CONTINUOUS FAMILIES OF ISOSPECTRAL

METRICS ON SIMPLY CONNECTED MANIFOLDS

Dorothee Schueth

Abstract. We construct continuous families of Riemannian metrics on certain simply connected manifolds with the property that the resulting Riemannian manifolds are pairwise isospectral for the Laplace operator acting on functions. These are the first examples of simply connected Riemannian manifolds without boundary which are isospectral, but not isometric. For example, we construct continuous isospectral families of metrics on the product of spheres S^{4} × S^{3} × S^{3}. The metrics considered are not locally homogeneous. For a big class of such families, the set of critical values of the scalar curvature function changes during the deformation. Moreover, the manifolds are in general not isospectral for the Laplace operator acting on 1-forms.

Introduction. Spectral geometry deals with the mutual influences between the geometry of a Riemannian manifold and the spectrum of the associated Laplace operator acting on smooth functions. Until 1964 it was not known whether the spectrum determines the geometry completely. Then J. Milnor constructed the first counterexample, namely, a pair of isospectral, non-isometric flat tori in dimension sixteen. Numerous other examples of isospectral manifolds followed in the 1980’s and 1990’s. See, e.g., [Vi], [Ik], [GW2], [Su], [BT], [Gt1] for examples of discrete families of isospectral manifolds, or [GW1], [DG1,2], [BG], [Sch], [Gt2] for isospectral deformations, that is, continuous families of isospectral manifolds.

In all the examples cited above, the isospectral manifolds arise as different quotients (by discrete groups of isometries) of a common Riemannian covering manifold. In particular, such quotients can neither differ in their local geometry nor be simply connected.

In 1993, C. Gordon [Go1,2] constructed the first pairs of isospectral closed Riemannian manifolds which are not locally isometric. S. Szabó [Sz] had constructed the first such examples in the case of manifolds with boundary even earlier, but did not publish them at that time. In 1996, even continuous families of isospectral metrics with different local geometries were obtained by C. Gordon and E. Wilson [GW3] in the case of manifolds with boundary. More precisely, in their examples the underlying manifolds are products of a ball with a torus. Recently it was discovered

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 58G25.

Key words and phrases. Laplacian, isospectral deformations, simply connected manifolds, products of spheres, scalar curvature, 1-form spectrum, heat invariants.

The author is partially supported by SFB 256, Bonn. Moreover, the author wishes to thank the GADGET program for supporting her participation in the NSF-CNRS workshop on Spectral Geometry in Grenoble (1997) where part of this work was carried out.

[GGSWW] that also the boundaries of these manifolds are isospectral, which led to the first examples of isospectral deformations with changing local geometry in the case of closed manifolds, namely, on products of a sphere with a torus. Szabó [Sz] independently gave finite families of isospectral manifolds which represent special cases of the examples in [GW3] and [GGSWW].

However, there were still no examples known, not even pairs, of simply connected, closed isospectral manifolds. Simply connected examples existed until now only in the case of manifolds with (non-smooth) boundary, most notably the famous pairs of isospectral planar domains by Gordon, Webb, and Wolpert [GWW].

In the present paper, we construct the first examples of simply connected, closed Riemannian manifolds which are isospectral, but not isometric; we even give continuous families of such manifolds. We will show:

Main Theorem. Let S^{n} be a sphere of dimension n ≥ 4, and let S be a compact, simply connected Lie group whose maximal torus is of dimension at least two (for example, S := SU(2) × SU(2) ≅ S^{3} × S^{3}). Then there exist continuous isospectral families of Riemannian metrics on S^{n} × S which are not locally isometric; more precisely, the set of critical values of the corresponding scalar curvature functions changes during the deformations.

In particular, we obtain such families of metrics on S^{n} ×S^{3} ×S^{3} for every n ≥ 4. These isospectral deformations can take place arbitrarily close to the standard metric on S^{n} × S^{3} × S^{3} (although none of our deformations is one of the standard metric itself).

Our construction is related to, and motivated by, the isospectral examples in [GGSWW], where the underlying manifolds are products of a sphere with a torus. The idea is to embed the non-simply connected factor, that is, the torus, into a compact, simply connected Lie group and to extend the metrics to the bigger manifolds in such a way that they remain isospectral.

Concerning the change in geometry during the deformations, it turns out that in many cases, the isospectral manifolds can be distinguished by the set of critical values assumed by the scalar curvature function. Note that this function is in general nonconstant here since the manifolds are not locally homogeneous.

Moreover, our isospectral deformations are in general not isospectral for the Laplace operator acting on 1-forms. More precisely, the manifolds can be distinguished already by the heat invariants of the Laplacian acting on 1-forms. This is interesting in several respects: There are only few examples known of manifolds which are isospectral on functions, but not on 1-forms. The only known examples are those given in [Ik] (lens spaces), [Go3] (Heisenberg manifolds), and [Gt1,2] (certain three-step nilmanifolds). Among these, only Gornet’s examples include continuous families of such manifolds [Gt2]. In all these examples, the isospectral manifolds arise as different quotients of some common Riemannian covering. In particular, these isospectral manifolds, even though they are not isospectral on 1-forms, always share the same heat invariants of the Laplacian acting on 1-forms. The isospectral families constructed in the present paper are the first ones where this is not the case.

We will construct a large class of continuous isospectral families in which the manifolds can be distinguished by either of the two criteria mentioned above (critical values of the scalar curvature, resp. spectrum on 1-forms).

The paper is organized as follows:

In section one, we introduce the Laplace operator and a general isospectrality principle (Proposition 1.2) which was discovered by C. Gordon [Go2] and is the key of the isospectrality proof in section two, as it was the case also for the manifolds constructed in [GW3] and [GGSWW]. We then prepare our construction of isospectral manifolds by introducing a certain class of metrics g_{j} on ℝ^{n+1} × S ⊃ S^{n} × S, where S is a compact, simply connected Lie group, and discussing some of their properties. These metrics are related to left invariant metrics on two-step nilpotent Lie groups diffeomorphic to ℝ^{n+1} × H, where H is a closed abelian subgroup of S. Each metric g_{j} is associated with a linear map j : ℝ^{r} → so(n + 1), where r is the dimension of H.

In section two, we consider the submanifolds S^{n} × S of ℝ^{n+1} × S, endowed with the restricted metrics, and show that under a certain condition on j and j′ which was introduced in [GW3], the Riemannian manifolds (S^{n} × S, g_{j}) and (S^{n} × S, g_{j′}) are isospectral (Theorem 2.2). We finish this section by giving a class of examples of continuous families of isospectral metrics in the case n = 4.

In section three, we study the geometry of the isospectral manifolds constructed in section two. We establish a formula for the scalar curvature function on these manifolds (Lemma 3.3). This allows us to describe the sets of critical points and critical values of the scalar curvature in terms of eigenvectors and eigenvalues of the Ricci tensor of the corresponding nilpotent Lie group which we mentioned above (Corollary 3.4) and thus to obtain a sufficient condition for non-isometry of two isospectral manifolds (S^{n} × S, g_{j}) and (S^{n} × S, g_{j′}).

In section four, we show that for n ≥ 4, any generic linear map j : ℝ^{2} → so(n+1) is contained in a continuous family j(t) of such maps with the property that the corresponding manifolds (S^{n} ×S, g_{j(t)}) are isospectral and pairwise satisfy the nonisometry condition established in section three. More precisely, the manifolds have pairwise different sets of critical values of the scalar curvature. The families j(t) are obtained as the flow lines of a certain vector field on the space of all linear maps from ℝ^{2} to so(n + 1).

Section five is devoted to the spectrum on 1-forms. Using the heat invariants for the Laplace operator on 1-forms and on functions, we show that in each of the special families constructed in section four, the spectrum on 1-forms changes during the deformation.

The author wishes to thank Carolyn Gordon and Ruth Gornet for interesting discussions. Concerning the 1-form spectrum, she also wishes to acknowledge that Ruth Gornet was the originator of the idea to study the heat invariants for the form spectra of the manifolds constructed in [GGSWW], thereby also inspiring section five of the present paper.

§1 Preliminaries

1.1 Definition. Let (M, g) be a compact Riemannian manifold, and let Δ_{g} be the Laplacian acting on functions by

(Δ_{g}f)(p) := − ∑_{i=1}^{n} \frac{d^{2}}{dt^{2}} |t = 0 f(c_{i}(t)) for p ∈ M,

where the c_{i} are geodesics starting in p such that {ċ_{1}(0), …, ċ_{n}(0)} is an orthonormal basis for T_{p}M. The discrete sequence 0 = λ_{0} < λ_{1} ≤ λ_{2} ≤ … → ∞ of the eigenvalues of Δ_{g}, counted with the corresponding multiplicities, is called the spectrum of (M, g); we will denote it by spec(M, g) or spec(Δ_{g}). If H ⊆ L^{2}(M, g) is a subspace of functions which is invariant under Δ_{g}, we will denote the corresponding spectrum of eigenvalues by spec(H). Two compact Riemannian manifolds are said to be isospectral if their spectra coincide.

In the following Proposition we formulate some version of a principle which was observed by C. Gordon [Go2] and is the key of the isospectrality proofs also for the manifolds constructed in [GW3] and [GGSWW].

Let T be a compact abelian Lie group, i.e., a torus, and let L : T × M → M be a smooth free action of T on a compact Riemannian manifold (M, g) by isometries. There is a unique Riemannian metric, denoted g^{T} , on the quotient manifold T \M such that the canonical projection π_{T} : (M, g) → (T \M, g^{T}) becomes a Riemannian submersion.

1.2 Proposition. Let (M, g) and (M′, g′) be two closed Riemannian manifolds, and let L : T × M → M and L′ : T × M′ → M′ be smooth free actions of a torus T by isometries. Assume that for every closed subgroup W of T which is either T itself or a subtorus of codimension 1 in T, the following holds:

(i) The fibers of the projections π_{W} : M → W \M and π′_{W} : M′ → W \M′ are totally geodesic submanifolds of (M, g) and (M′, g′), respectively.

(ii) The Riemannian manifolds (W \M, g^{W}) and (W \M′, g^{′ W}) are isospectral. Then (M, g) and (M′, g′) are isospectral.

Proof. Write T = ℝ^{r}/L, where L is a lattice of full rank in ℝ^{r}. For every z̄ in T choose z ∈ ℝ^{r} such that z̄ = z + L. Consider the unitary representation ρ of T on L^{2}(M, g) defined by ρz̄f = f ∘ Lz̄ for all z̄ ∈ T, f ∈ L^{2}(M, g). Since T is abelian, L^{2}(M, g) decomposes as an orthogonal sum ⨁_{μ∈L^{∗}} H_{μ}, where L^{∗} is the dual lattice of L, and

H_{μ} = {f ∈ L^{2}(M, g) | ρz̄f = e^{2πiμ(z)}f for all z̄ ∈ T }.

The spaces H_{μ} are invariant under Δ := Δ_{g} because T acts by isometries. Note that π_{T}^{∗} is a linear bijection from L^{2}(T \M, g^{T}) to H_{0} . Since π_{T} is a Riemannian submersion with totally geodesic fibers by (i), π_{T}^{∗} intertwines the corresponding Laplacians. Thus spec(H_{0}) = spec(T \M, g^{T}). Assumption (ii) for W = T implies, with the obvious analogous notations for (M′, g′), that spec(H_{0}) = spec(H′_{0}).

Now let W be a closed subgroup of codimension 1 in T. Denote by W̃ the corresponding subspace of ℝ^{r}. Then π_{W}^{∗} is a linear bijection from L^{2}(W \M, g^{W}) to ⨁\begin{matrix}μ∈L^{∗} \\ μ|W̃=0 \end{matrix} H_{μ} = H_{0} ⊕⨁\begin{matrix}μ∈L^{∗} \\ ker μ=W̃ \end{matrix} H_{μ}. From assumptions (i) and (ii) for W and the

fact that spec(H_{0}) = spec(H′_{0}) we conclude, using the same argument as before, that

spec ( ⨁_{\begin{matrix}μ∈L^{∗} \\ ker μ=W̃ \end{matrix}} H_{μ}) = spec ( ⨁_{\begin{matrix}μ∈L^{∗} \\ ker μ=W̃ \end{matrix}} H′_{μ}).

Since L^{∗} \ {0} is the disjoint union of the sets {μ ∈ L^{∗} | ker μ = W̃}, where W runs through the set of all subtori of codimension 1 in T, we conclude spec(M, g) = spec(M′, g′). □

1.3 Definitions and Notation.

(i) For positive integers m and r let 𝔳 := ℝ^{m} and 𝔷 := ℝ^{r}. Throughout this paper, we will assume that 𝔳 and 𝔷 are equipped with the euclidean standard metric. Moreover, j will always denote a linear map from 𝔷 to so(𝔳). With every such map j we associate a two-step nilpotent Lie algebra 𝔤_{j} = 𝔳⊕𝔷 whose Lie bracket is defined by ⟨z, [x, y]⟩ = ⟨j_{z}x, y⟩ for all x, y ∈ 𝔳, z ∈ 𝔷. Every two-step nilpotent Lie algebra arises in this way. Let G_{j} be the simply connected Lie group with Lie algebra 𝔤_{j} . The Lie group exponential map exp : 𝔤_{j} → G_{j} is a diffeomorphsim. We write the elements of G_{j} as (x, z) := exp(x + z) with x ∈ 𝔳 and z ∈ 𝔷. With this notation, multiplication in G_{j} is given by

(1)  (x, z) · (y, w) = (x + y, z + w + \frac{1}{2}[x, y]).

(ii) Let S be a simply connected compact Lie group whose maximal torus is of dimension not less than r. Denote the Lie algebra of S by 𝔰. Let H ⊂ S be an r-dimensional closed abelian subgroup with Lie algebra 𝔥 ⊂ 𝔰. Let S be equipped with some fixed biinvariant metric k, and consider the corresponding scalar product on 𝔰. Choose a linear bijection f : 𝔷 → 𝔥 which is an isometry with respect to the induced metric on 𝔥 and the standard metric on 𝔷. Define

E := 𝔳 × S,

and define actions of G_{j} on E from the left and of S on E from the right by

(2)  L_{(x,z)}(v, s) := (x + v, exp(f(z + \frac{1}{2}[x, v])) · s), R_{t}(v, s) := (v, st).

Here exp denotes the Lie group exponential map from 𝔰 to S. Using the multiplication law (1), one easily verifies that L is indeed a group action. Note that these actions of G_{j} and S commute. Let D_{j} := L_{G_{j}} · R_{S} . Then D_{j} is a transitive group of diffeomorphisms of E.

1.4 Lemma. If φ ∈ D_{j} fixes (0, e) ∈ 𝔳×S = E, then φ_{∗ (0,e)} is of the form (Id, ψ), where ψ = Ad_{h} for some h ∈ H.

Proof. Let φ = L_{(x,z)} · R_{s}. If φ fixes (0, e) then x = 0 and exp(f(z)) = s^{−1}. Thus s ∈ H, and φ acts on E by (x, t) ↦ (x, s^{−1}ts). Therefore φ_{∗ (0,e)} = (Id, Ad_{s}^{−1}). □

1.5 Definition. Define a metric g on T_{(0,e)}E = 𝔳 ⊕ 𝔰 by letting 𝔳 and 𝔰 be orthogonal and requiring that the induced metric on 𝔳, resp. 𝔰, be the standard metric on 𝔳, resp. the scalar product corresponding to the biinvariant metric k. Let g_{j} be the Riemannian metric on E which is invariant under D_{j} and coincides with g on T_{(0,e)}E.

Remark. Since k corresponds to a biinvariant metric on S, inner automorphisms of 𝔰 are orthogonal with respect to k. Thus Lemma 1.4 implies that g_{j} is well-defined.

1.6 Lemma. For all x ∈ 𝔳 and s ∈ S, the submanifolds {x} × S and {x} × Hs are totally geodesic in (E, g_{j}). The same holds for {x} × W s, where W is any closed subgroup of H.

Proof. Consider the involution σ : E → E given by σ(x, s) = (−x, s). Obviously σ_{∗ (0,e)} is an isometry of the metric g on T_{(0,e)}E. Moreover, σ ∘ R_{s} = R_{s} ∘ σ for all s ∈ S, and one easily checks by (2) that σ ∘ L_{(x,z)} = L_{(−x,z)} ∘ σ for all (x, z) ∈ G_{j} . These equations imply that σ^{∗}g_{j} is invariant under D_{j} and therefore equal to g_{j} . Thus σ is an isometry of (E, g_{j}); hence its fixed point set {0}×S is totally geodesic. Note that the metric induced on {0} × S by g_{j} corresponds to the metric k on S under the canonical bijection. Since H is totally geodesic in S with respect to this metric, the set {0} × H is totally geodesic in (E, g_{j}). Moreover, {0} × H is a flat torus. Hence any subtorus {0} × W corresponding to a closed subgroup W of H is totally geodesic in {0} × H and thus also in (E, g_{j}). For x ∈ 𝔳 and s ∈ S, the isometry L_{(x,0)} ∘ R_{s} carries {0} × S, {0} × H, and {0} × W to {x} × S, {x} × Hs, and {x} × W s, respectively, whence the statement follows. □

The following lemma is immediate from (2). Part (i) describes the g_{j}-orthogonal splitting of the tangent space T_{(x,s)}E at an arbitrary point (x, s) ∈ E into horizontal and vertical parts with respect to the projection E = 𝔳 × S → 𝔳.

1.7 Lemma. (i) For every (x, s) ∈ E, the g_{j}-isometry (L_{(x,0)} ∘ R_{s})_{∗} : T_{(0,e)}E = 𝔳 ⊕ 𝔰 → T_{(x,s)}E maps v ∈ 𝔳 to v_{(x,s)}^{j} := (v, r_{s∗}f(\frac{1}{2}[x, v])) and u ∈ 𝔰 to u_{(x,s)}^{j} := (0, r_{s∗}u), where r_{s} denotes right multiplication by s in S.

(ii) For v ∈ 𝔳 and h ∈ 𝔥 ⊂ 𝔰, the vector fields v^{j} and h^{j} are invariant under the group D_{j} = L_{G_{j}} · R_{S} . For a vector u in the orthogonal complement 𝔲 of 𝔥 in 𝔰, the vector field u^{j} is invariant under R_{S} , but in general not under L_{G_{j}} . However, the distribution 𝔲_{(x,s)} = {u_{(x,s)}^{j} | u ∈ 𝔲} is invariant under both actions.

Note that for v ∈ 𝔳, the vector field v^{j} on E depends on j since [ , ] does so, while u^{j} is actually independent of j for u ∈ 𝔰. We use the superscript just to distinguish between the vector field u^{j} on E and the vector u ∈ 𝔰.

1.8 Definition. Let C := S^{1}(𝔳) × S ⊂ E, where S^{1}(𝔳) is the unit sphere in 𝔳. By g_{j} we will in the following denote the above metric on E as well as its restriction to C.

1.9 Remark. Concerning the geometry of the manifolds (C, g_{j}), note that the canonical projection from (C, g_{j}) to its first factor S^{1}(𝔳), endowed with the round sphere metric, is a Riemannian submersion with totally geodesic fibers which are isometric to the compact Lie group (S, k). However, the inclusion S^{1}(𝔳) → S^{1}(𝔳) × {e} ⊂ (C, g_{j}) is not isometric, nor is its image orthogonal to the fibers {x} × S of the above submersion.

§2 The isospectral metrics

2.1 Definition ([GW3]). Two linear maps j, j′ : 𝔷 → so(𝔳) are called isospectral, denoted j ∼ j′, if for every z ∈ 𝔷 there exists an orthogonal map A_{z} : 𝔳 → 𝔳 such that j′_{z} = A_{z}j_{z}A_{z}^{−1}.

2.2 Theorem. If j ∼ j′ then (C, g_{j}) and (C, g_{j′}) are isospectral.

Proof. Let L be the set of those z ∈ 𝔷 for which L_{(0,z)} = Id. Thus L = f^{−1}{h ∈ 𝔥 | exp h = e} in the notation 1.3(ii). Since H is a torus, L is a lattice of full rank in 𝔷. The linear isometry f : 𝔷 → 𝔥 induces an isometry f̄ between the flat tori T := 𝔷/L and H. The torus T acts on E by Lz̄ := L_{(0,z)}, where z̄ = z + L ∈ T. Obviously this action is smooth, free, and by isometries with respect to g_{j} . The same holds for its restriction to the submanifold C which is invariant under the action of T. The orbits are of the form {v} × Hs ⊂ C with v ∈ S^{1}(𝔳) and s ∈ S, and the orbits of the action of any closed subgroup W of T have the form {v}×Us, where U = f̄(W) is the corresponding subtorus of H. By Lemma 1.6 these orbits are totally geodesic in (E, g_{j}) and thus also in (C, g_{j}).

Since these facts are true also with respect to g_{j′}, we are in the situation of Proposition 1.2 with condition (i) satisfied. Our goal now is to show that j ∼ j′ implies that also condition (ii) of the proposition is satisfied. We will actually show that the various pairs of quotients (W \C, g_{j}^{W}) and (W \C, g_{j′}^{W}) are not only isospectral here, but even isometric.

First consider W = T. Denote the Lie bracket of 𝔤_{j′} by [ , ]′. Then for all x, v ∈ 𝔳 and s ∈ S the difference between the vectors v_{(x,s)}^{j} and v_{(x,s)}^{j′} ∈ T_{(x,s)}E equals (0, r_{s∗}f(\frac{1}{2}[x, v]− \frac{1}{2}[x, v]′)) and thus is tangent to the L_{T} -orbit through (x, s). Lemma 1.7 now implies that Id : E → E induces an isometry between the quotient manifolds (T \E, g_{j}^{T}) and (T \E, g_{j′}^{T}). By restriction, Id : C → C induces the required isometry between (T \C, g_{j}^{T}) and (T \C, g_{j′}^{T}).

Now let W ⊂ T be a closed subgroup of codimension 1. Denote by W̃ the corresponding subspace of 𝔷, and choose z ∈ 𝔷 \ {0} such that z ⊥ W̃. Since j ∼ j′, there exists an orthogonal map A_{z} : 𝔳 → 𝔳 such that j′_{z} = A_{z}j_{z}A_{z}^{−1}. Define a diffeomorphism F_{z} : E → E by F_{z}(x, s) = (A_{z}x, s). For u ∈ 𝔰, the vector field u^{j} = u^{j′} from Lemma 1.7 is obviously carried to itself by F_{z∗}. Since the tangent space to the L_{W} -orbit through (x, s) ∈ E is just {u_{(x,s)}^{j} | u ∈ f(W̃)}, the map F_{z} carries L_{W} -orbits to L_{W} -orbits. Moreover, for every v ∈ 𝔳 we have

F_{z∗}_{|_{(x,s)}}(v^{j}) = (A_{z}v, r_{s∗}f(\frac{1}{2}[x, v]))  and
(A_{z}v)^{j′}_{|_{F_{z}(x,s)}} = (A_{z}v, r_{s∗}f(\frac{1}{2}[A_{z}x, A_{z}v]′)).

The choice of A_{z} implies that ⟨z, [x, v] − [A_{z}x, A_{z}v]′⟩ = ⟨j_{z}x, v⟩ − ⟨j′_{z}A_{z}x, A_{z}v⟩ = 0. Thus the difference between F_{z∗}_{|_{(x,s)}}(v^{j}) and (A_{z}v)^{j′}_{|_{F_{z}(x,s)}} is orthogonal to (0, r_{s∗}f(z)) and therefore tangent to the L_{W} -orbit through F_{z}(x, s). Using Lemma 1.7 we conclude that F_{z} induces an isometry between the quotients (W \E, g_{j}^{W}) and (W \E, g_{j′}^{W}). Since F_{z} preserves C, this isometry restricts to the required isometry between (W \C, g_{j}^{W}) and (W \C, g_{j′}^{W}). □

2.3 Remark. In our notation, the bounded isospectral manifolds constructed in [GW3] are of the form (B^{1}(𝔳) × H, g_{j}), resp. (B^{1}(𝔳) × H, g_{j′}), where B^{1}(𝔳) denotes the unit ball in 𝔳. Similarly, the isospectral manifolds in [GGSWW] are of the form (S^{1}(𝔳) × H, g_{j}), resp. (S^{1}(𝔳) × H, g_{j′}).

2.4 Remarks. (i) If there exists an orthogonal map A : 𝔳 → 𝔳 such that j′_{z} = Aj_{z}A^{−1} for all z ∈ 𝔷 then the isospectrality condition from Definition 2.1 is trivially satisfied. In this situation the manifolds (C, g_{j}) and (C, g_{j′}) are isometric; an isometry from the first to the latter is given by (x, s) ↦ (Ax, s). For example, this is always the case if dim 𝔷 = 1.

(ii) However, we will show that in case dim 𝔷 = 2 and dim 𝔳 ≥ 5, any generic linear map j : 𝔷 → so(𝔳) is contained in a continuous isospectral family of maps j(t) such that the manifolds (C, g_{j(t)}) are not pairwise isometric. Here, generic means that j is an element of a certain Zariski open subset of the space of all linear maps from 𝔷 to so(𝔳). Note that a similar genericity result, although excluding the case dim 𝔳 = 6, was obtained in [GW3, Theorem 2.2 and Proposition 1.4] and [GGSWW, Propositions 9 and 10] with respect to the isospectral manifolds considered there. That result was stronger than the one we prove here in the sense that it included the existence of multi-parameter families of isospectral metrics. On the other hand, our result includes the case dim 𝔳 = 6 in which the previously used arguments had failed.

(iii) For dim 𝔳 ≤ 4 and dim 𝔷 = 2, it is not hard to show that, as mentioned in [GW3], the condition j ∼ j′ can be satisfied only trivially (in the sense of (i) above). For dim 𝔳 ≤ 4 and dim 𝔷 ≥ 3 it is still possible to show that there are no continuous families of nontrivially isospectral maps j(t) : 𝔷 → so(𝔳) (although a pair of such maps is given in [GW3, Example 1.10]). Thus dim 𝔳 = 5 and dim 𝔷 = 2 is indeed the minimal choice of dimensions for which nontrivial continuous isospectral families j(t) : 𝔷 → so(𝔳) exist.

(iv) The lowest dimensional compact, simply connected Lie group S whose maximal torus has dimension r is the direct product of r copies of SU(2). For this choice of S, the manifold C is diffeomorphic to the product of S^{m−1} with r copies of S^{3}. By the previous remark, the lowest dimensional example of this form on which our method produces nontrivial, continuous families of isospectral metrics is S^{4} × S^{3} × S^{3}.

2.5 Example. This example is a modified version of Example 2.3 from [GW3]. The modification mainly consists in lowering the dimension of 𝔳 by one. Consider 𝔳 = ℝ^{5} and 𝔷 = ℝ^{2}. Let a, b ∈ so(𝔳) be given as matrices with respect to the standard basis by

a = ( \begin{matrix}0 & −a_{1} & 0 & 0 & 0 \\ a_{1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −a_{2} & 0 \\ 0 & 0 & a_{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ) , b = ( \begin{matrix}0 & 0 & b_{12} & 0 & b_{13} \\ 0 & 0 & 0 & 0 & 0 \\ −b_{12} & 0 & 0 & 0 & b_{23} \\ 0 & 0 & 0 & 0 & 0 \\ −b_{13} & 0 & −b_{23} & 0 & 0 \end{matrix} ) .

Define a linear map j_{a,b} : 𝔷 → so(𝔳) by letting j_{a,b}(z_{1}) = a and j_{a,b}(z_{2}) = b, where {z_{1}, z_{2}} is the standard basis of 𝔷. The characteristic polynomial det(λId−(sa+ub)) is equal to λ^{5}+(s^{2}(a_{1}^{2}+a_{2}^{2})+u^{2}(b_{12}^{2}+b_{13}^{2}+b_{23}^{2}))λ^{3}+(s^{4}a_{1}^{2}a_{2}^{2}+s^{2}u^{2}(a_{1}^{2}b_{23}^{2}+a_{2}^{2}b_{13}^{2}))λ. Thus j_{a,b} ∼ j_{a,b′} if and only if

b_{12}^{2} + b_{13}^{2} + b_{23}^{2} = b_{12}^{′ 2} + b_{13}^{′ 2} + b_{23}^{′ 2} and a_{1}^{2}b_{23}^{2} + a_{2}^{2}b_{13}^{2} = a_{1}^{2}b_{23}^{′ 2} + a_{2}^{2}b_{13}^{′ 2} .

Now assume 0 < a_{1} < a_{2} and b_{12} , b_{13} , b_{23} ≥ 0. Then with

b_{12}(t) := \sqrt{b_{12}^{2} + t(a_{2}^{2} − a_{1}^{2})} , b_{13}(t) := \sqrt{b_{13}^{2} + ta_{1}^{2}} , b_{23}(t) := \sqrt{b_{23}^{2} − ta_{2}^{2}} , the maps j_{a,b} and j_{a,b(t)} satisfy conditions (i) and (ii) for every t ∈ [max\{\frac{−b_{12}^{2}}{a_{2}^{2}−a_{1}^{2}} , \frac{−b_{13}^{2}}{a_{1}^{2}} \},

\frac{b_{23}^{2}}{a_{2}^{2}} ]. This interval has nonzero length if b_{23} > 0 or b_{12} , b_{13} > 0, and j(t) := j_{a,b(t)} is a continuous family of pairwise isospectral linear maps. This family is nontrivial; i.e., the maps j(t) are not pairwise equivalent in the sense of Remark 2.4(i). This can be seen, for example, by noting that the maps (j_{z_{1}}^{2} + j_{z_{2}}^{2})(t) = a^{2} + b(t)^{2} are not pairwise similar, since routine computation shows that their determinant is a linear polynomial in t with nonzero leading coefficient a_{1}^{4}a_{2}^{4}(a_{1}^{2} − a_{2}^{2}). In the next section, we will interpret the non-similarity of the various (j_{z_{1}}^{2} + j_{z_{2}}^{2})(t) geometrically in terms of the scalar curvature of the isospectral manifolds (C, g_{j(t)}) (see formula (10) and Proposition 3.5).

§3 Scalar curvature and a sufficient condition for non-isometry

In this section we will establish a sufficient condition for two manifolds (C, g_{j}) and (C, g_{j′}) (with j ∼ j′) to be non-isometric. We will show that the difference in geometry can in general be detected by studying the behaviour of the scalar curvature. The manifolds (C, g_{j}) are in general not locally homogeneous, and the scalar curvature function is in general nonconstant on a given manifold (C, g_{j}). In Proposition 3.5, we will formulate a condition on two isospectral linear maps j, j′ : 𝔷 → so(𝔳) which will imply that (C, g_{j}) and (C, g_{j′}) have different sets of critical values of the scalar curvature function. In section four we will then show that in case dim 𝔷 = 2 and dim 𝔳 ≥ 5, any generic linear map j : 𝔷 → so(𝔳) admits a continuous isospectral deformation j(t) such that the maps j(t) pairwise satisfy this condition.

We need some preparations in order to compute the scalar curvature of (C, g_{j}). In the following, let G_{j} be endowed with the left invariant metric m_{j} induced by the standard scalar product on 𝔤_{j} = 𝔳 ⊕ 𝔷.

3.1 Lemma. The map ℓ : G_{j} ∋ (x, z) ↦ L_{(x,z)}(0, e) = (x, e^{f(z)}) ∈ L_{G_{j}}(0, e) ⊂ (E, g_{j}) is a local isometry. Moreover, the submanifold L_{G_{j}}(0, e) is totally geodesic in (E, g_{j}); consequently, this holds also for its right translates under R_{S} .

Proof. Two elements of G_{j} have the same image under ℓ if and only if they differ by an element of exp L, where L is the central lattice f^{−1}{h ∈ 𝔥 | exp h = e} ⊂ 𝔷 as in the proof of Theorem 2.2. Thus ℓ is a local diffeomorphism. For v ∈ 𝔳, the left invariant vector field on G_{j} induced by v is mapped by ℓ_{∗} to v^{j}_{|_{L_{G_{j}} (0,e)}} , and for z ∈ 𝔷, the corresponding left invariant vector field on G_{j} is mapped by ℓ_{∗} to h^{j}_{|_{L_{G_{j}} (0,e)}} , where h = f(z). Lemma 1.7 now implies that ℓ is a local isometry from (G_{j}, m_{j}) to L_{G_{j}}(0, e).

To see that L_{G_{j}}(0, e) is totally geodesic, first note that its tangent space at (0, e) is 𝔳 ⊕ 𝔥 ⊂ 𝔳 ⊕ 𝔰 = T_{(0,e)}E, and L_{G_{j}}(0, e) is the integral manifold through (0, e) of the distribution {v^{j} + h^{j} | v ∈ 𝔳, h ∈ 𝔥}. In particular, for all x, v ∈ 𝔳 and every u ∈ 𝔲, where 𝔲 denotes the orthogonal complement of 𝔥 in 𝔰, we have

(3)  ⟨[x^{j}, v^{j}]_{(0,e)} , u⟩ = 0.

Extend every u ∈ 𝔲 ⊂ T_{(0,e)}E to a vector field ũ on E whose flow F t ; ũ consists of the right translations R_{e^{tu}} from 1.3(ii), and consider the vector field v^{j} associated with v ∈ 𝔳 as in Lemma 1.7. One easily checks that the flow F_{v^{j}}^{t} commutes with the maps R_{e^{tu}}, and thus

(4)  [v^{j}, ũ] = 0.

Moreover, for x, v ∈ 𝔳 the function ⟨x^{j}, v^{j}⟩ is constant on E by Lemma 1.7, and ⟨x^{j}, ũ⟩ = ⟨v^{j}, ũ⟩ = 0 on E. By the Koszul formula for the Levi-Cività connection ∇ on (E, g_{j}), this implies together with (3) and (4) that

(5)  ⟨∇xv^{j}, u⟩ = 0.

We know that ∇hh^{j} = 0 for h ∈ 𝔥, since h^{j} is tangent to the flat submanifold {0} × H which is totally geodesic by Lemma 1.6. Finally, for all v ∈ 𝔳, h ∈ 𝔥, and u ∈ 𝔲, we have ⟨∇vh^{j}, u⟩ = ⟨∇σ_{∗}vσ_{∗}h^{j}, σ_{∗}u⟩ = ⟨∇−vh^{j}, u⟩, where σ : E ∋ (x, s) ↦ (−x, s) ∈ E is the involutive isometry from the proof of 1.6. Thus ⟨∇vh^{j}, u⟩ = 0, and similarly ⟨∇hv^{j}, u⟩ = 0. In summary, ∇yy^{j} ∈ 𝔳 ⊕ 𝔥 for every y ∈ 𝔳 ⊕ 𝔥, whence L_{G_{j}}(0, e) is totally geodesic. □

Denote the scalar curvatures of (E, g_{j}), (G_{j}, m_{j}), and (S, k) by scal^{E,j}, scal^{G_{j}}, and scal^{S}, respectively. Since all three manifolds are homogeneous, their scalar curvatures are constant.

3.2 Lemma.

scal^{E,j} = scal^{G_{j}} + scal^{S}.

Proof. We compute the scalar curvature in the point (0, e) ∈ E. Recall the orthogonal splitting T_{(0,e)}E = 𝔳 ⊕ 𝔥 ⊕ 𝔲 from the proof 3.1. Since 𝔳 ⊕ 𝔥 and 𝔥 ⊕ 𝔲 = 𝔰 are the tangent spaces to L_{G_{j}}(0, e) and {0} × S which are both totally geodesic by 3.1 and 1.6, and since the sectional curvature of planes tangent to 𝔥 is zero, it suffices to show that

(6)  K(v, u) = 0

for all v ∈ 𝔳 and u ∈ 𝔲, where K denotes the sectional curvature on (E, g_{j}).

Using again the involutive isometry σ : E ∋ (x, s) ↦ (−x, s) ∈ E, we have ⟨∇uv^{j}, w⟩ = ⟨∇u(−v^{j}), w⟩ = 0 for every w ∈ 𝔰. Thus ∇uv^{j} ∈ 𝔳. But the same argument which led to (5) also shows ⟨∇uv^{j}, x⟩ = 0 for every x ∈ 𝔳. Hence ∇uv^{j} = 0, and by Lemma 1.7,

(7)  ∇u^{j} v^{j} = 0 for all u ∈ 𝔲, v ∈ 𝔳.

Since u^{j} equals ũ along the curve t ↦ (tv, e), where ũ is the vector field from the proof of 3.1, we get ∇v∇ũv^{j} = 0. By (4) we also know ∇[ũ, v^{j}]v^{j} = 0 and

thus R(u, v)v = ∇u∇v^{j} v^{j}. Note that ∇v^{j} v^{j} = 0 by Lemma 3.1 and the fact that ∇vv = −^{t}ad_{v}v = 0 for the corresponding left invariant vector field on (G_{j}, m_{j}), since v ⊥ [𝔤_{j}, 𝔤_{j}]. Hence R(u, v)v = 0, and the statement follows. □

We can now compute the scalar curvature of the submanifold (C, g_{j}) of (E, g_{j}):

3.3 Lemma. The scalar curvature scal^{C,j} of (C, g_{j}) at the point (v, s) ∈ C is given by

scal_{(v,s)}^{C,j} = scal^{G_{j}} + scal^{S} + (m − 1)(m − 2) − Ric^{G_{j}}(v, v),

where m = dim(𝔳), and scal^{G_{j}} and Ric^{G_{j}} denote the scalar curvature and the Ricci tensor of (G_{j}, m_{j}).

Proof. Let N be the outward unit normal field to (C, g_{j}) in (E, g_{j}). A standard computation involving only the Gauss equations and the fact that (C, g_{j}) is a codimension one submanifold of (E, g_{j}) shows that

(8)  scal_{p}^{C,j} = scal^{E,j} − 2Ric(N_{p} , N_{p}) + (tr(∇N_{|_{T_{p}C}}))^{2} − ∥∇N_{|_{T_{p}C}}∥^{2}

for all p ∈ C, where ∇ and Ric denote the Levi-Cività connection and the Ricci tensor of (E, g_{j}), and ∥ . ∥ denotes the euclidean norm on tensors. Note that for p = (v, s), the tangent space T_{p}C is the orthogonal sum of {x_{(v,s)}^{j} | x ∈ 𝔳, x ⊥ v} and {u_{(v,s)}^{j} | u ∈ 𝔰}. Thus N_{(v,s)} = v_{(v,s)}^{j}. Moreover, one easily checks that for each x ⊥ v in 𝔳 and each u ∈ 𝔰, we have

∇x_{(v,s)}^{j}N = x_{(v,s)}^{j} + ∇x_{(v,s)}^{j}v^{j} , ∇u_{(v,s)}^{j}N = ∇u_{(v,s)}^{j}v^{j}.

Recall that L_{G_{j}}-orbits are totally geodesic in (E, g_{j}) by Lemma 3.1. Since x^{j} , v^{j} , and h^{j} for h ∈ 𝔥 are tangent to these, we get for all x ∈ 𝔳 and h ∈ 𝔥, with z := f^{−1}(h) ∈ 𝔷,

∇x_{(v,s)}^{j}N = x_{(v,s)}^{j} + (∇x^{G_{j}}v)_{(v,s)}^{j} , ∇h_{(v,s)}^{j}N = (∇z^{G_{j}}v)_{(v,s)}^{j} ,

whereas ∇u^{j} N = 0 for u ∈ 𝔲 = 𝔥^{⊥} ∩ 𝔰 by (7). We have (see e.g. [Eb])

(9)  ∇x^{G_{j}}v = \frac{1}{2}[x, v], ∇z^{G_{j}}v = −\frac{1}{2}j_{z}v

for all x, v ∈ 𝔳 and z ∈ 𝔷. Thus ∇^{G_{j}}v sends 𝔳 to 𝔷 and vice versa; hence

tr(∇N_{|_{T_{(v,s)}C}}) = m − 1, ∥∇N_{|_{T_{(v,s)}C}}∥^{2} = m − 1 + ∥∇^{G_{j}}v∥^{2}.

Moreover,

(10)  Ric^{G_{j}}_{|_{𝔳}} = \frac{1}{2} ∑_{i=1}^{r} j_{z_{i}}^{2} ,

where {z_{1}, …, z_{r}} is an orthonormal basis of 𝔷 (see [Eb]). One routinely derives from (9) and (10) that

∥∇^{G_{j}}v∥^{2} = −Ric^{G_{j}}(v, v).

Finally note that

Ric(v^{j}, v^{j}) = Ric^{G_{j}}(v, v)

for all v ∈ 𝔳. This follows using again Lemma 3.1 and the fact that the sectional curvatures K(v, u) vanish by (6) whenever v ∈ 𝔳 and u ∈ 𝔲 = 𝔥^{⊥} ∩𝔰. The statement is now immediate from (8) and Lemma 3.2. □

The scalar curvature function scal^{C,j} can thus be viewed as a quadratic function in the first coordinate v of (v, s) ∈ S^{1}(𝔳) × S = C. Consequently, we have:

3.4 Corollary. The set of critical points of the scalar curvature on (C, g_{j}) is given by {(v, s) ∈ C | v is an eigenvector of Ric^{G_{j}}_{|_{𝔳}}}, and the set of critical values is {scal^{G_{j}} + scal^{S} + (m − 1)(m − 2) − λ | λ is an eigenvalue of Ric^{G_{j}}_{|_{𝔳}}}.

We can now establish a sufficient condition for non-isometry of two metrics g_{j} and g_{j′} on C. Note that by the discussion in 2.5, this condition is satisfied for the families given there.

3.5 Proposition. Let j ∼ j′. If Ric^{G_{j}}_{|_{𝔳}} and Ric^{G_{j′}} _{|_{𝔳}} have different sets of eigenvalues, then scal^{C,j} and scal^{C,j′} have different sets of critical values. In particular, (C, g_{j}) and (C, g_{j′}) are not isometric. More generally, if Ric^{G_{j}}_{|_{𝔳}} and Ric^{G_{j′}} _{|_{𝔳}} do not have the same collection of eigenvalues, counted with multiplicities (i.e., if Ric^{G_{j}}_{|_{𝔳}} and Ric^{G_{j′}} _{|_{𝔳}} are not similar), then (C, g_{j}) and (C, g_{j′}) are not isometric.

Proof. The first statement follows from Corollary 3.4 by noting that j ∼ j′ implies scal^{G_{j}} = scal^{G_{j′}}. In fact, by [Eb] we have Ric^{G_{j}}(z, w) = −\frac{1}{4}tr(j_{z}j_{w}) for all z, w ∈ 𝔷. Together with (10) we get scal^{G_{j}} = \frac{1}{4} ∑_{i=1}^{r} tr(j_{z_{i}}^{2}), where \{z_{1}, …, z_{r}\} is an orthonormal basis of 𝔷. But if j ∼ j′ then obviously tr(j_{z}^{2}) = tr(j_{z}^{′ 2}) for each z ∈ 𝔷, and thus scal^{G_{j}} = scal^{G_{j′}} as claimed.

The last statement follows from the fact that the multiplicity n_{λ} of an eigenvalue λ of Ric^{G_{j}}_{|_{𝔳}} can be read off from the dimension dim S + (n_{λ} − 1) of the set of critical points in which scal^{C,j} assumes the critical value scal^{G_{j}} + scal^{S} + (m − 1)(m − 2) − λ. □

3.6 Remark. Since the total scalar curvature is a spectral invariant, (C, g_{j}) and (C, g_{j′}) must have the same total scalar curvature if j ∼ j′. This follows also from 3.3 by direct computation; note that ∫_{S^{1}(𝔳)} Ric^{G_{j}}(v, v) dv equals vol(S^{m−1}) · \frac{1}{m} · tr(Ric^{G_{j}}_{|_{𝔳}}) which coincides for j ∼ j′ by formula (10).

§4 Generic families

Throughout this section we assume dim 𝔷 = 2. As mentioned in Remark 2.4(iii), nontrivial pairs of isospectral maps j : ℝ^{2} → so(m) do not exist if m ≤ 4. However, we will show in this section that in case m ≥ 5, any generic linear map j : ℝ^{2} → so(m) is contained in a nontrivial, continuous isospectral family of linear maps j(t) : ℝ^{2} → so(m) which pairwise satisfy the non-isometry condition from Proposition 3.5. The idea is to construct the families j(t) as the (locally defined) flow lines of a smooth vector field on the space of all linear maps from ℝ^{2} to so(m).

Taking into account Theorem 2.2 and Proposition 3.5, the following proposition not only completes the proof of the Main Theorem stated in the Introduction, but also shows that there is an abundant class of examples to it.

4.1 Proposition. Let m ≥ 5, and denote by J the space of all linear maps from ℝ^{2} to so(m). Then there exists a Zariski open subset U ⊂ J such that every j ∈ U is contained in a continuous isospectral family j(t) with j(0) = j such that the function t ↦ ∥Ric^{G_{j(t)}}_{|_{𝔳}}∥^{2} has nonzero derivative at t = 0. In particular, for every j ∈ U there exists ε(j) > 0 such that for −ε(j) < t < ε(j), the isospectral manifolds (C, g_{j(t)}) have pairwise different sets of critical values of the scalar curvature function.

Proof. We write j ∈ J in the form j = (j_{1}, j_{2}), where j_{1} := j_{z_{1}} , j_{2} := j_{z_{2}} , and {z_{1}, z_{2}} is an orthonormal basis of 𝔷 = ℝ^{2}. The isospectrality condition j ∼ j′ says that for all s, t ∈ ℝ, the skew-symmetric map sj_{1} + tj_{2} has the same collection of eigenvalues, counted with multiplicities, as sj′_{1} + tj′_{2}. As mentioned in the proof of [GW3, Theorem 2.2], it is easy to see that this is equivalent to the condition tr((sj_{1}+ tj_{2})^{k}) = tr((sj′_{1} + tj′_{2})^{k}) for all even numbers k in {1, …, m}, or equivalently, for all k ∈ ℕ. Let a, b ∈ ℕ_{0} such that a + b > 0. Then the coefficient at the monomial s^{a}t^{b} in tr((sj_{1} + tj_{2})^{a+b}) equals

(11)  p_{a,b}(j) := ∑_{σ∈𝔖_{a,b}} tr(j_{σ(1)} … j_{σ(a+b)}),

where 𝔖_{a,b} denotes the set of all maps σ : {1, …, a + b} → {1, 2} which satisfy #σ^{−1}(1) = a, #σ^{−1}(2) = b. Thus we get

(12)  j ∼ j′ ⇔ p_{a,b}(j) = p_{a,b}(j′) for all a, b ∈ ℕ_{0} with a + b > 0.

On the other hand, we consider the polynomial j ↦ ∥Ric^{G_{j}}_{|_{𝔳}}∥^{2} = ∥\frac{1}{2}(j_{1}^{2} + j_{2}^{2})∥^{2} = \frac{1}{4}(tr(j_{1}^{4}) + 2tr(j_{1}^{2} j_{2}^{2}) + tr(j_{2}^{4})). The condition j ∼ j′ implies tr(j_{1}^{4}) = tr(j_{1}^{′ 4}) and tr(j_{2}^{4}) = tr(j_{2}^{′ 4}). Thus in case j ∼ j′ we have

(13)  ∥Ric^{G_{j}}_{|_{𝔳}}∥^{2} − ∥Ric^{G_{j′}} _{|_{𝔳}}∥^{2} = \frac{1}{2}(q(j) − q(j′)),

where

(14)  q(j) := tr(j_{1}^{2} j_{2}^{2}).

By Lemma 4.2 below, the algebraic vector field Y on J, given by Y (j) = (j_{1}^{3} j_{2} − j_{2} j_{1}^{3} , 0), is everywhere orthogonal to the gradient of each of the polynomials p_{a,b}, while it is not everywhere orthogonal to the gradient of q. The first property implies by (12) that the flow lines F_{Y}^{t} (j) consist of pairwise isospectral maps. By the second property and (13), the function t ↦ ∥Ric^{G_{j(t)}}_{|_{𝔳}}∥^{2} (where j(t) := F_{Y}^{t} (j)) has nonzero derivative at t = 0 for every j in the Zariski open subset U := J \ {j ∈ J | dq_{|_{j}}(Y) = 0} ≠ ∅. The last statement of the Proposition now follows by Proposition 3.5 and the fact that ∥Ric^{G_{j}}_{|_{𝔳}}∥^{2} equals λ_{1}^{2} + … λ_{m}^{2} , where λ_{1}, …, λ_{m} are the eigenvalues of Ric^{G_{j}}_{|_{𝔳}}. □

4.2 Lemma. Let p_{a,b} and q be the polynomials on the space J of all linear maps from ℝ^{2} to so(m) which were defined in (11) and (14). Then the vector field Y on J, given by Y (j) = (j_{1}^{3} j_{2} −j_{2} j_{1}^{3} , 0), satisfies dp_{a,b}(Y) = 0 for all a, b ∈ ℕ_{0} such that a + b > 0, while the polynomial j → dq_{|_{j}}(Y) does not vanish identically on J if m ≥ 5.

Proof. First note that for all a, b ∈ ℕ, we have dp_{a,0}(Y) = a·tr(j_{1}^{a−1}(j_{1}^{3} j_{2}−j_{2} j_{1}^{3})) = 0, and trivially dp_{0,b}(Y) = 0. Thus we can assume in the following that both a and b are nonzero. Let j ∈ J, and let ε = (ε_{1}, ε_{2}) ∈ T_{j}J ≅ J. Then

dp_{a,b}_{|_{j}}(ε) = ∑_{σ∈𝔖_{a,b}} ∑_{i=1}^{a+b} tr(j_{σ(1)} … j_{σ(i−1)} ε_{σ(i)} j_{σ(i+1)} … j_{σ(a+b)})

= ∑_{i=1}^{a+b} ∑_{σ∈𝔖_{a,b}} tr(ε_{σ(i)} j_{σ(i+1)} … j_{σ(a+b)} j_{σ(1)} … j_{σ(i−1)})

= (a + b) · ( ∑_{τ∈𝔖_{a−1,b}} tr(ε_{1} j_{τ(1)} … j_{τ(a+b−1)})

+ ∑_{τ∈𝔖_{a,b−1}} tr(ε_{2} j_{τ(1)} … j_{τ(a+b−1)})).

Showing dp_{a,b}(Y) = 0 thus reduces to showing that

(15) ∑_{τ∈𝔖_{a−1,b}} tr(j_{1}^{3} j_{2} j_{τ(1)} … j_{τ(a+b−1)}) = ∑_{τ∈𝔖_{a−1,b}} tr(j_{2} j_{1}^{3} j_{τ(1)} … j_{τ(a+b−1)})

for all j ∈ J. We now define a bijection φ from 𝔖_{a−1,b} to itself such that for ρ = φ(τ),

tr(j_{1}^{3} j_{2} j_{τ(1)} … j_{τ(a+b−1)}) = tr(j_{2} j_{1}^{3} j_{ρ(1)} … j_{ρ(a+b−1)}).

Namely, given τ ∈ 𝔖_{a−1,b}, let c = max{i | τ(i) = 2}; define ρ = φ(τ) by letting ρ(i) = 1 for i = 1, …, a + b − c − 1, ρ(a + b − c) = 2, and ρ(a + b − c + i) = τ(i) for i = 1, …, c − 1. Obviously φ : 𝔖_{a−1,b} → 𝔖_{a−1,b} is a bijection; moreover,

tr(j_{1}^{3} j_{2} j_{τ(1)} … j_{τ(a+b−1)}) = tr(j_{1}^{3} j_{2} j_{τ(1)} … j_{τ(c−1)} j_{2} j_{1}^{a+b−c−1})
= tr(j_{2} j_{1}^{3} j_{1}^{a+b−c−1} j_{2} j_{τ(1)} … j_{τ(c−1)})
= tr(j_{2} j_{1}^{3} j_{ρ(1)} … j_{ρ(a+b−1)}),

as required. Equation (15) now follows; hence dp_{a,b}(Y) = 0 as claimed. Moreover,

dq_{|_{j}}Y = tr((j_{1}^{3} j_{2} − j_{2} j_{1}^{3})j_{1} j_{2}^{2} + j_{1}(j_{1}^{3} j_{2} − j_{2} j_{1}^{3})j_{2}^{2}) = tr(j_{1}^{3} j_{2} j_{1} j_{2}^{2} − j_{1} j_{2} j_{1}^{3} j_{2}^{2}).

For m ≥ 5, this polynomial does not vanish identically on J; e.g., for

(16)  j_{1} = ( \begin{matrix}0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ −1 & −1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 \end{matrix} ) and j_{2} = ( \begin{matrix}0 & 1 & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} )

it equals 2 ≠ 0. □

4.3 Remarks. (i) In case m ≤ 4, we have tr(j_{1}^{3} j_{2} j_{1} j_{2}^{2} − j_{1} j_{2} j_{1}^{3} j_{2}^{2}) = 0 for all j_{1}, j_{2} in so(m), which is not hard to show. We also remark that on the other hand, this is implied by the above arguments in connection with Remark 2.4(iii).

(ii) In Example 2.5, it happens to be the case that ∥Ric^{G_{j(t)}}_{|_{𝔳}}∥^{2} is constant in t. In other words, even though det(Ric^{G_{j(t)}}_{|_{𝔳}}) is nonconstant in t (as observed in 2.5) and thus Ric^{G_{j(t)}} has nonconstant eigenvalue spectrum, not only the trace of Ric^{G_{j(t)}}_{|_{𝔳}} is constant in t (which is automatic by (10) since the j(t) are pairwise isospectral), but even the sum of the squares of its eigenvalues is constant. Hence the families given in 2.5 do not represent examples for Proposition 4.1. Examples for 4.1 would be obtained by computing flow lines of the vector field Y through suitable starting points j = (j_{1}, j_{2}), e.g., the one given by (16). However, this seems hard to do explicitly. A different explicit example of an isospectral family j(t) = (j_{1}(t), j_{2}(t)) : ℝ^{2} → so(5) with ∥Ric^{G_{j(t)}}_{|_{𝔳}}∥^{2} ≠ const, although not representing a flow line of Y, is given by

j_{1}(t) := ( \begin{matrix}0 & 0 & −t & 0 & 0 \\ 0 & 0 & 0 & t−1 & 0 \\ t & 0 & 0 & 0 & −φ(t) \\ 0 & 1−t & 0 & 0 & −ψ(t) \\ 0 & 0 & φ(t) & ψ(t) & 0 \end{matrix} ) , j_{2}(t) := ( \begin{matrix}0 & 1 & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ) ,

where φ(t) = ((t^{4} − 3t^{2} + 1)/(1 − 2t))^{1/2} and ψ(t) = ((−t^{4} + 4t^{3} − 3t^{2} − 2t + 1)/(1 −

2t))^{1/2}. This deformation is defined for t ∈ [\frac{1}{2}(1 − \sqrt{5}), \frac{1}{2}(3 − \sqrt{5})]. The j(t) are pairwise isospectral since det(λId−(sj_{1}(t)+uj_{2}(t))) = λ^{5}+(3s^{2}+2u^{2})λ^{3}+(s^{2}+u^{2})λ is independent of t. On the other hand, ∥Ric^{G_{j(t)}}_{|_{𝔳}}∥^{2} = t^{2} − t + \frac{13}{2} is nonconstant in t.

§5 The spectrum on 1-forms

Let (M, g) be a compact Riemannian manifold without boundary. The HodgedeRham Laplacian Δ_{g}^{p} acts on the space of smooth p-forms on (M, g) by Δ_{g}^{p}(α) = −(dδ +δd)α, where d denotes the exterior derivative and δ its adjoint operator with respect to the canonical scalar product on p-forms induced by g. Like Δ_{g} = Δ_{g}^{0} , the operators Δ_{g}^{p} are elliptic, positive semi-definite and self-adjoint, and have a discrete spectrum of eigenalues tending to infinity. Two manifolds (M, g) and (M′, g′) are called isospectral on p-forms if spec(Δ_{g}^{p}) = spec(Δ_{g′}^{p}).

5.1 Proposition. If j ∼ j′ and ∥Ric^{G_{j}}_{|_{𝔳}}∥^{2} ≠ ∥Ric^{G_{j′}} _{|_{𝔳}}∥^{2} then the isospectral manifolds (C, g_{j}) and (C, g_{j′}) are not isospectral on 1-forms. In particular, in each of the families (C, g_{j(t)}) from Proposition 4.1, the manifolds are not pairwise isospectral on 1-forms.

We will prove this proposition by using the heat invariants for the spectrum on functions and on 1-forms; see [Gi, §4.8] for details. The heat invariants a_{i}^{p}(g) are the coefficients appearing in the asymptotic expansion

tr(exp(−tΔ_{g}^{p})) ∼ (4πt)^{−dimM/2} ∑_{i=0}^{∞} a_{i}^{p}(g)t^{i} for t ↘ 0.

If (M, g) and (M′, g′) are isospectral on p-forms, then necessarily a_{i}^{p}(g) = a_{i}^{p}(g′) for all i ∈ ℕ_{0}. We will use this fact for the heat invariants a_{2}^{0}(g) and a_{2}^{1}(g) which are given explicitly as follows.

5.2 Lemma ([Gi, Theorem 4.8.18]). Let (M, g) be a closed Riemannian manifold, and let c_{s}(g) := ∫_{M} scal^{2}(p) dvol_{g}(p), c_{Ric}(g) := ∫_{M} ∥Ric∥_{p}^{2} dvol_{g}(p), and c_{R}(g) := ∫_{M} ∥R∥_{p}^{2} dvol_{g}(p), where scal denotes the scalar curvature, Ric and R denote the Ricci and curvature tensors of (M, g), respectively, and the norm ∥ . ∥_{p} is the euclidean norm on tensors, corresponding to the scalar product induced by g on T_{p}M. Then

(i) a_{2}^{0}(g) = \frac{1}{360}(5c_{s} − 2c_{Ric} + 2c_{R})(g),

(ii) a_{2}^{1}(g) = \frac{1}{360}((5 dimM − 60)c_{s} − (2 dimM − 180)c_{Ric} + (2 dimM − 30)c_{R})(g).

5.3 Corollary. If (M, g) and (M′, g′) are isospectral on functions and if a_{2}^{1}(g) = a_{2}^{1}(g′) (in particular, if (M, g) and (M′, g′) are isospectral both on functions and on 1-forms), then

c_{s}(g) − c_{s}(g′) + 10(c_{Ric}(g) − c_{Ric}(g′)) = 0.

Proof. Since a_{2}^{0}(g)−a_{2}^{0}(g′) = 0 by isospectrality on functions, we have by Lemma 5.2 that c_{R}(g) − c_{R}(g′) = −\frac{5}{2}(c_{s}(g) − c_{s}(g′)) + c_{Ric}(g) − c_{Ric}(g′). Substituting this into a_{2}^{1}(g) − a_{2}^{1}(g′) = 0 yields the required formula. □

The next lemma is crucial for the proof of Proposition 5.1.

5.4 Lemma. If j ∼ j′ then for the metrics g_{j} and g_{j′} on C = S^{1}(𝔳) × S the following holds:

(i) c_{s}(g_{j})−c_{s}(g_{j′}) = vol(S)·vol(S^{m−1})· \frac{2}{m(m + 2)} ·(∥Ric^{G_{j}}_{|_{𝔳}}∥^{2} −∥Ric^{G_{j′}} _{|_{𝔳}}∥^{2}),

(ii) c_{Ric}(g_{j})−c_{Ric}(g_{j′}) = vol(S)·vol(S^{m−1})· \frac{m − 4}{m} ·(∥Ric^{G_{j}}_{|_{𝔳}}∥^{2} −∥Ric^{G_{j′}} _{|_{𝔳}}∥^{2}).

Here m = dim 𝔳, and vol(S) is the volume of (S, k), where k is the biinvariant metric used in 1.3(ii).

Proof of Proposition 5.1. By Lemma 5.4,

c_{s}(g_{j}) − c_{s}(g_{j′}) + 10(c_{Ric}(g_{j}) − c_{Ric}(g_{j′}))

= vol(S) · vol(S^{m−1}) · \frac{10m^{2} − 20m − 78}{m(m + 2)} · (∥Ric^{G_{j}}_{|_{𝔳}}∥^{2} − ∥Ric^{G_{j′}} _{|_{𝔳}}∥^{2}).

This is nonzero since the third factor is nonzero for all integers m and the last factor is nonzero by assumption. Since (C, g_{j}) and (C, g_{j′}) are isospectral on functions by Theorem 2.2, Corollary 5.3 implies that a_{2}^{1}(g_{j}) ≠ a_{2}^{1}(g_{j′}); hence the two manifolds are not isospectral on 1-forms. □

Proof of Lemma 5.4. (i) By Lemma 3.3,

∫_{C}(scal_{p}^{C,j})^{2} dvol_{g_{j}}(p) = vol(S) · ∫_{S^{1}(𝔳)}(Ric^{G_{j}}(v, v)^{2} − 2φ(j)Ric^{G_{j}}(v, v) + φ(j)^{2}) dv,

where φ(j) = scal^{G_{j}} + scal^{S} + (m − 1)(m − 2) does not depend on v. Recall from the proof of 3.5 that j ∼ j′ implies φ(j) = φ(j′). Moreover, for every A ∈ End(ℝ^{m}) we have the general formulas

(17)

∫_{S^{m−1}} ⟨Av, v⟩ dv = tr(A) · ∫_{S^{m−1}} v_{1}^{2} dv = tr(A) · vol(S^{m−1}) · \frac{1}{m} ,
∫_{S^{m−1}} ⟨Av, v⟩^{2} dv = (tr(A)^{2} + ∥A∥^{2} + ⟨A,^{t}A⟩) · ∫_{S^{m−1}} v_{1}^{2}v_{2}^{2} dv

= (tr(A)^{2} + ∥A∥^{2} + ⟨A,^{t}A⟩) · vol(S^{m−1}) · \frac{1}{m(m + 2)} ,

where v_{1}, …, v_{m} are the standard coordinates on ℝ^{m}. If j ∼ j′ then tr(Ric^{G_{j}}_{|_{𝔳}}) = \frac{1}{2} ∑_{i=1}^{r} tr(j_{z_{i}}^{2}) equals tr(Ric^{G_{j′}} _{|_{𝔳}}) = \frac{1}{2} ∑_{i=1}^{r} tr(j_{z_{i}}^{′ 2}). Since Ric^{G_{j}}_{|_{𝔳}} : 𝔳 → 𝔳 is symmetric, (17) therefore implies that c_{s}(g_{j})−c_{s}(g_{j′}) = vol(S)·vol(S^{m−1})· \frac{1}{m(m+2)} · (2∥Ric^{G_{j}}_{|_{𝔳}}∥^{2} − 2∥Ric^{G_{j′}} _{|_{𝔳}}∥^{2}), as claimed.

(ii) In order to access c_{Ric}(g_{j}) we first establish explicit formulas for the Ricci tensor of (C, g_{j}), denoted Ric^{C,j}. Recall from the proof of 3.3 that for any (v, s) ∈ S^{1}(𝔳)×S = C, the tangent space T_{(v,s)}C is given as {(x^{j} +h^{j} +u^{j})_{(v,s)} | x ∈ 𝔳, x ⊥ v, h ∈ 𝔥, u ∈ 𝔲 = 𝔥^{⊥} ∩ 𝔰}, where x^{j} etc. are the vector fields defined in 1.7. Using Lemmas 3.1 and 1.6, the fact that R(u^{j}, v^{j})v^{j} = 0 in (E, g_{j}) by the proof of 3.2, the results on the shape operator ∇N from the proof of 3.3, formula (7), and the Gauss equations, one easily shows for all x, x′ ∈ 𝔳 such that x, x′ ⊥ v, all u, u′ ∈ 𝔲, all h, h′ ∈ 𝔥 and the corresponding vectors z = f^{−1}(h), z′ = f^{−1}(h′) ∈ 𝔷:

Ric_{(v,s)}^{C,j}(x^{j}, x^{′ j}) = ⟨Ric^{G_{j}}x, x′⟩ + \frac{1}{2} ⟨[x, v], [x′, v]⟩ + (m − 2)⟨x, x′⟩,
Ric_{(v,s)}^{C,j}(x^{j}, h^{j}) = \frac{m − 2}{2} ⟨j_{z}x, v⟩,
Ric_{(v,s)}^{C,j}(x^{j}, u^{j}) = 0,
Ric_{(v,s)}^{C,j}(h^{j}, h^{′ j}) = −\frac{1}{4}tr(j_{z} j_{z′}) − \frac{1}{2} ⟨j_{z}v, j_{z′}v⟩ + Ric^{S}(h, h′),
Ric_{(v,s)}^{C,j}(h^{j}, u^{j}) = Ric^{S}(h, u),
Ric_{(v,s)}^{C,j}(u^{j}, u^{′ j}) = Ric^{S}(u, u′),

where Ric^{S} denotes the Ricci tensor of (S, k). Thus

(18)

∥Ric_{(v,s)}^{C,j}∥^{2} = ∑_{a,b=1}^{m}(⟨Ric^{G_{j}}x_{a} , x_{b}⟩ + \frac{1}{2} ⟨[x_{a} , v], [x_{b} , v]⟩ + (m − 2)⟨x_{a} , x_{b}⟩)^{2}

− 2 ∑_{a=1}^{m}(⟨Ric^{G_{j}}x_{a} , v⟩ + (m − 2)⟨x_{a} , v⟩)^{2}

+ (⟨Ric^{G_{j}}v, v⟩ + (m − 2))^{2}

+ 2 ∑_{a=1}^{m} ∑_{i=1}^{r}(\frac{m − 2}{2} ⟨j_{z_{i}}x_{a} , v⟩^{2})

+ ∑_{i,k=1}^{r}(−\frac{1}{4}tr(j_{z_{i}} j_{z_{k}}) − \frac{1}{2} ⟨j_{z_{i}}v, j_{z_{k}}v⟩)^{2}

+ 2 ∑_{i,k=1}^{r}(−\frac{1}{4}tr(j_{z_{i}} j_{z_{k}}) − \frac{1}{2} ⟨j_{z_{i}}v, j_{z_{k}}v⟩) · Ric^{S}(h_{i} , h_{k})

+ ∥Ric^{S}∥^{2},

where {x_{1}, …, x_{m}} and {z_{1}, …, z_{r}} are orthonormal bases of 𝔳 and 𝔷, respectively, and h_{i} = f(z_{i}) for i = 1, …, r. By ∥Ric^{S}∥ we denote the pointwise norm of Ric^{S} which is constant since (S, k) is homogeneous; moreover, this norm does not depend on j. Note also that j ∼ j′ implies tr(j_{z} j_{w}) = tr(j′_{z} j′_{w}) as well as tr(2 j_{z}^{2} j_{w}^{2} + j_{z} j_{w} j_{z} j_{w}) = tr(2 j_{z}^{′ 2} j_{w}^{′ 2}+j′_{z} j′_{w} j′_{z} j′_{w}) for all z, w ∈ 𝔷; this follows by considering the coefficients at su, resp. s^{2}u^{2}, in tr((sj_{z} +uj_{w})^{2}), resp. tr((sj_{z} +uj_{w})^{4}). Using these facts and the formulas (10) and (17), one derives from (18) by direct computation (of which we omit the details here) that the only summands in ∫_{C} ∥Ric^{C,j}∥_{p}^{2} dvol_{g_{j}}(p) which in case j ∼ j′ do possibly not coincide for j and j′ consist of linear multiples of ∥Ric^{G_{j}}_{|_{𝔳}}∥^{2} ; moreover, the respective coefficients sum up to vol(S) · ∫_{S^{1}(𝔳)}(1 − 4v_{1}^{2} + 0 v_{1}^{2}v_{2}^{2}) dv = vol(S) · vol(S^{m−1}) · (1 − \frac{4}{m}). □

5.5 Final Remarks. (i) The proof of Proposition 5.1 shows that the heat invariant a_{2}^{1}(g_{j(t)}) changes nontrivially during each of the isospectral deformations given in Proposition 4.1. As mentioned in the Introduction, these families consitute the first examples of manifolds which are isospectral on functions and for which nonisospectrality on 1-forms turns out already from considering heat invariants only. Not also that the terms c_{s}(g_{j(t)}), c_{Ric}(g_{j(t)}), and c_{R}(g_{j(t)}) change nontrivially as t varies. The heat invariant a_{2}^{0}(g_{j(t)}) which is a linear combination of these (see 5.2(i)) is nevertheless constant in t by isospectrality on functions, while a_{2}^{1}(g_{j(t)}) (another linear combination of the same terms, see 5.2(ii)) is not constant in t.

(ii) Note that Proposition 5.1 does not exclude the possible existence of isospectral manifolds (C, g_{j}) and (C, g_{j′}) with j ∼ j′ which are also isospectral on 1-forms. We do not know whether such examples exist here. The proposition only says that in this case, the norms of Ric^{G_{j}}_{|_{𝔳}} and Ric^{G_{j′}} _{|_{𝔳}} must coincide, which by 4.1 is frequently not the case.

(iii) Each of the conditions j ∼ j′ and ∥Ric^{G_{j}}_{|_{𝔳}}∥^{2} ≠ ∥Ric^{G_{j′}} _{|_{𝔳}}∥^{2} on a pair j, j′ of linear maps from 𝔷 to so(𝔳) is invariant under rescaling both maps by the same real factor. Lemma 1.7 implies that the metrics g_{αj} converge to the product metric on C = S^{1}(𝔳) × S for α → 0, where S^{1}(𝔳) is the round unit sphere in 𝔳 = ℝ^{m}, and S is endowed with the biinvariant metric k used in 1.3(ii). Therefore, it follows from Proposition 4.1 that there are nontrivial isospectral families of metrics g_{j(t)} on S^{n} × S for every n ≥ 4 which are arbitrarily close to the product metric g_{0} . In particular, nontrivial isospectral deformations of the form (S^{n} × S^{3} × S^{3}, g_{j(t)}) can take place arbitrarily close to the standard metric on S^{n} × S^{3} × S^{3} for every n ≥ 4.

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

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