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Revista de la Unión Matemática Argentina

versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006

 

Spectral properties of elliptic operators on bundles of  k ℤ 2 -manifolds

Ricardo A. Podestá

Abstract: We present some results on the spectral geometry of compact Riemannian manifolds having holonomy group isomorphic to ℤk2 , 1 ≤ k ≤ n - 1 , for the Laplacian on mixed forms and for twisted Dirac operators.

2000 Mathematics Subject Classification. Primary 58J53; Secondary 58C22, 20H15.

Key words and phrases. compact flat manifolds, spectrum, Laplacian, spin structures, Dirac operator, isospectrality, spectral asymmetry, eta invariants

Supported by Conicet

Introduction

This expository article is based on a homonymous talk I gave during the "II Encuentro de Geometría" which took place in La Falda, Sierras de Córdoba, from June 6th to 11th of 2005. It summarizes previous results from [MP], [MP2], [MPR], and [Po], answering standard questions in spectral geometry by using a special class of compact Riemannian manifolds.

Spectral Geometry. It is a kind of mixture between Spectral Theory and Riemannian Geometry. The general situation is to consider (pseudo) differential operators acting on sections of bundles of Riemannian manifolds. However, one usually considers a compact Riemannian manifold M , a vector bundle E → M and an elliptic self-adjoint differential operator D acting on smooth sections of E , i.e.  ∞ ∞ D : Γ (E ) → Γ (E ) . Since M is compact, D has a discrete spectrum, denoted by SpecD (M ) , consisting of real eigenvalues of finite multiplicity which accumulate only at infinity. In symbols, we have

  •  ∞ SpecD (M ) = {{ λ ∈ ℝ : Df = λf, f ∈ Γ (E )}} ⊂ ℝ,
  • 0 ≤ |λ | ≤ |λ | ≤ ⋅⋅⋅ ≤ |λ | ↗ ∞ 1 2 i , λ ∈ Spec (M ), i ∈ ℕ i D .
  • d λ = dim(H λ) < ∞ , H λ = {f : Df = λf } = λ -eigenspace.

We can also think of Spec (M ) D as being the set { (λ, d )} ⊂ ℝ × ℕ λ 0 .

Two manifolds  ′ M, M are called isospectral with respect to D , or simply D -isospectral, if SpecD (M ) = SpecD (M ′) . That is, if M, M ′ have the same set of eigenvalues with the same corresponding multiplicities. It is a general fact that the spectrum determines the dimension and the volume of M . In other words, if M, M ′ are D -isospectral, then  ′ dim (M ) = dim (M ) and  ′ vol (M ) = vol(M ) . The spectrum is said to be asymmetric if dλ ⁄= d -λ for some λ ∈ SpecD (M ) \ {0 } .

Main Aim. The goal of Spectral Geometry is to study the spectrum of M and the interrelations between this object and the geometry or topology of M . That is, knowing the spectrum, what can be said geometrically about M ? Conversely, which spectral data can be deduced provided that we know the geometry of M ? This can be summarize in the following diagram

 SpecD (M ) / / \ // \ Geom (M ) ooo oo oo oo oooo oo oo oo oooo oo oo oo oooo oo oo oo oooo oo oo oTop (M )

Incidentally, by using the diagonal "maps" one could say something about the horizontal "map".

Main Problems. There are several different ways of studying the spectrum. In my opinion, the following are the three most important and interesting ones. In this paper we shall collect results concerning all of them. ∙ Computation of SpecD (M ) . The problem is to determine the eigenvalues λ and their multiplicities dλ . This is in general a difficult task in the sense that this cannot always be done. Indeed, there are few classes of manifolds with explicitly known spectrum for some given operator. The simplest case is the Laplacian Δ acting on smooth functions on the torus 𝕋n = ℤn\ℝn .

∙ Isospectrality. Physically, it is a problem with more than a century old and inquires about the possibility of changing shape while sounding the same. Mathematically, it begun in 1964 with the famous Kac's question Can one hear the shape of a drum? ([Ka]) and the negative answer given by Milnor to a related question ([Mi]). There are basically two antagonistic approaches to this problem: criteria vs. counterexamples. In the first case, one seeks sufficient conditions ensuring that two manifolds are isospectral. This is what some people have called Optimistic Spectral Geometry. On the contrary, in the second case, one tries to produce examples of pairs of isospectral manifolds which are very similar to each other but differing in some geometrical or topological property P . In this case, we say that this particular property P cannot be heard or that we cannot hear property P . This has been fairly called Pessimistic Spectral Geometry, but I would faintly call it Deafferential Geometry. One interesting challenge here is to construct big families (the bigger the best) with respect to the dimension n , of isospectral n -manifolds which are topologically very different (the more different the best) to each other. One purpose of this might be to tightly highlight the fact that if some property cannot be heard, it is not merely an isolated casualty but a concrete reality we cannot ignore.

∙ Spectral asymmetry: following the acoustic jargon before, one studies now when our drum (the manifold) is out of tune. That is, when the positive and the negative spectra differ. The usual devices designed to detect this phenomenon are the eta series and the eta invariant. One wants to compute them explicitly.

Summary of results. Here we give a list of the principal results obtained for ℤk 2 -manifolds (that is, compact flat manifolds having holonomy group isomorphic to  k ℤ2 ) relative to the problems mentioned before. The results will be properly stated and explained in the body of the paper.

A. Full Laplacian Δ F : (1) all ℤk 2 -manifolds covered by the same torus (or by isospectral tori) are isospectral on differential forms of mixed degree; (2) There are big families of ΔF -isospectral manifolds.

B. Spin structures: (3) we give necessary and sufficient conditions for their existence; (4) there are families of ℤk 2 -manifolds which are spin while there are others which are not spin; (5) we answer Webb's question: "Can one hear the property of being spin on a compact Riemannian manifold?".

C. Dirac spectrum: (6) we compute the multiplicities of the eigenvalues of twisted Dirac operators D ρ for an arbitrary spin ℤk 2 -manifold.

D. Dirac isospectrality: (7) we obtain several examples of pairs M, M ′ of D ρ -isospectral manifolds having different topological, geometrical or spectral properties; (8) there are big families of D ρ -isospectral manifolds.

E. Spectral asymmetry. (9) we give a characterization of those manifolds having asymmetric Dirac spectrum; (10) explicit expressions for the eta series and the η -invariant are given; (11) we answer Schueth's question: "Can one hear the η -invariants of a compact Riemannian manifold?".

1.  k ℤ2 -manifolds

What are we talking about?. A Bieberbach group is a crystallographic group without torsion. That is, a discrete, cocompact, torsion-free subgroup Γ ⊂ I(ℝn ) of the isometries of ℝn . Such Γ acts properly discontinuously on  n ℝ , thus  n M Γ = Γ \ ℝ is a compact flat Riemannian manifold having fundamental group Γ . Any element  n n γ ∈ I(ℝ ) = O (n) ⋉ ℝ decomposes uniquely as γ = BLb , where B ∈ O(n ) and Lb denotes translation by b ∈ ℝn .

By the classical Bieberbach's theorems we have the following two basic results: (i) the translations in Γ form a normal maximal abelian subgroup L Λ of finite index, with Λ a lattice in  n ℝ which is B -stable for each BLb ∈ Γ (as usual, one identifies L Λ with Λ ) and (ii) the restriction to Γ of the canonical projection  n r : I(ℝ ) → O(n ) given by BLb ↦→ B is a group homomorphism with kernel Λ and F := r(Γ ) is a finite subgroup of O (n) . It turns out that Γ satisfies the exact sequence of groups

 r 0 → Λ → Γ → F → 1.

The group F ≃ Λ \Γ is called the holonomy group of Γ and it is isomorphic to the linear holonomy group of the Riemannian manifold M Γ . The action of F on Λ by conjugation defines an integral representation F → GLn (ℤ) which is usually called the integral holonomy representation of Γ . Note that this representation does not determine the group Γ , i.e. there may be many non-isomorphic Bieberbach groups with the same holonomy representation.

A compact Riemannian manifold with holonomy group isomorphic to F will be called an F -manifold (see [Ch]). We shall only be concerned with ℤk 2 -manifolds which, by the Cartan-Ambrose-Singer theorem, are necessarily flat.

Compact flat manifolds. A flat manifold is a closed, connected, Riemannian manifold M , whose curvature identically vanishes. Notably, by the Killing-Hopf theorem any compact flat manifold M is isometric to a quotient M = Γ \ℝn Γ , with Γ a Bieberbach group. Putting the Bieberbach theorem's into Riemannian language (see [Wo] or [Ch]) we get that: (i) M Γ is covered by the associated flat torus  n TΛ = Λ \ℝ and the covering π : TΛ → M Γ is a local isometry, (ii) M Γ is affinely equivalent to M Γ ′ if and only if  ′ Γ ≃ Γ and (iii) there is a finite number of classes of affine equivalence of compact flat manifolds, in each dimension.

Up to equivalence, in dimension 1 there is only one compact flat manifold, the circle  1 𝕊 = ℤ \ℝ , while in dimension 2 there are two, the torus  2 T and the Klein bottle  2 K :

 2 2 2 -1 0 2 T = ⟨Le1,Le2⟩\ℝ , K = ⟨[ 0 1]L e2,Le1, Le2⟩\ℝ . 2

The number of compact flat manifolds grows rapidly with the dimension and a classification is unfortunately known only up to dimension 6.

There are two nice results concerning compact flat manifolds. One says that there are a plethora of them while the other says that all these manifolds bound: (1) Every finite group F can be realized as the holonomy group of a compact flat manifold ([AK]) and (2) If M is a compact flat n -manifold then there is a compact (n+1 ) -manifold ˜ M such that ∂ ˜M = M ([HR]).

Into the jungle. A ℤk2 -manifold is just a compact flat n -manifold whose holonomy group is isomorphic to ℤk 2 , with 1 ≤ k ≤ n - 1 . Thus, it is of the form M Γ = Γ \ℝn where Γ = ⟨γ1,...,γk,Λ ⟩ , with Λ = ℤλ1 ⊕ ⋅⋅⋅ ⊕ ℤλn and γi = BiLbi satisfying Bi ∈ O(n ) , Bi Λ = Λ ,  n bi ∈ ℝ and  2 B i = Id , BiBj = BjBi for 1 ≤ i,j ≤ k .

Some friendly tribes. We now introduce some particularly interesting classes of ℤk2 -manifolds that will be used in the rest of the paper.

∘ ℤ2 -manifolds. They generalize the Klein Bottle in the sense that they are quotients of tori divided by a ℤ2 -action. They are determined by the integral holonomy representation which can be parametrized by the block matrices

Bj,h = diag(J◟,..◝.◜,J◞,-◟-1,.◝.◜.,--1◞,1◟,.◝.◜.,1◞), J = [01 1 0], j≥0 h≥0 l≥1

with 2j + h + l = n and j + h ⁄= 0 . The corresponding diffeomorphism classes are represented by  n Mj,h = ⟨Bj,hL e2n,Λ⟩\ℝ , Λ the canonical lattice. One can compute their first integral homology groups and their Betti numbers. Indeed, H1 (Mj,h,ℤ ) ≃ ℤj+l ⊕ ℤh2 and  ∑ ( )( ) βp(Mj,h) = [ip=∕20] j+2hi pj+-l2i , for 0 ≤ p ≤ n . (See [MP]).

∘ Primitive  2 ℤ2 -manifolds. We recall that primitive means that β1 (M Γ ) = 0 , that is F has trivial center. By a construction due to Calabi (see [Ca], [Wo]), any compact flat manifold can be obtained from a primitive one. Primitive  2 ℤ2 -manifolds are also determined by the integral holonomy representation, which decomposes as a sum of integral representations of rank ≤ 3 ([Ti]).

∘ Diagonal type. A compact flat manifold M Γ is of diagonal type if there is an orthonormal ℤ -basis {ei,...,en} of Λ satisfying Bei = ±ei , 1 ≤ i ≤ n , for every BLb ∈ Γ . In this case we say that Γ have diagonal holonomy representation. One can assume that Λ is the canonical lattice and that b ∈ 1Λ 2 . They necessarily have holonomy group F ≃ ℤk2 . (See [MR3]).

∘ Hantzsche-Wendt manifolds. (Or, HW-manifolds, for short). They are the orientable ℤn2-1 -manifolds in odd dimension n . Any such manifold M Γ is given by Γ = ⟨B1Lb ,...,BnLbn, Λ ⟩ 1 where Bi fixes ei , Biej = - ej for 1 ≤ i ≤ n , if j ⁄= i , and Λ = ℤe ⊕ ...⊕ ℤe 1 n (note that B = B B ⋅⋅⋅B n 1 2 n- 1 ). They generalize the only orientable  2 ℤ2 -manifold existing in dimension 3, historically called the Hantzsche-Wendt manifold, and were studied in [MR]. They are primitive, of diagonal type and, furthermore, they are rational homology spheres, i.e. H *(M, ℚ) = H *(𝕊n,ℚ ) for every HW-manifold M . Also, one can associate certain directed graphs to them.

∘ Generalized Hantzsche-Wendt manifolds. (Or GHW-manifolds). They are simply the ℤn2-1 -manifolds in dimension n . They share many properties with HW-manifolds but they are not primitive in general. There are [n+21-] different integral holonomy representations, all of diagonal type. (See [RS]).

A little bit of numerology. As we have seen, we have the following natural inclusions HW ⊂ GHW ⊂ Diagonal type ⊂ ℤk2-manifolds . In the table below we compare the cardinality of these families. We see that, at least in low dimensions, the class of ℤk 2 -manifolds represents more than half of the compact flat manifolds.

|-------------|------|------|------|-------|------|-------| |#--manifolds-|dim-1-|dim-2-|dim-3-|dim-4--|dim--5-|-dim--6-| |compact--flat-|----1-|----2-|---10-|---74--|1.060--|38.746-| | ℤk | - | 1 | 6 | 43 | 650 |27.515 | |------2------|------|------|------|-------|------|-------| |---GHW-------|------|----1-|----3-|---12--|-123--|-2.536-| | HW | - | -| 1 | - | 2 | - | |-------------|------------------------------------------- | |

2. The full Laplacian

Consider the Laplacian on p -forms Δp . It is a first order elliptic differential operator acting on smooth sections of the p -exterior bundle  p Λ (T M ) of M . The spectrum of this operator on compact flat manifolds was studied in [MR2] (see also [MR3], [MR4]). The multiplicity of the eigenvalue 4π2 μ of Δp has the expression

 1 ∑ dp,μ(Γ ) = |F-| χp (B) eμ,γ γ=BLb∈Λ\Γ

where  ∑ eμ,γ = v∈ Λ*μ:Bv=v e-2πiv⋅b , with Λ *μ = {v ∈ Λ * : ∥v ∥ = μ} , and χp is the character of the p -exterior representation. For Γ of diagonal type, this character is given by integer values of certain polynomials. In fact, for BL ∈ Γ b , we have

 ∑ p ( )( ) χp(B ) = Knp(n - nB ) with Knp (x) := (- 1)t x n-x t=0 t p-t

where  n B nB = dim (ℝ ) and  n K p(x) is the (binary) Krawtchouk polynomial of order n and degree p . They are discrete orthogonal polynomials (see [KL]). The first ones have the expressions Kn0(x) = 1 , Kn1(x) = - 2x + n ,  ( ) Kn2 (x) = 2x2 - 2nx + n2 ,  ( ) Kn3(x ) = - 4x3 + 2nx2 - (n2 - n + 2)x + n 3 3 3 , etc.

From now on in this section we refer to [MPR]. One can simply define a Laplacian on arbitrary forms by considering the p -Laplacians altogether, that is, we can take

 n ∑ ΔF := ⊕ Δp. p=0

This full Laplacian is again a first order elliptic differential operator which acts on sections of the full exterior bundle  ⊕ Λ(T M ) = n Λp(T M ) p=0 of M . The eigenvalues are still of the form  2 4π μ , but their multiplicities are now given by the sum  ∑n dF,μ(Γ ) = p=0 dp,μ(Γ ) . Clearly, p -isospectrality (i.e. isospectrality with respect to Δp ) for all p implies ΔF -isospectrality, but the converse is far from being true, as will be shown in Example 2.2 below.

We have the following curious "optimistic" result from [MPR]:

Theorem 2.1. Let Γ be a Bieberbach group of dimension n with translation lattice Λ and holonomy group ℤk2 . Then, the eigenvalue 4 π2μ of ΔF has multiplicity dF,μ(Γ ) = 2n- k|Λ *μ| where Λ *μ = {v ∈ Λ * : ||v|| = μ} .

Thus, two ℤk 2 -manifolds M ,M ′ Γ Γ are Δ F -isospectral if and only if the translation lattices  ′ Λ, Λ are isospectral. In particular, for fixed Λ and k , all  k ℤ2 -manifolds with covering torus TΛ are ΔF -isospectral.

Sketch of proof. Let F = ⟨B1, ...,Bk⟩ ≃ ℤk2 . Then, the Bi 's diagonalize simultaneously with eigenvalues ± 1 . Thus, every Bi is conjugate in GLn (ℝ ) to the diagonal matrix DB := diag(- In-n ,In ) B B where Im is the identity matrix in  m ℝ . Thus  n χp (B ) = χp (DB ) = Kp (n - nB) . Hence, we have  -k ((n) * ∑ n ) dp,μ(Γ ) = 2 p |Λ μ| + γ∈Λ\Γ \{Id} K p(n - nB) eμ,γ and, adding over p

 n n- k * - k ∑ ( ∑ n ) dF,μ(Γ ) = 2 |Λ μ| + 2 K p(n - nB ) eμ,γ. γ∈Λ\Γ \{Id} p=0

Now, for j ⁄= 0 , one can show that ∑n n p=0 K p(j) = 0 (⋆ ) . Since n - nB = 0 if and only if B = Id , we finally get that dF,μ(Γ ) = 2n-k|Λ*μ| , as asserted. □

Note 1. The proof seems to be of an entirely combinatorial nature since it only depends on (⋆ ), and the Krawtchouk polynomials at integer values Kn (j) p have some combinatorial interpretations in the literature.

Example 2.2 (ℤ2 -manifolds of dim 3). We illustrate the theorem in the simplest non-trivial case, i.e. when n = 3 and k = 1 . Up to diffeomorphism, there are only three ℤ2 -manifolds in dimension 3 (see [Wo]). They are M1,0,M0,2 and M0,1 , in the notation of page 138.

In the tables below we give the multiplicities for Δp , with 0 ≤ p ≤ 3 , and also for ΔF , of the 2 lowest non trivial eigenvalues.

 

The values in the tables show that the manifolds are not p -isospectral to each other for any 0 ≤ p ≤ 3 . However, we can see how all these multiplicities balance, that is how they manage to distribute themselves in order to have equal sums for each eigenvalue, in each case.

We really need the hypothesis F ≃ ℤk2 in the theorem. The "magical" averaging phenomenon, present when considering all the p -Laplacians Δp simultaneously, only seems to work in the case considered. The result does not hold, in general, for holonomy groups different from  k ℤ 2 . There is a pair of 6-dimensional orientable ℤ4 -manifolds, M Γ ,M Γ ′ , which are not ΔF -isospectral ([MPR, Ex. 3.5]), even though they are isospectral on functions. In fact, let Λ = ℤe1 ⊕ ⋅⋅⋅ ⊕ ℤe6 and take Γ = ⟨B1Lb1, Λ ⟩ and Γ ′ = ⟨B ′1Lb′,Λ⟩ 1 where B1 = diag ( ˜J,J˜,1, 1) , b1 = e5- 4 ,  ′ ˜ B 1 = diag(J,1,- 1,- 1,1) ,  ′ e6 b1 = 4 with  ˜ 0 1 J = [-1 0 ]. Following [MR2] one can prove that M Γ ,M Γ ′ are 0 -isospectral (and hence 6 -isospectral, by orientability) but they are not p -isospectral for 1 ≤ p ≤ 5 . Since dp,0(Γ ) = βp(M ) we have that  ∑ dF,0(M ) = 6p=0 βp(M ) . By [Hi], the Betti numbers βp , 0 ≤ p ≤ 6 , for M and M ′ are respectively given by 1,2,5,8,5,2,1 and 1,2,3,4,3,2,1. In this way we get that dF,0(M ) = 24 while  ′ dF,0(M ) = 16 , hence  ′ M, M are not isospectral on forms.

Note 2. One can also consider the Laplacian on even/odd forms given by  ∑ Δe = p even Δp and Δ = ∑ Δ o p odd p . The results in Theorem 2.1 hold mutatis mutandis but now with  1 do,μ(Γ ) = de,μ(Γ ) = 2dF,μ(Γ ) . In particular,  ∑n p χ (M Γ ) = p=0(- 1) βp(M Γ ) = do,0(Γ ) - de,0(Γ ) = 0 .

Big families of ΔF -isospectral manifolds. As a straight consequence of the result in the previous theorem, we can exhibit several big families of ΔF -isospectral ℤk2 -manifolds in arbitrary dimension n . Here, big alludes to the fact that the cardinality of the family grows polynomially —or even exponentially— with respect to n and, also, that all the manifolds in each family are not homeomorphic to each other. It is worth noting that for the Laplacian on functions, or on p -forms, there are not known examples of such exponential families. In the famous isospectral deformations of Gordon and Wilson (see [GW]), the manifolds have different metrics but they are all homeomorphic to each other.

Consider the following families: F1 = {ℤ2 -manifolds} , F2 = { primitive  2 ℤ2 -manifolds} and F3 = {HW -manifolds } . For simplicity, we assume that all the groups have the canonical lattice of translations Λ = ℤn . Thus, by Theorem 2.1, all manifolds belonging to each family are mutually isospectral on forms. We now indicate the order of the cardinality of each family. We have that #F1 = o(n2) and #F2 = o(n5) (see [MP], [Ti]). By using a small subfamily of HW-manifolds it is proved in [MR] that  2n-3 #F3 > n-1 . Furthermore, based on an example given in [LS], Rossetti constructed a very big family F4 of GHW-manifolds with  √ -- 2 #F4 = o(( 2 )n ) (see [MPR]).

3. Spin structures

Spin structures play a role in geometry and physics. One relevant fact is that they allow to define Dirac operators. On an arbitrary Riemannian manifold M , the Laplacian on functions and the Laplacian on p -forms are always defined. On the other hand, for the Dirac operator to be defined, M needs to have an extra geometric structure. More precisely, to each spin structure on M one can construct the so called spinor bundle and a Dirac operator acting on sections of it. However, one needs to have some care here, since not every Riemannian manifold admits a spin structure (see [LS]).

Let M be an oriented Riemannian n -manifold and let  π B (M ) → M be the SO (n) -principal bundle of oriented frames. Inside the group of units of the Clifford algebra Cl (n) of ℝn lies the compact connected Lie group

Spin(n) = {v ⋅⋅⋅v : v ∈ 𝕊1(ℝn ),j = 1,...,2k} 1 2k j

where  1 n n 𝕊 (ℝ ) = {x ∈ ℝ : ||x || = 1} . This group satisfies the exact sequence

 μ 0 → ℤ2 → Spin(n) → SO (n ) → 1

where μ(g)(x) = gxg -1 . Thus, μ is a double covering and since Spin (n) is simply connencted, for n ≥ 3 , it is the universal covering of SO (n ) .

A spin structure on an orientable manifold M is an equivariant 2-fold covering  p ˜B (M ) → B (M ) such that π ∘ p = ˜π , where  ˜π B˜(M ) → M is a Spin(n ) -principal bundle. A manifold endowed with a spin structure is called a spin manifold.

Fortunately, for compact flat manifolds we can get rid of this complicated geometrical-topological definition by using the following result. The spin structures of M Γ are in a 1-1 correspondence (see [LM] or [Fr2]) with the group homomorphisms ɛ commuting the diagram

 Spin(n) ɛ / // | / / / |μ / / Γ ------r-------SO (n )
(3.1)

This gives a purely algebraic alternative definition, simpler than the original one, which is in fact a criterion to decide the existence of spin structures. It can be used not only to construct such structures, but also to count them.

Spin structures on ℤk 2 -manifolds. In this subsection we refer to [MP]. Let  n M Γ = Γ \ℝ be a  k ℤ 2 -manifold and ɛ a spin structure as in (3.1). Since r(Λ) = Id , then ɛ(L λ) ∈ {±1 } for any λ ∈ Λ . Let λ1,...,λn be a ℤ -basis of Λ and put δi = ɛ(Lλi) . For  ∑ λ = imi λi ∈ Λ , with mi ∈ ℤ , we have ɛ(L λ) = δm11δm22 ⋅⋅⋅δmnn . For any γ = BLb ∈ Γ we fix a distinguish (though arbitrary) element uB ∈ μ- 1(B ) . Then,

ɛ(γ) = σB uB
(3.2)

where σB ∈ {±1 } depends on γ and on the choice of uB .

The map ɛ is determined by its action on the generators of Γ , and so we can identify it with the (n + k) -tuple  n+k ɛ ≡ (δ1,...,δn,σ1,...,σk) ∈ {±1 } . Since ɛ is a group homomorphism it must satisfy, for every γ = BLb , the following conditions

(⋆1 ) ɛ(γ2) = uB2, (⋆2 ) ɛ(LB λ) = ɛ(Lλ) (λ ∈ Λ).

Note that, since B2 = Id , these are conditions for ɛ over Λ , i.e. for the character δɛ = ɛ|Λ ∈ Hom (Λ, {±1 }) . We define the set
 { Λˆ(Γ ) := χ ∈ Hom (Λ,{±1 } ) : χ satisfies (⋆1 ) and (⋆2 )}.

The following theorem says that the above necessary conditions (⋆ ) 's for the existence of spin structures on ℤk 2 -manifolds, are also sufficient .

Theorem 3.1. Let Γ = ⟨γ1,...,γk,Λ ⟩ be an n -dimensional Bieberbach group with holonomy group F = ⟨B1,...,Bk ⟩ ≃ ℤk2 , F ⊂ SO (n ) , and let σ1,...,σk be as in (3.2). Then, the map

ɛ ↦→ (ɛ|Λ, σ1,...,σk)

defines a bijective correspondence between the spin structures of M Γ and the set  k ˆΛ (Γ ) × {±1 } . Hence, the number of spin structures of M Γ is either 0 or  r 2 for some r ≥ k .

Applications. By applying Theorem 3.1 we can: (1) study the existence of spin structures in particular families of ℤk2 -manifolds, (2) give a simple method to obtain spin manifolds and (3) determine the audibility of the spin structures, that is, whether spin structures can be heard or not.

Spin structures in families. (i) Every orientable ℤ2 -manifold is spin and orientable ℤ2 -manifolds of diagonal type have  n 2 spin structures (see [MP]), the same as for any n -torus (see [Fr]). (ii) Orientable primitive ℤ22 -manifolds are spin. (iii) The 3-dimensional HW-manifold is spin. HW-manifolds of dimension n , n ≥ 5 , are not spin. See [Po] for the case n = 4k + 1 . The general case, i.e. n odd, was proved independently by J. P. Rossetti ([Ro], by using a criterion in [MP]) and by S. Console ([Co], by computing the second Stiefel-Whitney classes ω2 ).

How to get spin manifolds easily?. By using the doubling procedure in [JR] or [BDM]. Let Γ be an n -dimensional Bieberbach group with translation lattice Λ and holonomy group F . The double of Γ is the Bieberbach group defined by dΓ = ⟨dB L(b,b),L(λ,λ ) : BLb ∈ Γ ,λ1,λ2 ∈ Λ⟩ 1 2 , with dB = [B 0] 0 B . It follows that dΓ has translation lattice Λ ⊕ Λ and holonomy group F . The associated manifold M dΓ has dimension 2n and is Kähler (see [DM]). Now, doubling an orientable manifold of diagonal type gives a spin manifold (see [MP2]). If the manifold is not orientable, then one has to double twice.

Spin structures are not audible. Take Λ = ℤe1 ⊕ ⋅⋅⋅ ⊕ ℤe6 and consider the manifolds M = Γ \ℝ6 and M ′ = Γ ′\ℝ6 where Γ = ⟨B1Lb1,B2Lb2, Λ ⟩ and Γ ′ = ⟨B1Lb ′,B2Lb′,Λ ⟩ 1 2 are Bieberbach groups of diagonal type given, in diagonal notation (see [MR2], [MP]), in the following table. For example, the 1∕2 in the first column means that  e3- Lb1 = 2 . Also, B3 = B1B2 and b3 = B2b1 + b2 ,  ′ ′ ′ b3 = B2b 1 + b2 .

|-------------|--------------|-------------| |B1--Lb1--Lb-′1|B2---Lb2--Lb′2-|B3--Lb3--Lb-′3| | 1 | 1 1/2 1/2 | 1 1/2 1/2 | | 1 1/2 | 1 1/2 | 1 1/2 1/2 | | 1 1/2 | -1 | -1 1/2 | | -1 | 1 1/2 | -1 1/2 | | -1 | 1 | -1 | |-1--------------1--------------1------------

It is easy to see that  ′ M, M are orientable  2 ℤ2 -manifolds of dimension 6 and that they are p -isospectral for every p , 0 ≤ p ≤ 6 . Note that they are not primitive since β1(M ) = β1(M ′) = 2 . Using Theorem 3.1 we can check that M has no spin structures while M ′ has 25 spin structures of the form ɛ = (- 1,- 1,δ3,1,δ5,δ6;σ1,σ2) , with δi,σj ∈ {±1 } . Thus, we cannot hear the spin structures of Riemannian manifolds!

4. Dirac spectrum

We begin with the ingredients necessary to define twisted Dirac operators on Riemannian manifolds. Let M Γ be an orientable compact flat manifold endowed with a spin structure ɛ as in (3.1), denoted by (M Γ ,ɛ) from now on. Let Ln : Spin(n) → GL (Sn) be the spin representation, that is the restriction to Spin(n) of any irreducible complex representation of the complexified Clifford algebra Cl(n ) ⊗ ℂ . It is wellknown that  [n∕2] dim ℂ(Sn) = 2 and that Ln is irreducible if n is odd while, if n is even, Ln splits into two inequivalent irreducible representations (L±n ,S±n) of the same dimension, called the half-spin representations. Let ρ : Γ → U (V ) be a unitary representation such that ρ|Λ = 1 . As usual, we take χ (γ) = Trρ (γ ) ρ and d = dim (V ) ρ .

Now, the morphism ɛ allows to construct the spinor bundle twisted by ρ

 n n Sρ(M Γ ,ɛ) := Γ \ (ℝ × (Sn ⊗ V )) → Γ \ ℝ

with action given by γ ⋅ (x,w ⊗ v) = (γx,Ln (ɛ(γ))(w) ⊗ ρ(γ)(v)) . One can identify the space of smooth sections of this bundle, Γ ∞(S ρ(M Γ ,ɛ)) , with the set {f : ℝn → Sn ⊗ V smooth : f(γx ) = (Ln ∘ ɛ ⊗ ρ)(γ)f (x)} .

With the above identification, the Dirac operator twisted by ρ on compact flat manifolds is Dρ : Γ ∞(S ρ(M Γ ,ɛ)) → Γ ∞ (Sρ(M Γ ,ɛ)) given by

 ∑n ∂f- D ρ f(x) = ei ⋅∂xi(x) i=1

where ei acts by Ln (ei) ⊗ Id in Sn ⊗ V and e1,...,en is an orthonormal basis of ℝn . If (ρ,V ) = (1,ℂ ) we have the classical Dirac operator D . Dρ is a first order elliptic differential operator, symmetric and essentially self-adjoint. It does not depend on the choice of the orthonormal basis of  n ℝ . Also, it is a formal square root of the Laplacian, that is  2 D ρ = - Δs,ρ , called the twisted spinor Laplacian. If f ∈ ker Dρ , f is called a harmonic spinor.

Dρ has a discrete spectrum consisting of real eigenvalues ± 2πμ , μ ≥ 0 , of finite multiplicity d±ρ,μ . Explicit expressions for d±ρ,μ for an arbitrary pair (M Γ ,ɛ) were obtained in [MP2]. We now recall this result.

Let F1 = {B ∈ F = r(Γ ) : nB = 1} where nB = dim ker(B - Id) . Put Λ *ɛ = {u ∈ Λ* : ɛ(λ ) = e2πiλ⋅u,λ ∈ Λ } , with Λ * the dual lattice of Λ , and

 * * Λ ɛ,μ = {u ∈ Λ ɛ : ∥u ∥ = μ}.

Now, for each γ = BLb ∈ Γ , let  * B (Λɛ,μ) denotes the set of elements fixed by B in  * Λ ɛ,μ , that is (Λ *ɛ,μ)B = {u ∈ Λ *ɛ,μ : Bv = v} . Furthermore, for γ ∈ Γ , let xγ be a fixed (though arbitrary) element in the maximal torus of Spin(n - 1) , conjugate in Spin(n) to ɛ(γ) . Finally, define a sign σ (u, x ) γ , depending on u and on the conjugacy class of x γ in Spin (n - 1) , in the following way. If γ = BLb ∈ Λ \Γ and  * B u ∈ (Λɛ) \ {0} , let hu ∈ Spin (n) such that  -1 hu uh u = ∥u∥en . Hence,  - 1 huɛ(γ)hu ∈ Spin (n - 1) . Take σ ɛ(u, xγ) = 1 if huɛ(γ )h -u1 is conjugate to x γ in Spin (n - 1) and σ ɛ(u, xγ) = - 1 otherwise. As a consequence, σ (- u, xγ) = - σ(u,xγ) and σ(αu, xγ) = σ(u, xγ) for every α > 0 (see Definition 2.3, Remark 2.4 and Lemma 6.2 in [MP2] for details).

For n odd, the multiplicity of the eigenvalue ± 2πμ , for μ > 0 , is given by

 ( ± 1 ∑ ∑ -2πiu⋅b dρ,μ(Γ ,ɛ) = |F|- χρ(γ) e ⋅ χL ± (x γ) + γ ∈ Λ\Γ u∈(Λ*ɛ,μ)B n- 1 B ⁄∈ F1 ) ∑ ∑ -2πiu⋅b χ ρ(γ) e ⋅ χL±nσ-(u1,xγ)(xγ) , γ ∈ Λ\Γ u∈(Λ*ɛ,μ)B B ∈ F1
(4.1)

while for n even, it is given by the first term in (4.1), where the sum is taken over all γ ∈ Λ \Γ , with L ±n-1 replaced by Ln -1 . For μ = 0 , with n even or odd, we have that  1--∑ F d0(Γ ,ɛ) = |F| γ∈Λ\Γ χ ρ(γ)χLn (ɛ (γ )) = dim S , if ɛ|Λ = 1 , and d0(Γ ,ɛ) = 0 , otherwise.

Dirac spectrum of  k ℤ 2 -manifolds. In the particular case when F ≃ ℤk2 , the formula (4.1) becomes more tractable and allows one to give shorter expressions for the multiplicities. Also, one can characterize all the spin ℤk 2 -manifolds (M Γ ,ɛ) having asymmetric twisted Dirac spectrum. To wit

Theorem 4.1. Let (M Γ ,ɛ) be an n -dimensional spin ℤk2 -manifold.

(i) If F = ∅ 1 , then the spectrum Spec (M ,ɛ) Dρ Γ is symmetric and the non-zero eigenvalue ± 2πμ of D ρ has multiplicity

d±ρ,μ(Γ ,ɛ) = 2m -k-1 dρ|Λ*ɛ,μ |.
(4.2)

(ii) If F1 ⁄= ∅ , then SpecD ρ(M Γ ,ɛ) is asymmetric if and only if: n = 4r + 3 and there exists γ = BLb ∈ Γ with nB = 1 and χρ(γ) ⁄= 0 such that B |Λ = - δɛId . In this case, the asymmetric spectrum is the set

 -1 A = {±2 πμj : μj = (j + 12)∥f∥ , j ∈ ℕ0 }

where ΛB = ℤf and, if we put σγ := σ(⟨f,2b ⟩f, gm) , we have:

 { m-k-1 ( * r+j ) d± (Γ ,ɛ) = 2 dρ|Λɛ,μ| ± 2σ γ(- 1) χρ(γ ) μ = μj, ρ,μ 2m-k-1 dρ|Λ*ɛ,μ| μ ⁄= μj.

Also, by (iii), M Γ has no non-trivial harmonic spinors.

If Spec (M ,ɛ) Dρ Γ is symmetric then d ± (Γ ,ɛ) ρ,μ is given by (4.2).

(iii) The number of independent harmonic spinors is given by

 m-k dρ,0(Γ ,ɛ) = 2 dρ

if ɛ|Λ = 1 , and by d ρ,0(Γ ,ɛ) = 0 , otherwise. If k > m then M Γ has no spin structures of trivial type, hence, M Γ has no harmonic spinors. Furthermore, if M Γ has exactly 2md ρ harmonic spinors then M = T Γ Λ and ɛ = 1 .

5. Dirac isospectrality

We now deal with the isospectral problem for twisted Dirac operators D ρ . We claim the existence of twisted Dirac isospectral manifolds having different spectral, geometrical or topological properties. We will compare D ρ -isospectrality with other notions of isospectrality such as isospectrality with respect to the spinor Laplacian Δs and the p -Laplacian Δp . We will also look at the spectrum of closed geodesic with and without multiplicities, that is the so called [L] -spectrum and L -spectrum, respectively.

By using  k ℤ2 -manifolds one can obtain the following results from [MP2].

Theorem 5.1. There are families F of Riemannian n -manifolds, pairwise non homeomorphic, which are mutually D ρ -isospectral for each ρ , but they are neither isospectral on functions nor L -isospectral. Furthermore, F can be chosen satisfying any of the following extra properties:

(i) Every M ∈ F has (or has no) harmonic spinors.

(ii) All M 's in F have the same p -Betti numbers for 1 ≤ p ≤ n and they are p -isospectral to each other for any p odd.

Theorem 5.2. There are pairs of non-homeomorphic spin manifolds which, for each ρ , are:

(i) Δs,ρ -isospectral but not D ρ -isospectral; or

(ii) p -isospectral for 0 ≤ p ≤ n and [L ] -isospectral such that they are D ρ -isospectral, or not, depending on the choice of the spin structure; or

(iii) D ρ -isospectral and p -isospectral for 0 ≤ p ≤ n , which are L -isospectral but not [L] -isospectral.

We can summarize the previous results in the table below

D-isospectrality vs. other types of isospectrality

|--------|--------|----------------|-----|----|-------|----|------------| | | | | | | | | | |--D-ρ---|-Δs,ρ---|------Δp--------|-[L-]-|-L--|-dim.--|-F--|--[MP2--]---| | | | | | | | | | | Yes | Yes |No (generically) | No |No | n ≥ 3 |ℤ2 |Ex. 4.3 (i) | |--------|--------|----------------|-----|----|-------|----|------------| | Yes | Yes | Yes (if p odd )| No |No |n = 4t|ℤ2 |Ex. 4.3 (iii) | |--------|--------|----------------|-----|----|-------|----|------------| | No | Yes | No | No |No | n ≥ 7 |ℤ2 |Ex. 4.4 (i) | |--------|--------|----------------|-----|----|-------|----|------------| |Yes/No |Yes/No |Yes (0 ≤ p ≤ n )|Yes |Yes | n ≥ 4 |ℤ2 |Ex. 4.5 (i) | |--------|--------|----------------|-----|----|-------|--2-|------------| | | | | | | | 2 | | -Yes/No---Yes/No---Yes-(0-≤-p-≤-n-)--No---Yes---n-≥-4--ℤ-2--Ex.-4.5-(ii)-- | |

Theorem 5.3. There are big families of D ρ -isospectral manifolds. More precisely, there exists a family F of pairwise non-homeomorphic Riemannian n -manifolds that are all mutually D ρ -isospectral, for many different choices of spin structures, with the cardinality of F depending exponentially on n or, even better, on n2 .

6. Eta series and eta invariants

Let A be a self-adjoint elliptic differential operator of order d on a compact n -manifold M . To study the spectral asymmetry of A , Atiyah, Patodi and Singer introduced in [APS] the so called eta series defined by

 ∑ ηA (s) = sign(λ) |λ |- s. 0⁄= λ∈SpecA

This series converges for Re(s) > n d and defines a holomorphic function η (s) A which has a meromorphic continuation to ℂ with simple poles (possibly) at s = n - k , k ∈ ℕ0 . It is a non trivial fact that this function is really finite at s = 0 (See [APS2] for n odd, [Gi] for n even, and [Wod] using different methods). The number ηA(0) is a spectral invariant, called the eta invariant, which does not depend on the metric, although ηA(s) does. It gives a measure of the spectral asymmetry of A and it is important because it appears in the "correction term" of some Index Theorems for manifolds with boundary. For example, if A = D , the classical Dirac operator, and M is a compact spin manifold with N = ∂M , under certain global boundary conditions, the index of D is given by

 ∫ Ind(D ) = Aˆ(p) - d0-+-ηN- M 2

where  ˆ A (p ) is the Hirzebruch  ˆ A -polynomial in the Pontrjagin forms pi , ηN is the eta-invariant associated to D |N , and d0 = dim kerD |N (see [APS2]). Note the beauty of the above expression relating topological, geometrical and spectral data!

For A = D ρ , the twisted Dirac operator, we have the following result:

Theorem 6.1 ([MP2]). Let (M Γ ,ɛ) be a spin  k ℤ2 -manifold of dimension n = 2m + 1 = 4r + 3 . If SpecD ρ(M Γ ,ɛ) is asymmetric then:

 r m-k ∥f∥s ( 1 3 ) η(Γ ,ρ,ɛ)(s ) = (- 2) σγ χ ρ(γ) 2 (4π)s ζ(s,4 ) - ζ(s,4 )

where  ∑ ∞ --1--- ζ(s,α) = j=0 (j+ α)s , with Re (s) > 1 and α ∈ (0,1] , denotes the Hurwitz zeta function and f and σ γ are as defined in Theorem 4.1. In particular, η (Γ ,ρ,ɛ)(s) has an analytic continuation to ℂ that is everywhere holomorphic.

Furthermore, the eta invariant is given by η = (- 1)r σ χ (γ) 2m -k (Γ ,ρ,ɛ) γ ρ .

Eta invariants are not audible. There are 7-dimensional  2 ℤ2 -manifolds  ′ M, M which are p -isospectral for 0 ≤ p ≤ 7 such that η(M ) = 0 but η(M ′) = 2 (see [Po]). The trick is to pick one manifold having symmetric spectrum while the other not. It turns out that 7 is the minimum dimension in which this can be done. The moral is that we cannot hear the η -invariant of compact Riemannian manifolds!

Acknowledgements. I wish to thank the organizers of the "II Encuentro de Geometría" for having invited me to participate in the event, where so many good mathematicians were present. It was also a pleasure for me to write this extended version of the talk given there.

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Ricardo A. Podestá
FaMAF-CIEM
Universidad Nacional de Córdoba
Córdoba, Argentina.
podesta@mate.uncor.edu

Recibido: 10 de agosto de 2005
Aceptado: 22 de septiembre de 2006