φ_{λ′}(F_{μ∗}X) up to an error in 𝔴_{F_{μ}(x)} . But F_{μ∗}X is g_{0}-orthogonal to 𝔴_{F_{μ}(x)} ; thus Y is the projection of F_{μ∗}V to the g_{λ′}-orthogonal complement of 𝔴_{F_{μ}(x)} . Our assertion now follows from ∥Y ∥_{g_{λ′}} = ∥F_{μ∗}X∥_{g_{0}} = ∥X∥_{g_{0}} = ∥V ∥_{g_{λ}} . □

§2 A sufficient condition for nonisometry

Throughout this section we let (M, g_{0}), T, 𝔷 be as in Notation 1.5, and we define the principal T -bundle π : M̂ → M̂/T as in Notation 1.3. By λ, λ′ we will always denote admissible 𝔷-valued 1-forms on M. In Proposition 2.4 below we will establish a sufficient condition for (M, g_{λ}), (M, g_{λ′}) to be nonisometric.

2.1 Notation and Remarks.

(i) 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 = T_{e}T induced by conjugation by F. 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 1.5(i).

(ii) 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 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}).

(iii) We define D := {Ψ_{F} | F ∈ Aut_{g_{0}}^{T}(M)} ⊂ Aut(𝔷). 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).

(iv) 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. Then the connection form on M̂ associated with g_{λ} is obviously given by ω_{λ} := ω_{0} + λ.

(v) 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.

(vi) Since λ is T -invariant and horizontal it induces some 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 Lemma. Suppose that F : (M, g_{λ}) → (M, g_{λ′}) is a T -preserving isometry.

(i) F preserves the g_{0}-norm of vectors tangent to the T -orbits, and it induces an isometry F̄ of (M̂/T, g_{0}^{T}). In particular, F ∈ Aut_{g_{0}}^{T}(M) and Ψ_{F} ∈ D.

(ii) F^{∗}ω_{λ′} = Ψ_{F} ∘ ω_{λ}; in particular, F^{∗}dω_{λ′} = Ψ_{F} ∘ dω_{λ}.

(iii) The isometry F̄ of (M̂/T, g_{0}^{T}) satisfies

(3)  F̄^{∗}Ω_{λ′} = Ψ_{F} ∘ Ω_{λ} .

Proof. (i) This follows immediately from 1.5(iv).

(ii) We have ω_{λ′}(Z^{∗}) = ω_{λ}(Z^{∗}) = Z for all Z ∈ 𝔷, hence ω_{λ′}(F_{∗}(Z^{∗})) = ω_{λ′}(Ψ_{F} (Z)^{∗}) = Ψ_{F} (Z) = Ψ_{F} (ω_{λ}(Z^{∗})). The equation thus holds when applied to vectors tangent to the T -orbits. Since F is an isometry and maps orbits to orbits, it must map g_{λ}-horizontal vectors to g_{λ′}-horizontal vectors. Hence both sides of the asserted equation vanish when applied to a g_{λ}-horizontal vector.

(iii) This follows from (ii) and 2.1(v). □

2.3 Proposition. Let λ be an admissible 𝔷-valued 1-form on M such that the associated curvature form Ω_{λ} on M̂/T satisfies the following genericity condition:

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

Then T is a maximal torus in Isom(M, g_{λ}).

Proof. Let F_{t} ∈ Isom(M, g_{λ}) be a 1-parameter family of isometries commuting with T. In particular, the maps F_{t} are T -preserving and thus induce a 1-parameter family F̄_{t} ∈ Isom(M̂/T, g_{0}^{T}). The corresponding Ψ_{F_{t}} ∈ D satisfy Ψ_{F_{t}} ≡ Id because D is discrete and Ψ_{F_{0}} = Id. Thus according to (3), each F̄_{t} preserves Ω_{λ}. The assumed property (G) of Ω_{λ} now implies F̄_{t} ≡ Id. We conclude that F_{t} restricts to a gauge transformation of the principal T -bundle M̂. On the other hand, F_{t}^{∗}ω_{λ} ≡ ω_{λ} by 2.2(ii). But a gauge transformation which preserves the connection form of a principal bundle must act as an element of the structural group on each connected component of the bundle. Since an isometry of the connected Riemannian manifold (M, g_{λ}) is determined by its restriction to any nonempty open subset, it follows that the family F_{t} is contained in T. □

2.4 Proposition. Let λ, λ′ be admissible 1-forms on M such that Ω_{λ′} has property (G). Furthermore, assume that

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

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

Proof. Suppose that there were an isometry F : (M, g_{λ}) → (M, g_{λ′}). By Proposition 2.3, T is a maximal torus in Isom(M, g_{λ′}). Since all maximal tori are conjugate, we can assume F – after possibly combining it with an isometry of (M, g_{λ′}) – to be T -preserving. But then Lemma 2.2 implies F̄^{∗}Ω_{λ′} = Ψ_{F} ∘ Ω_{λ} with F̄ ∈ \overline{Aut}_{g_{0}}^{T} (M) and Ψ_{F} ∈ D, which contradicts our assumption (N). □

2.5 Remark. Note that equally valid (but weaker) versions of Propositions 2.3 and 2.4 would be obtained by replacing \overline{Aut}_{g_{0}}^{T} (M) by the possibly larger group Isom(M̂/T, g_{0}^{T}).

§3 Examples

3.1 Notation.

(i) Throughout the new examples given in 3.2 and 3.3 below we consider the two-dimensional torus T := ℝ^{2}/L with L := 2πℤ × 2πℤ. We denote the standard basis of its Lie algebra 𝔷 ≅ ℝ^{2} by {Z_{1} , Z_{2}}.

(ii) We let E be the group of the four linear isomorphisms of 𝔷 which preserve

each of the sets {±Z_{1}} and {±Z_{2}}. (iii) Let m ∈ ℕ. We identify the real vector spaces ℂ^{m+1} = ℂ^{m} ⊕ ℂ and ℝ^{2m+2}

via the linear isomorphism which sends {e_{1} , ie_{1} , …, e_{m+1} , ie_{m+1}} (in this order) to the standard basis of ℝ^{2m+2}, where {e_{1} , …, e_{m+1}} denotes the standard basis of ℂ^{m+1}. We let the torus T act on this space by

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

for all a, b ∈ ℝ, p ∈ ℂ^{m}, q ∈ ℂ. This action preserves the unit sphere S^{2m+1} ⊂ ℝ^{2m+2} as well as the unit ball B^{2m+2}; by restriction we thus obtain an action ρ of T on S^{2m+1}, respectively on B^{2m+2}.

3.2 Example: Continuous isospectral families of metrics on S^{2m+1≥7} and on B^{2m+2≥8}.

3.2.1 Notation. For each linear map j : 𝔷 ≅ ℝ^{2} → 𝔰𝔲(m) we define a 𝔷-valued 1-form λ = (λ^{1}, λ^{2}) on ℝ^{2m+2} ≅ ℂ^{m} ⊕ ℂ by letting

(4)  λ_{(p,q)}^{k}(X, U) = |p|^{2}⟨j_{Z_{k}}p, X⟩ − ⟨X, ip⟩⟨j_{Z_{k}}p, ip⟩

for k = 1, 2 and all (X, U) ∈ T_{p}ℝ^{2m} ⊕ T_{q}ℝ^{2}. Here ⟨ . , . ⟩ denotes the standard euclidean inner product on ℝ^{2m}, and the skew-hermitian maps j_{Z_{k}} := j(Z_{k}) act on p ∈ ℝ^{2m} via the above identification ℝ^{2m} ≅ ℂ^{m}.

By restriction we obtain a smooth 𝔷-valued 1-form λ on the unit sphere S^{2m+1}, respectively on the unit ball B^{2m+2}.

3.2.2 Remark. We observe that λ is admissible (see Notation 1.5(ii)) with respect to the action ρ of T defined above in 3.1(iii). In fact, invariance of λ under the action of T is immediate because multiplication with the complex scalar factor e^{ia} commutes with j_{Z_{k}} ∈ 𝔰𝔲(m) and preserves the euclidean inner product. It remains to check that λ vanishes on the spaces 𝔷_{(p,q)} = span {(ip, 0), (0, iq)} ⊂ T_{p}ℝ^{2m}⊕T_{q}ℝ^{2}. Indeed we have λ_{(p,q)}^{k}(ip, 0) = |p|^{2}⟨j_{Z_{k}}p, ip⟩ − ⟨ip, ip⟩⟨j_{Z_{k}}p, ip⟩ = 0 for k = 1, 2, and λ_{(p,q)}(0, iq) = 0 by definition.

3.2.3 Definition. Let g_{0} be the round standard metric on S^{2m+1}, respectively the standard metric on B^{2m+2}, and let g_{λ} be the metric associated with λ and g_{0} as in Notation 1.5(iii).

Summarizing, for each linear map j : ℝ^{2} → 𝔰𝔲(m) we have an associated Riemannian metric g_{λ} on S^{2m+1}, respectively on B^{2m+2}, via the corresponding 1-form λ as defined in (4).

3.2.4 Definition. Let j, j′ : 𝔷 ≅ ℝ^{2} → 𝔰𝔲(m) be two linear maps.

(i) We call j and j′ isospectral, denoted j ∼ j′, if for each Z ∈ 𝔷 there exists A_{Z} ∈ SU(m) such that j′_{Z} = A_{Z}j_{Z}A_{Z}^{−1} .

(ii) Let Q : ℂ^{m} → ℂ^{m} denote complex conjugation. We call j and j′ equivalent, denoted j ≅ j′, if there exists A ∈ SU(m) ∪ SU(m) ∘ Q and Ψ ∈ E (see Notation 3.1(ii)) such that j′_{Z} = Aj_{Ψ(Z)}A^{−1} for all Z ∈ 𝔷.

(iii) We say j is generic if no nonzero element of 𝔰𝔲(m) commutes with both j_{Z_{1}} and j_{Z_{2}} .

3.2.5 Proposition. Let j, j′ : 𝔷 ≅ ℝ^{2} → 𝔰𝔲(m) be two linear maps, and let g_{λ} , g_{λ′} be the associated pair of Riemannian metrics on S^{2m+1}, or on B^{2m+2}, as above. If j ∼ j′ then the Riemannian manifolds (S^{2m+1}, g_{λ}) and (S^{2m+1}, g_{λ′}) are isospectral, and (B^{2m+2}, g_{λ}) and (B^{2m+2}, g_{λ′}) are Dirichlet and Neumann isospectral. If j and j′ are not equivalent and if at least one of them is generic, then in both of these pairs the two manifolds are not isometric.

Proof. We will prove the nonisometry statement in Section 4. To prove isospectrality we use Theorem 1.6. Fix an arbitrary μ ∈ L^{∗}. We have to show that there exists an isometry F_{μ} of g_{0} which commutes with the action of T and satisfies μ ∘ λ = F_{μ}^{∗}(μ ∘ λ′). Let Z ∈ 𝔷 be the vector corresponding to μ ∈ L^{∗} ⊂ 𝔷^{∗} under the canonical identification of 𝔷 with 𝔷^{∗} associated with the basis {Z_{1} , Z_{2}}. Choose A_{Z} ∈ SU(m) ⊂ SO(2m) as in Definition 3.2.4(i), and let F_{μ} := (A_{Z} , Id) ∈ SO(2m + 2). Then F_{μ} is an isometry of g_{0} and satisfies

(F_{μ}^{∗}(μ ∘ λ′))_{(p,q)}(X, U) = (μ ∘ λ′)_{(A_{Z}p,q)}(A_{Z}X, U)
= |A_{Z}p|^{2}⟨j′_{Z}A_{Z}p, A_{Z}X⟩ − ⟨A_{Z}X, iA_{Z}p⟩⟨j′_{Z}A_{Z}p, iA_{Z}p⟩
= |p|^{2}⟨A_{Z}^{−1}j′_{Z}A_{Z}p, X⟩ − ⟨X, ip⟩⟨A_{Z}^{−1}j′_{Z}A_{Z}p, ip⟩
= |p|^{2}⟨j_{Z}p, X⟩ − ⟨X, ip⟩⟨j_{Z}p, ip⟩ = (μ ∘ λ)_{(p,q)}(X, U),

as desired. □

The following result shows that there are in fact many examples to Proposition 3.2.5.

3.2.6 Proposition. Let m ≥ 3 and {Z_{1} , Z_{2}} be the standard basis of ℝ^{2}.

(i) There exists a nonempty Zariski open subset U of the space J of linear maps j : ℝ^{2} → 𝔰𝔲(m) such that for each j ∈ U there is a continuous family j(t) in J, defined on some open interval around t = 0, such that j(0) = j and: 1.) The maps j(t) are pairwise isospectral in the sense of 3.2.4(i). 2.) The function t ↦ ∥j_{Z_{1}}(t)^{2} + j_{Z_{2}}(t)^{2}∥^{2} = tr((j_{Z_{1}}(t)^{2} + j_{Z_{2}}(t)^{2})^{2}) is not constant in t in any interval around zero. In particular, the maps j(t) are not pairwise equivalent in the sense of 3.2.4(ii). 3.) The maps j(t) are generic in the sense of 3.2.4(iii).

(ii) For m = 3, an explicit example of an isospectral family j(t) : ℝ^{2} → 𝔰𝔲(3) with ∥j_{Z_{1}}(t)^{2} + j_{Z_{2}}(t)^{2}∥^{2} ≠ const is given by

j_{Z_{1}}(t) := ( \begin{matrix}−i & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & i \end{matrix} ) , j_{Z_{2}}(t) := ( \begin{matrix}0 & cos t & \sqrt{2} sin t \\ − cos t & 0 & cos t \\ −\sqrt{2} sin t & − cos t & 0 \end{matrix} ) .

The j(t) are pairwise isospectral since det(λId−(sj_{Z_{1}}(t)+uj_{Z_{2}}(t))) = λ^{3} + (s^{2} +2u^{2})λ is independent of t. However, ∥j_{Z_{1}}(t)^{2} +j_{Z_{2}}(t)^{2}∥^{2} = 14+4 sin^{2}t is nonconstant in t. The map j(t) is generic in the sense of 3.2.4(iii) if and only if cos t ≠ 0.

Proof. (i) The same statement, but without requirement 3.), was shown in [Sch3, Proposition 3.6]. Since the section of two nonempty Zariski open sets is again such a set, it just remains to show that the subset O ⊂ J of those elements which are generic in the sense of 3.2.4(iii) is nonempty and Zariski open. Note that the map j ∈ J given by

j_{Z_{1}} := ( \begin{matrix}iα_{1} \\ & iα_{2} \\ & & iα_{3} \\ & & & . & . & . \\ & & & & & & iα_{m} \end{matrix} ) , j_{Z_{2}} := ( \begin{matrix}0 & −1 \\ 1 & 0 & −1 \\ & 1 & ... & ... \\ & & ... & 0 & −1 \\ & & & 1 & 0 \end{matrix} )

with pairwise different α_{1} , … α_{m} ∈ ℝ satisfying α_{1} + … + α_{m} = 0 is an element of O; thus O is nonempty. To see that O is Zariski open, note that it is equal to the set of those j ∈ J for which the map F_{j} : 𝔰𝔲(m) ∋ τ ↦ ([j_{Z_{1}} , τ], [j_{Z_{2}} , τ]) ∈ 𝔰𝔲(m) ⊕ 𝔰𝔲(m) has maximal rank r_{m} := dim(𝔰𝔲(m)). But the latter condition can be expressed as the nonvanishing of a certain polynomial in the coefficients of j, namely, the sum of the squared determinants of the (r_{m} × r_{m})-minors of a matrix representation of F_{j} .

(ii) This can be checked by straightforward calculation. □

3.2.7 Corollary. For every m ≥ 3 there exist continuous families of isospectral metrics on S^{2m+1} and continuous families of Dirichlet and Neumann isospectral metrics on B^{2m+2}. In particular, there exist such families on S^{7}, resp. on B^{8}. An explicit example is given by the metrics g_{λ(t)} associated with the family j(t) from 3.2.6(ii).

Remark. Carolyn Gordon [Go3] has previously given continuous families of isospectral metrics on each S^{n≥8} and B^{n≥9} using a related construction (see 3.4(i)). Those were the first examples of continuous isospectral families of metrics on balls and spheres.

3.3 Example: Isospectral pairs of metrics on S^{5} and on B^{6}.

3.3.1 Notation. In the context of Notation 3.1 we consider now the case m = 2.

(i) 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).

(ii) 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 λ = (λ^{1}, λ^{2}) on ℝ^{6} ≅ ℂ^{2} ⊕ ℂ by letting

(5)  λ_{(p,q)}^{k}(X, U) = ⟨c_{Z_{k}}P (p) × P (p), P_{∗p}(X)⟩

for k = 1, 2 and all (X, U) ∈ T_{p}ℝ^{2m} ⊕ T_{q}ℝ^{2}, where ⟨ . , . ⟩ denotes the standard euclidean inner product on ℝ^{3}, and × denotes the vector product in ℝ^{3}. By restriction we obtain a smooth 𝔷-valued 1-form λ on the unit sphere S^{5}, respectively on the unit ball B^{6}.

3.3.2 Remark. We observe that λ is admissible with respect to the action ρ of T on S^{5} ⊂ ℂ^{2} ⊕ ℂ, resp. B^{6} ⊂ ℂ^{2} ⊕ ℂ. Invariance of λ under the action of T is immediate since P (e^{ia}p) = P (p) for all a ∈ ℝ, p ∈ ℂ^{2}. By the same reason we have P_{∗p}(ip) = 0, which implies that λ vanishes on the spaces 𝔷_{(p,q)} .

3.3.3 Definition. Let g_{λ} be the metric associated with λ and the standard metric g_{0} on S^{5}, resp. B^{6}. Thus for each linear map c : ℝ^{2} → Sym_{0}(ℝ^{3}) we have an associated Riemannian metric g_{λ} on S^{5}, respectively on B^{6}, via the corresponding 1-form λ as defined in (5).

3.3.4 Definition. Let j, j′ : 𝔷 ≅ ℝ^{2} → Sym_{0}(ℝ^{3}) be two linear maps.

(i) We call c and c′ isospectral, denoted c ∼ c′, if for each Z ∈ 𝔷 there exists E_{Z} ∈ SO(3) such that c′_{Z} = E_{Z}c_{Z}E_{Z}^{−1} .

(ii) We call c and c′ equivalent, denoted c ≅ c′, if there exists E ∈ O(3) and

Ψ ∈ E (see Notation 3.1(ii)) such that c′_{Z} = Ec_{Ψ(Z)}E^{−1} for all Z ∈ 𝔷. (iii) We say c is generic if no nonzero element of 𝔰𝔬(3) commutes with both c_{Z_{1}}

and c_{Z_{2}} .

3.3.5 Proposition. Let c, c′ : 𝔷 ≅ ℝ^{2} → Sym_{0}(ℝ^{3}) be two linear maps, and let g_{λ} , g_{λ′} be the associated pair of Riemannian metrics on S^{5}, or on B^{6}, as above. If c ∼ c′ then the Riemannian manifolds (S^{5}, g_{λ}) and (S^{5}, g_{λ′}) are isospectral, and (B^{6}, g_{λ}) and (B^{6}, g_{λ′}) are Dirichlet and Neumann isospectral. If c and c′ are not equivalent and if at least one of them is generic, then in both of these pairs the two manifolds are not isometric.

Proof. Again, we postpone the proof of the nonisometry statement to Section 4, and we use Theorem 1.6 to prove isospectrality. Fix μ ∈ L^{∗} and let Z be the dual vector in 𝔷 as in the proof of 3.2.5. Choose E_{Z} ∈ SO(3) as in Definition 3.3.4(i), and choose A_{Z} ∈ SU(2) ⊂ SO(4) such that for the Hopf projection P : S^{3} → S_{1/2}^{2} from 3.3.1(i) we have P ∘ A_{Z} = E_{Z} ∘ P. Let F_{μ} := (A_{Z} , Id) ∈ SO(6). Then F_{μ} is an isometry of g_{0} and satisfies

(F_{μ}^{∗}(μ ∘ λ′))_{(p,q)}(X, U) = (μ ∘ λ′)_{(A_{Z}p,q)}(A_{Z}X, U)
= ⟨c′_{Z}P (A_{Z}p) × P (A_{Z}p), P_{∗A_{Z}p}(A_{Z}X)⟩
= ⟨c′_{Z}E_{Z}P (p) × E_{Z}P (p), E_{Z}P_{∗p}(X)⟩
= ⟨E_{Z}^{−1}c′_{Z}E_{Z}P (p) × P (p), P_{∗p}(X)⟩
= ⟨c_{Z}P (p) × P (p), P_{∗p}(X)⟩ = (μ ∘ λ)_{(p,q)}(X, U),

as desired. □

3.3.6 Proposition. There exist pairs of linear maps c, c′ : ℝ^{2} → Sym_{0}(ℝ^{3}) such that c and c′ are isospectral in the sense of 3.3.4(i), not equivalent in the sense of 3.3.4(ii), and both generic in the sense of 3.3.4(iii). An example of such a pair 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 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) , c′_{Z_{2}} = ( \begin{matrix}0 & 0 & \sqrt{2} \\ 0 & 0 & 0 \\ \sqrt{2} & 0 & 0 \end{matrix} ) ,

where {Z_{1} , Z_{2}} is the standard basis of ℝ^{2}.

Proof. One easily checks that for each fixed pair s, u ∈ ℝ, the characteristic polynomials of sc_{Z_{1}} + uc_{Z_{2}} and sc′_{Z_{1}} + uc′_{Z_{2}} are equal (namely, to λ^{3} + (s^{2} + 2u^{2})λ), which implies isospectrality. That c and c′ are not equivalent can be seen from the fact that ∥c_{Z_{1}}^{2} + c_{Z_{2}}^{2}∥^{2} = 14, while ∥c_{Z_{1}}^{′ 2} + c_{Z_{2}}^{′ 2}∥^{2} = 18. The maps are generic in the sense of 3.3.4(iii) because not even the map c_{Z_{1}} = c′_{Z_{1}} alone commutes with any nonzero element of 𝔰𝔬(3). □

3.3.7 Corollary. There exist nontrivial pairs of isospectral metrics on S^{5}, and there exist nontrivial pairs of Dirichlet and Neumann isospectral metrics on B^{6}.

Remark. Our above pairs of isospectral 5-spheres constitute the lowest dimensional examples of isospectral spheres which have been constructed so far. The analogous statement holds for our isospectral 6-dimensional balls.

3.4 Survey of related examples. As a complement to our above new examples of isospectral spheres and balls we now give a short survey of related examples. Among them, (ii) and (v) are new; (i), (iii), and (iv) are already known, but were constructed first in slightly different settings.

Although we will not present any proofs here, we note that in each of the examples below the Riemannian manifolds can be described within the setting of 1.5 and proven to be isospectral by Theorem 1.6. Moreover, for each of them there is a nonisometry proof using the general results from Section 2 and following the lines of Section 4 below, where we will prove nonisometry for the above examples 3.2 / 3.3. Since the formulas defining λ, λ′ in the examples below (except (v)) are less complicated than those in 3.2 or 3.3, the corresponding computations are simpler than those in Section 4. Also note that in some of the examples the situation is simplified by the fact that M̂ = M (in (iii) and (v)), or that Ω_{0} = 0 (in (i)–(iv)). For example, in (iii) below, the curvature form Ω_{λ} equals dλ̄ on M/T = S^{2m−1}, where dλ̄^{k} is of the form (X, Y) ↦ 2⟨j_{Z_{k}}X, Y ⟩.

The notation we use is similar to the one we used above. In particular, T again denotes the torus ℝ^{2}/(2πℤ × 2πℤ), and {Z_{1} , Z_{2}} is the standard basis of its Lie algebra 𝔷 ≅ ℝ^{2}. On all manifolds M which we consider below there is a canonical standard metric which will in each case play the role of g_{0} . When we call two manifolds with boundary isospectral we always mean Dirichlet and Neumann isospectral.

(i) Continuous families of isospectral metrics on S^{n≥8} and on B^{n≥9} [Go3]. Let M = S^{m+3} ⊂ ℝ^{m} ⊕ ℂ ⊕ ℂ, resp. M = B^{m+4} ⊂ ℝ^{m} ⊕ ℂ ⊕ ℂ. Let the action of T on M be induced by its canonical action on the ℂ ⊕ ℂ component (generated by multiplication with i on each summand). For each linear map j : ℝ^{2} → 𝔰𝔬(m) we define a 𝔷 ≅ ℝ^{2}-valued 1-form on ℝ^{m} ⊕ ℂ ⊕ ℂ ≅ ℝ^{m} ⊕ ℝ^{4} (and hence on M) by letting

λ_{(p,q)}^{k}(X, U) = ⟨j_{Z_{k}}p, X⟩

for k = 1, 2 and all (X, U) ∈ T_{p}ℝ^{m} ⊕ T_{q}ℝ^{4}. The following conditions on a pair j, j′ imply that the associated Riemannian manifolds (M, g_{λ}), (M, g_{λ′}) are isospectral and not isometric:

1.) j and j′ are isospectral: For each Z ∈ 𝔷 there exists A_{Z} ∈ O(m) such that

j′_{Z} = A_{Z}j_{Z}A_{Z}^{−1}. 2.) j and j′ are nonequivalent: There is no A ∈ O(m) and no Ψ ∈ Ẽ such that j′_{Z} = Aj_{Ψ(Z)}A^{−1} for all Z ∈ 𝔷; here Ẽ denotes the set of the eight

automorphisms of ℝ^{2} which preserve the set {±Z_{1} , ±Z_{2}}.

3.) At least one of j, j′ is generic; j is called generic if no nonzero element

of 𝔰𝔬(m) commutes with both j_{Z_{1}} and j_{Z_{2}} . It is known [GW2] that for m ≥ 5 there are continuous families – even multiparameter families – j(t) whose elements pairwise satisfy these conditions. The associated manifolds (M, g_{λ(t)}) are Carolyn Gordon’s examples of isospectral deformations on spheres and balls.

(ii) Isospectral pairs of metrics on S^{6} and on B^{7}. This is just a slight modification of Example 3.3 which does not involve the Hopf projection P anymore; in turn, an additional dimension is needed for the construction. Let M = S^{6} ⊂ ℝ^{3} ⊕ℂ⊕ℂ, resp. M = B^{7} ⊂ ℝ^{3} ⊕ℂ⊕ℂ. Let T act canonically on the ℂ ⊕ ℂ component as in (i). For each linear map c : ℝ^{2} → Sym_{0}(ℝ^{3}) we define a 𝔷 ≅ ℝ^{2}-valued 1-form on ℝ^{3} ⊕ ℂ ⊕ ℂ ≅ ℝ^{3} ⊕ ℝ^{4} (and hence on M) by letting

λ_{(p,q)}^{k}(X, U) = ⟨c_{Z_{k}}p × p, X⟩

for k = 1, 2 and all (X, U) ∈ T_{p}ℝ^{3} ⊕ T_{q}ℝ^{4}. The conditions on a pair c, c′ which cause (M, g_{λ}) and (M, g_{λ′}) to be isospectral and not isometric are the same as in 3.3, except that in the nonequivalence condition the group E has to be replaced by the larger group Ẽ as in (i).

The specific pair c, c′ given in 3.3.6 satisfies these conditions and thus yields a pair of isospectral metrics g_{λ} , g_{λ′} on S^{6} (resp. on B^{7}). Moreover, one can compute the loci N, N′ of the maximal scalar curvature in (S^{6}, g_{λ}) and (S^{6}, g_{λ′}). It turns out that N′ contains a 4-sphere, while N is a union of 2- and 3-spheres.

(iii) Continuous families of isospectral metrics on S^{m−1≥4} × T^{2} [GGSWW] and on B^{m≥5}×T^{2} [GW2]; pairs of isospectral metrics on S^{2}×T^{2} [Sch3] and on B^{3}×T^{2}. For a, b, c > 0 with a^{2} + b^{2} + c^{2} ≤ 1 we define M_{a,b,c} := {(p, u, v) ∈ ℝ^{m} ⊕ ℂ ⊕ ℂ | |p|^{2} = a^{2}, |u|^{2} = b^{2}, |v|^{2} = c^{2}}. Then M_{a,b,c} is diffeomorphic to S^{m−1} × T^{2} and is a submanifold of B^{m+4} ⊂ ℝ^{m} ⊕ ℂ ⊕ ℂ. The restrictions of two metrics g_{λ} , g_{λ′} from (i) to one of these submanifolds are again isospectral and nonisometric under the same conditions as in (i) on the underlying pair of maps j, j′. In particular, one obtains continuous isospectral families g_{λ(t)} on S^{m−1} × T^{2} for m ≥ 5. These are just the examples constructed in [GGSWW].

Similarly, for m = 3 the metrics g_{λ} , g_{λ′} from (ii) restrict to isospectral nonisometric metrics on M_{a,b,c} which is now diffeomorphic to S^{2} × T^{2}; these pairs were first constructed in [Sch3].

In both cases we can replace the condition |p|^{2} = a^{2} by |p|^{2} ≤ a^{2} in the definition of M_{a,b,c} which then becomes diffeomorphic to B^{m} × T^{2}. We obtain continuous families of isospectral metrics g_{λ(t)} on B^{m} × T^{2} for m ≥ 5 [GW2] and pairs of such metrics on B^{3} × T^{2}.

(iv) Isospectral pairs of metrics on S^{2} × S^{3} [Ba]. Instead of considering the submanifolds M_{a,b,c} ≈ S^{2} × T^{2} of B^{7} ⊂ ℝ^{3} ⊕ ℂ ⊕ ℂ as in the middle part of (iii) above, we now consider M_{a,b} := {(p, q) ∈ ℝ^{3} ⊕ ℝ^{4} | |p|^{2} = a^{2}, |q|^{2} = b^{2}} with a, b > 0. Then M_{a,b} is diffeomorphic to S^{2} × S^{3}, and again the metrics g_{λ} , g_{λ′} from (ii) restrict to isospectral nonisometric metrics on M_{a,b}.

(v) Continuous isospectral families on S^{2m−1≥5} × S^{1} and on B^{2m≥6} × S^{1}; pairs of isospectral metrics on S^{3} × S^{1} and on B^{4} × S^{1}. We return to the context of our examples 3.2 / 3.3 above and consider the submanifolds M_{a,b} := {(p, q) ∈ ℂ^{m} ⊕ ℂ | |p|^{2} = a^{2}, |q|^{2} = b^{2}} of B^{2m}. For a, b > 0 the manifold M_{a,b} is diffeomorphic to S^{2m−1} × S^{1}. If j, j′ : ℝ^{2} → 𝔰𝔲(m) satisfy the assumptions of Proposition 3.2.5 then the associated pair of metrics g_{λ} , g_{λ′} on B^{2m} restricts to a pair of isospectral nonisometric metrics on M_{a,b} as well. In particular, for m ≥ 3 we obtain continuous families g_{λ(t)} of such metrics as in 3.2.6 / 3.2.7.

Similarly, for m = 2 each pair c, c′ : ℝ^{2} → Sym_{0}(ℝ^{3}) satisfying the assumptions of Proposition 3.3.5 also yields a pair of isospectral nonisometric metrics on the manifold S^{2m−1} × S^{1} = S^{3} × S^{1}.

In both cases we can again replace the condition |p|^{2} = a^{2} by |p|^{2} ≤ a^{2} in the definition of M_{a,b} which then becomes diffeomorphic to B^{2m} × S^{1}. We obtain continuous families of isospectral metrics g_{λ(t)} on B^{2m} × S^{1} for m ≥ 3, and pairs of such metrics on B^{4} × S^{1}.

§4 Nonisometry of the examples from 3.2 / 3.3

Our strategy for proving the nonisometry statements of Proposition 3.2.5 / 3.3.5 consists in showing that the nonequivalence condition on j, j′ (resp. on c, c′) implies condition (N) of Proposition 2.4, while the genericity condition on the pair of maps implies Property (G) for the associated curvature forms. See Proposition 4.3 below for these assertions. The desired nonisometry statements then follow immediately from Proposition 2.4.

4.1 Notation and Remarks. Throughout this section we denote by (M, g_{0}) either S^{2m+1} ⊂ ℝ^{2m} ⊕ ℝ^{2} or B^{2m+2} ⊂ ℝ^{2m} ⊕ ℝ^{2}, endowed with the standard metric. We consider the action ρ of T = ℝ^{2}/(2πℤ×2πℤ) on M which was defined in 3.1(iii).

(i) For a, b ≥ 0 let M_{a,b} := {(p, q) ∈ ℝ^{2m} ⊕ ℝ^{2} | |p|^{2} = a^{2}, |q|^{2} = b^{2}}. Thus S^{2m+1}, resp. B^{2m+2}, is the disjoint union of the submanifolds M_{a,b} with a^{2} + b^{2} = 1, resp. a^{2} + b^{2} ≤ 1. Note that in either case, M̂ is the union of those M_{a,b} ⊂ M with a, b > 0. Each M_{a,b} is obviously T -invariant.

(ii) For M_{a,b} ⊂ M̂ the manifold (M_{a,b}/T, g_{0}^{T}) is isometric to (ℂP^{m−1}, a^{2}g_{FS}),

where g_{FS} denotes the Fubini-Study metric. (iii) The first component of the 𝔷 ≅ ℝ^{2}-valued form which the curvature form Ω_{0}

induces on (M_{a,b}/T, g_{0}^{T}) = (ℂP^{m−1}, a^{2}g_{FS}) is a scalar multiple of the standard Kähler form on ℂP^{m−1}, and the second component is zero. In fact, for (X, Z), (Y, W) ∈ T_{(p,q)}M_{a,b} = T_{p}S_{a}^{2m−1} ⊕ T_{q}S_{b}^{1} we have ω_{0}^{1}(X, Z) = ⟨X, ip⟩/a^{2}, ω_{0}^{2}(X, Z) = ⟨Z, iq⟩/b^{2}, hence dω_{0}^{1}((X, Z), (Y, W)) = 2⟨iX, Y ⟩/a^{2} and dω_{0}^{2}((X, Z), (Y, W)) = 2⟨iZ, W ⟩/b^{2} = 0, where the last equation follows from the fact that T_{q}S_{b}^{1} is one-dimensional. The statement now follows because Ω_{0} is induced by dω_{0} .

(iv) Any isometry of (ℂP^{m−1}, a^{2}g_{FS}) = S_{a}^{2m−1}/S^{1} is induced by some ℂ-linear or ℂ-antilinear isometry A ∈ SU(m) ∪ SU(m) ∘ Q (see 3.2.4) of ℂ^{m}.

4.2 Lemma. If F ∈ Aut_{g_{0}}^{T}(M) then F preserves each of the submanifolds M_{a,b} of M̂. Moreover, Ψ_{F} ∈ E (see Notation 3.1(ii)). In particular, for the group D defined in 2.1(ii) we have D ⊂ E.

Proof. Note that the T -orbit through (p, q) ∈ M_{a,b} ⊂ M̂, endowed with the metric induced by g_{0} , is a rectangular torus with side lengths 2πa and 2πb. It follows that F preserves the sets M_{a,b} ∪ M_{b,a} ⊂ M̂. We have to show that F cannot switch the components M_{a,b} and M_{b,a} if a ≠ b. If it did then it would induce an isometry from (M_{a,b}/T, g_{0}^{T}) to (M_{b,a}/T, g_{0}^{T}). But by 4.1(ii) these manifolds have different volume. Thus F preserves M_{a,b}. Now choose 0 < a < b such that M_{a,b} ⊂ M̂, and let (p, q) ∈ M_{a,b}. Since F preserves M_{a,b}, both of the orbits T · (p, q) and T · F (p, q) are rectangular tori on which the shortest closed loops have length 2πa and are precisely the flow lines of Z_{1}^{∗}. This, together with Z_{1}^{∗} ⊥ Z_{2}^{∗} and ∥Z_{2}^{∗}_{|_{(p,q)}}∥ = ∥Z_{2}^{∗}_{|_{F (p,q)}}∥ = b, implies that F_{∗}(Z_{1}^{∗}) ∈ {±Z_{1}^{∗}} and F_{∗}(Z_{2}^{∗}) ∈ {±Z_{2}^{∗}}; hence Ψ_{F} ∈ E. Now D ⊂ E follows from the definition of D. □

In view of Proposition 2.4, the following result implies immediately the nonisometry statements of Proposition 3.2.5 / 3.3.5 :

4.3 Proposition. Let λ, λ′ be of the type defined in (4) (resp. in (5)), and let j, j′ : ℝ^{2} → 𝔰𝔲(m) (resp. c, c′ : ℝ^{2} → Sym_{0}(ℝ^{3})) be the pair of linear maps with which λ, λ′ are associated.

(i) If j and j′ (resp. c and c′) are not equivalent in the sense of 3.2.4(ii) (resp. 3.3.4(ii)), then Ω_{λ} and Ω_{λ′} satisfy condition (N) of Proposition 2.4.

(ii) If j (resp. c) is generic in the sense of 3.2.4(iii) (resp. 3.3.4(iii)), then Ω_{λ} has Property (G).

Proof. We choose one of the submanifolds M_{a,b} =: L of M̂ and denote by ω_{λ}^{L} , Ω_{λ}^{L} the forms induced by ω_{λ} , Ω_{λ} on L ⊂ M̂ and L/T ⊂ M̂/T, respectively.

(i) Suppose that condition (N) were not satisfied. Let Ψ ∈ D ⊂ E (recall 4.2) and F̄ ∈ \overline{Aut}_{g_{0}}^{T} (M) such that Ω_{λ} = Ψ∘F̄^{∗}Ω_{λ′} . Lemma 4.2 implies that F̄ preserves L/T; hence Ω_{λ}^{L} = Ψ∘F̄^{∗}Ω_{λ′}^{L}. Recall from 2.1(vi) that Ω_{λ} = Ω_{0}+dλ̄, Ω_{λ′} = Ω_{0}+dλ̄′. Each closed 2-form on ℂP^{m−1} is uniquely decomposable into an exact component and a multiple of the Kähler form. Therefore, the description of Ω_{0}^{L} given in 4.1(iii) now implies that

(6)  dλ̄^{L} = Ψ ∘ F̄^{∗}dλ̄′^{L}

and hence

(7)  dλ^{L} = Ψ ∘ A^{∗}dλ′^{L}

for the map A ∈ SU(m) ∪ SU(m) ∘ Q which induces the isometry F̄ of (L/T, g_{0}^{T}) = (ℂP^{m−1}, a^{2}g_{FS}) (recall 4.1(iv)). Here λ^{L}, λ′^{L} and λ̄^{L}, λ̄′^{L} denote the forms which λ, λ′ and λ̄, λ̄′ induce on L and L/T, respectively. We are going to show that

(8)  j′_{Ψ(Z)} = Aj_{Z}A^{−1}

for all Z ∈ 𝔷 in the case of forms of type (4), and

(9)  c′_{Φ(Z)} = Ec_{Z}E^{−1}

for all Z ∈ 𝔷 in the case of forms of type (5), where E := F̄_{|_{L/T}} ∈ O(3) and Φ := det(E)Ψ ∈ {±Ψ} ⊂ E. This will contradict the assumed nonequivalence of j and j′ (resp. of c and c′).

In the first case, we have for all (X, Z), (Y, W) ∈ T_{(p,q)}L = T_{p}S_{a}^{2m−1} × T_{q}S_{b}^{1} and for k = 1, 2:

λ^{k}(X, Z) = a^{2}⟨j_{k}p, X⟩ − ⟨j_{k}p, ip⟩⟨X, ip⟩, hence
dλ^{k}((X, Z), (Y, W)) = 2a^{2}⟨j_{k}X, Y ⟩ − 2⟨j_{k}X, ip⟩⟨Y, ip⟩ + 2⟨j_{k}Y, ip⟩⟨X, ip⟩
− 2⟨j_{k}p, ip⟩⟨iX, Y ⟩
= 2a^{2}⟨j_{k}X^{h}, Y^{h}⟩ − 2⟨j_{k}p, ip⟩⟨iX, Y ⟩,

where j_{k} := j_{Z_{k}} and X^{h} denotes the g_{0}-orthogonal projection of X to (ip)^{⊥}. Let ε_{k} ∈ {±1} be such that Ψ(Z_{k}) = ε_{k}Z_{k}. Then one derives from equation (7):

\begin{matrix}ε_{k}(a^{2}⟨j′_{k}(AX)^{h}, (AY)^{h}⟩ − ⟨j′_{k}Ap, iAp⟩⟨iAX, AY ⟩) \\ = a^{2}⟨j_{k}X^{h}, Y^{h}⟩ − ⟨j_{k}p, ip⟩⟨iX, Y ⟩ \end{matrix}

for k = 1, 2 and all X, Y ∈ T_{p}S_{a}^{2m−1}. Since (AX)^{h} = AX^{h} and since A either commutes or anticommutes with i, we get, letting τ_{k} := ε_{k}A^{−1}j′_{k}A − j_{k} ∈ 𝔰𝔲(m):

a^{2}⟨τ_{k}X^{h}, Y^{h}⟩ − ⟨τ_{k}p, ip⟩⟨iX, Y ⟩ = 0.

In particular, we obtain for all p ∈ S_{a}^{2m−1} and all nonvanishing X ∈ span {p, ip}^{⊥}, by letting Y := iX:

⟨τ_{k}X, iX⟩/|X|^{2} = ⟨τ_{k}p, ip⟩/|p|^{2}.

The hermitian map iτ_{k} must therefore be a scalar multiple of the identity, and τ_{k} be a scalar multiple of iId. This implies τ_{k} = 0 because of τ_{k} ∈ 𝔰𝔲(m); equation (8) now follows.

In the second case (namely, with λ, λ′ of type (5)) we have

λ̄^{k}(X) = a^{3}⟨c_{k}x × x, X⟩

for k = 1, 2 and all X ∈ T_{x}(L/T) = T_{x}S_{a/2}^{2} , where c_{k} := c_{Z_{k}} . The factor a^{3} is due to the fact that if π : S_{a}^{3} → S_{a/2}^{2} denotes the Riemannian submersion, then for the projection P as defined in 3.3.1 we have |P (p)| = a|π(p)|. We obtain

dλ̄^{k}(X, Y)/a^{3} = ⟨c_{k}X × x + c_{k}x × X, Y ⟩ − ⟨c_{k}Y × x + c_{k}x × Y, X⟩
= ⟨c_{k}X × x, Y ⟩ + 2⟨c_{k}x × X, Y ⟩ − ⟨c_{k}Y × x, X⟩

= ⟨c_{k}X × x, Y ⟩ + 2⟨c_{k}x × X, Y ⟩ + ⟨Y × c_{k}x, X⟩ + ⟨Y × x, c_{k}X⟩

= 3⟨c_{k}x × X, Y ⟩.

Note that in the third equation we have used tr(c_{k}) = 0. Equation (6) now implies for F̄|_{S_{a/2}^{2}} = E ∈ O(3):

ε_{k}⟨c′_{k}Ex × EX, EY ⟩ = ⟨c_{k}x × X, Y ⟩

for k = 1, 2 and all X, Y ∈ T_{x}S_{a/2}^{2} , where ε_{k} is defined as above. Letting τ_{k} := ε_{k} det(E)E^{−1}c′_{k}E − c_{k} ∈ Sym_{0}(ℝ^{3}) we obtain

⟨τ_{k}x × X, Y ⟩ = 0

for all X, Y ⊥ x, which implies τ_{k}x ⊥ x for all x ∈ S_{a/2}^{2} . Since τ_{k} is symmetric, it must be zero; equation (9) now follows.

(ii) Suppose to the contrary that λ does not have Property (G); i.e., there is a 1-parameter family F̄_{t} ∈ \overline{Aut}_{g_{0}}^{T} (M) such that F̄_{t}^{∗}Ω_{λ} ≡ Ω_{λ}. Proceeding as in the proof of (i) (in the special situation Ψ = Id and λ = λ′) we now obtain 1-parameter families A_{t} ∈ SU(m) (resp. E_{t} ∈ SO(3)) preserving dλ^{L} (resp. dλ̄^{L}), and we derive that j_{Z} ≡ A_{t}j_{Z}A_{t}^{−1} (resp. c_{Z} ≡ E_{t}c_{Z}E_{t}^{−1}) for each Z ∈ 𝔷 (cp. (8), (9)). But this contradicts the genericity assumption made on j (resp. c). □

§5 Isospectral metrics on spheres and balls which are equal to the standard metric on large subsets

The main idea of this section is to simultaneously multiply λ, λ′ by some smooth scalar function f on the sphere (resp. the ball) which has small support but is chosen such that both isospectrality and nonisometry of the associated metrics g_{fλ} and g_{fλ′} continue to hold.

The following observation is trivial.

5.1 Remark. In the context of Theorem 1.6, let λ, λ′ be two admissible 𝔷-valued 1-forms on M which satisfy (∗). Let f ∈ C^{∞}(M) be any function which is invariant under T and under each of the isometries F_{μ} occurring in (∗). Then fλ and fλ′ are again admissible and satisfy (∗). In particular, (M, g_{fλ}) and (M, g_{fλ′}) are isospectral.

5.2 Proposition. Let (M, g_{0}) = S^{2m+1} ⊂ ℝ^{2m} ⊕ ℝ^{2}, resp. (M, g_{0}) = B^{2m+2} ⊂ ℝ^{2m} ⊕ ℝ^{2}. Choose any φ ∈ C^{∞}([0, 1]^{2}) which does not vanish identically; in case M = S^{2m+1} we assume that φ does not vanish identically on the set {(s, 1 − s) | s ∈ [0, 1]}. Let f ∈ C^{∞}(M) be defined by f(p, q) := φ(|p|^{2}, |q|^{2}) for (p, q) ∈ M ⊂ ℝ^{2m} ⊕ ℝ^{2}. Then under the assumptions of Proposition 3.2.5 (resp. 3.3.5) on the pair j, j′ (resp. c, c′) which defines λ, λ′ as in (4) (resp. (5)), the manifolds (M, g_{fλ}) and (M, g_{fλ′}) are isospectral and not isometric.

Proof. The function f is obviously invariant under the action of T = ℝ^{2}/L as defined in 3.1(iii). Moreover, f is invariant under those F_{μ} occurring in the isospectrality proofs of Proposition 3.2.5 / 3.3.5: Recall that the F_{μ} used there were of the form (A_{Z} , Id) with A_{Z} ∈ SO(2m). The isospectrality statement thus follows from Remark 5.1 above.

For the nonisometry proof we must only slightly modify Proposition 4.3 and its proof. In both statements of that proposition we replace each occurrence of Ω_{λ} by Ω_{fλ}, and similarly Ω_{λ′} by Ω_{fλ′} . In the proof, we now choose L := M_{a,b} ⊂ M̂ not arbitrarily, but such that C := φ(a^{2}, b^{2}) ≠ 0. This is possible by the assumptions made on φ. The forms induced by fλ, fλ′ on L are then equal to the ones induced by Cλ and Cλ′, respectively. But replacing λ^{L}, λ^{′ L} by Cλ^{L}, Cλ^{′ L} does not afflict any of the subsequent arguments of the proof. Thus both statements of Proposition 4.3, modified as described, remain true. □

5.3 Theorem. Let ε > 0 be given, and let m ≥ 2. Then there exist nonisometric pairs of isospectral metrics on S^{2m+1} whose volume element is the standard one and which are equal to the round standard metric outside a subset of volume smaller than ε. For m ≥ 3 there are even continuous families of such metrics. The analogous statements hold for B^{2m+2} (with metrics which are both Dirichlet and Neumann isospectral ). The mentioned subset of small volume can be chosen as a tubular neighborhood around any of the submanifolds S_{a}^{2m−1}×S_{b}^{1} with 0 ≤ a, b ≤ 1 and a^{2} + b^{2} = 1 (in the case of S^{2m+1}), resp. a^{2} + b^{2} ≤ 1 (in the case of B^{2m+2}).

Proof. We can choose the function φ ∈ C^{∞}([0, 1]^{2}) in Proposition 5.2 such that its support is contained in a sufficiently small rectangle around (a^{2}, b^{2}). Concerning the coincidence of the volume elements of g_{0} and g_{fλ} recall 1.5(iii). □

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

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