1 Spectral estimates on 2-tori 0 0 2 n Bernd Ammann ∗ a J 8 March 2000 ] G D . h t Abstract a m [ Weprove upper and lower bounds for the eigenvalues of the Dirac operator and the Laplace operator on 2-dimensional tori. In particluar we give a lower bound 1 v for the first eigenvalue of the Dirac operator for non-trivial spin structures. It is the 1 only explicit estimate for eigenvalues of the Dirac operator known so far that uses 6 information aboutthespinstructure. 0 1 As a corollary we obtain lower bounds for the Willmore functional of a torus 0 embedded intoS3. 1 InthefinalsectionwecompareDiracspectrafortwodifferentspinstructures on 0 / anarbitrary Riemannianspinmanifold. h t a m Keywords: Dirac operator, Laplace operator, spectrum, conformal metrics, two-dimen- : sionaltorus,spinstructures, Willmorefunctional v i X Mathematics Classification2000: Primary: 53C27,Secondary: 58J5053C80 r a 1 Introduction The Dirac operator is an elliptic differential operator of order one playing an important role both in modern physics and in mathematics. In physics, particles with non-integer spin, so-called fermions, are described by the Dirac equation. Let us assume that the space-time M is stationary, M = R N and that the spatial component N is compact × and admitsaspinstructure. Then stationaryfermionshaveawavefunctionoftheform Ψ(t,x) = eiEtΨ (x) t R, x N 0 ∈ ∈ ∗[email protected] 1 whereΨ isaneigenspinorofD2 ,thesquareoftheDiracoperatoronN, thatbelongsto 0 N theeigenvalueλ . Theenergy E andtheeigenvalueλ arerelated viatheformula E2 = λ+m2 withmbeingtherestmassoftheparticle. Knowingthespectrumthereforemeansknow- ing possible energies. The first eigenvalue is of particular interest as it characterizes the energy of the state of lowest energy — the vacuum. On an arbitrary Riemannian mani- fold, exact calculation of the spectrum is impossible,thus one tries try to find bounds for theeigenvalues. BoundingeigenvaluesoftheDiracoperatoronacompactRiemannianmanifoldN isalso an importanttool in differential geometry and topology. If N is spin and carries a metric whosescalarcurvatureisgreaterthanorequaltos > 0ateverypoint,thenwiththehelp 0 of the Schro¨dinger-Lichnerowicz formula it is easy to prove that the first eigenvalue λ 1 of D2 is bounded from below by s /4. On the other hand Atiyah-Singer index theorem 0 tellsusthatpositivityofthefirst eigenvalueofD2 onacompactRiemannianmanifoldN implies that the Aˆ-genus vanishes. Therefore any compact spin manifold admitting a positivescalarcurvaturemetrichas vanishingAˆ-genus. LowerboundsforDiraceigenvaluescanalsobeappliedtoproblemsinclassicaldifferen- tial geometry. For any immersion F : N Rn of a compact manifold N, Christian Ba¨r → [Ba¨r98b]proved H 2 µ area(N). (1) 1 | | ≥ ZN Here N carries the induced metric, µ is the first eigenvalue of the square of a twisted 1 Dirac operator and H is the mean curvature vector field of F(N) Rn. If N is the 2- ⊂ dimensional torus T2, then the left hand side of (1) is the so-called Willmore functional. TheWillmoreconjecturestates H 2 2π2 ZT2 | | ≥ for any immersion F : T2 Rn. This conjecture first appeared in [Wil65] for the → case n = 3. In the meantime the conjecture has been verified for several classes of immersions, for example for immersions with rotational symmetry [LS84] or for non- injective immersions [LY82]. Nevertheless the conjecture remains open until now. For further information on this conjecture the reader may read the introductions of [Top98b] or[Amm00]. Now assume for simplicity that F is an embedding and F(T2) S3 R4. In this case, ⊂ ⊂ the twisting bundle is trivial, and µ is the first eigenvalue of the square of the classical 1 Dirac operator associated to a non-trivial spin structure. Our goal is to use inequality (1) in order to derive lower bounds for the Willmorefunctional. If the induced metric on T2 is flat, the spectrum of D has been explicitely calculated [Fri84] and we obtain a lower boundfor H 2. T2| | R 2 Obtaining lower eigenvalue estimates for non-flat tori is much harder. John Lott [Lot86, Proposition 1] proved the existence of a constant C > 0 depending on the spin- Lott conformal typeofthetorussuchthat µ area C . (2) 1 Lott ≥ Unfortunately, Lott’s article does not give an explicit value and it seems hard to express such a constant C in terms of meaningful geometric data. Lott’s estimate uses the Lott Lp-boundedness of zero order pseudo-differential operators and Sobolev embedding the- orems, hence correspondingconstants are hard to interpret withoutusing explicitcoordi- nates. Thestartingpointoftheauthor’sPhDthesis[Amm98]andofthepresentarticleistofind an explicitlowerbound forµ that uses informationabout thespin structure. All explicit 1 lowerestimatesknownbeforedid notuseanyinformationaboutthespinstructure. For general compact Riemannian manifolds the problem of finding such estimates is rather difficult. It is not clear at all what kind of data from the spin structure could be usedinordertogetanadditionalterminalowereigenvalueestimate. Takeforexamplea compact manifoldwithnon-vanishingAˆ-genus. It hasµ = 0foranyspinstructure, thus 1 thecontributionofthespinstructureintheestimatehas tovanish. As the general case is hard to handle, most of the article will specialize to the 2-dimen- sional torus T2. By the uniformization theorem any 2-dimensional torus is conformally equivalenttoaflat torus. Weusethisfact inordertocontrolthegeometry. Animportant, but also very technical step for this is the estimate of the oscillation of the conformal factor (Section 9). Although our main goal was to find lower estimates for the Dirac eigenvalues,itturnsoutthatthismethodgivesupperandlowerboundsforalleigenvalues both of theLaplace operator and the Dirac operator and for any spin structure. We prove differentversionsoftheestimates. Theorem2.2forexamplestatesforthefirsteigenvalue µ ofthesquareoftheDirac operator 1 µ area C κ (3) 1 Ammann ≥ · where C > 0 is an explicit constant depending on the spin-conformal class and Ammann κ 1 is a curvature expression that satisfies κ = 1 if the metric is flat. This estimate is ≤ sharp foranyflat metric. In view of Lott’s result (2), it is tempting to conjecture that we can drop the curvature term, i.e. µ area C . This is false however: we can prove by example at the i Ammann ≥ end of section 12 that for many spin-conformal structures the optimal constant in Lott’s estimateis notattainedby aflat torus. In section 12 we will prove some lower bounds for the Willmore functional that are strongly related to our lower estimates of the Dirac eigenvalues. In particular we prove 3 for embeddings T2 S3 that under a curvature condition the Willmore functional con- → verges to if the spin-conformal type of the embedding converges to one end of the ∞ spin-conformalmodulispace(Corollary 12.5). The results in this paper about the Willmore conjecture are strongly related to another preprint of the author [Amm00]. The results of the present article are stronger near one of the ends of the spin-conformal moduli space but they have other drawbacks. Namely, they do notgeneralize easilyto highercodimensionsand theyimposea restrictionon the spin-conformalclass. The structure of the article is as follows: In section 2 we will state our spectral estimates on 2-tori. Sections 3 to 11 provide proofs of the statements in section 2. We then ap- ply Theorem 9.1 once again and derive an application to the Willmore functional that is related to ourlowereigenvalueestimates. Finally in section 13 we will prove a result for arbitrary spin manifolds M. Let M carry twodifferentspinstructuresϑandϑ′. Thedifferenceofthesespinstructuresχ := ϑ ϑ′ − is an element in H1(M,Z2) = HomZ(H1(M,Z),Z2). Assume that χ vanishes on the torsion part of H (M,Z). We will define a norm χ , the stablenormof χ. We prove 1 k kL∞ that the eigenvalues (ρi)i∈Z of the Dirac operator corresponding to ϑ and the eigenvalues (ρ′i)i∈Z correspondingtoϑ′ can benumbered sothat ρ ρ′ 2π ϑ ϑ′ . | i − i| ≤ k − kL∞ Ifthespectrumisknownforϑandif ρ > 2π ϑ ϑ′ foranyi Z,thenthisyields | i| k − kL∞ ∈ a lowerboundforanyρ′. i At the end of the introduction we want to mention some other publications that treat the interplay between spin structures and the spectrum of the Dirac operator. However, they do not derive expliciteigenvalue bounds for generic metrics. We will restrict to the most recentones. Forfurtherreferencesandagoodoverviewofthesubjectwereferto[Ba¨r00]. Dahl [Dah99] shows that the difference of the eta-invariants corresponding to two dif- ferent spin structures is an integer, if the difference of the spin structures viewed as an element in HomZ(H1(M,Z),Z2) vanishes on the torsion part. Ba¨r [Ba¨r98a] calculated the essential spectrum of hyperbolic2- and 3-manifolds of finite volume. In these exam- ples, the essential spectrum depends on the spin structure at the cusps. Pfa¨ffle [Pfa¨99] calculated the spectrum and the η-invariants of flat Bieberbach manifolds. These spectra also dependon thespinstructure. Severalresultsinthepresentarticlealreadyappearedintheauthor’sPhDthesis[Amm98]. 4 2 Main results InthissectionwesummarizeourresultsaboutthespectraofDiracandLaplaceoperators on 2-tori. ThespectrumoftheDiracoperatordependsonthespinstructure. Atfirst,werecallsome importantfactsaboutspinstructuresandintroducesomenotation. Spinstructureswillbe discussedin moredetail insection4. Let M be a compact orientable manifold with vanishing second Stiefel-Whitney class w (TM) = 0. Such manifolds admit a spin structure. However, the spin structure is 2 not unique in general. The group H1(M,Z ) acts freely and transitively on the set of 2 spin structures Spin(M), i.e. Spin(M) is an affine space associated to the vector space H1(M,Z ). After fixing a spin structure and a Riemannian metric on M we can define 2 thespinorbundleΣM M and aDiracoperatorD : Γ(ΣM) Γ(ΣM). → → We are mainly interested in the case M = T2. The 2-dimensional torus T2 is spin. Be- cause of #Spin(T2) = #H1(T2,Z ) = 4 there are 4 spin structures on T2. There is 2 exactly one spin structure in Spin(T2) for which 0 lies in the spectrum of D, regardless of the underlying metric g. This spin structure will be called trivial (see section 4 for other characterizations). We will identify the trivial spin structure with 0 H1(T2,Z ). 2 ∈ Thisidentificationyieldsan identificationoftheaffinespaceSpin(T2)withH1(T2,Z ). 2 On the other hand, we will identify H1(T2,Z2) with HomZ(H1(T2,Z),Z2). Hence spin structures on T2 are in a canonical one-to-one relation to such homomorphisms. Fre- quently, we will use the term “spin homomorphism” instead of “spin structure” in order to indicatethat weregardthespinstructureas an elementin HomZ(H1(T2,Z),Z2). If the torus T2 carries a flat metric, it is very helpful to write the torus as R2/Γ with a lattice Γ = H (T2,Z). We always assume that R2/Γ carries the metric induced by the ∼ 1 Euclidean metric on R2. Let Γ∗ be the lattice dual to Γ. Elements χ HomZ(Γ,Z2) are ∈ represented byvectorsα (1/2)Γ∗ withtheproperty ∈ χ(x) = ( 1)2α(x) x Γ. − ∀ ∈ Notethatχdeterminesα onlyup toelementsinΓ∗. We definethefunction : [0,4π[ [0, [ ]1, [ ]0, ] ]0, ]by S × ∞ × ∞ × ∞ → ∞ p ′ 1 2 ′ ( , ′,p, ) := K + log 1 K + K log K + KV S K K V p−1 "4π 2(cid:12)(cid:12) (cid:18) − 4π(cid:19)(cid:12)(cid:12) 8π −2K K !# 8 (cid:12) (cid:12) for > 0 and (0, ′,p, ) := 0. (cid:12) (cid:12) K S K V Let Area bethearea of(T2,g). g THEOREM 2.1. Let(T2,g)beaRiemannian2-toruswithspinhomomorphismχ. Choose alatticeΓinR2withvol(R2/Γ) = 1togetherwithaconformalmapA : R2/Γ (T2,g). → 5 Assume that A∗(χ) is represented by α (1/2)Γ∗. Let 0 ℓ ℓ ℓ ... be the 0 1 2 ∈ ≤ ≤ ≤ ≤ sequence of lengths of Γ∗ + α (with multiplicities), and let (µ i = 1,2,...) be the i | spectrum ofD2 on (T2,g,χ). Then e−2oscu4π2ℓ2 µ Area e2oscu4π2ℓ2 . [i−1] ≤ i g ≤ [i−1] 2 2 If K < 4π,then k gkL1(T2,g) oscu K , K Area 1−(1/p),p,σ (T2,g)−2 (4) ≤ S k gkL1(T2,g) k gkLp(T2,g) g 1 (cid:16) (cid:17) with σ (T2,g) := inf length(β) β Γ 0 . 1 | ∈ −{ } n o Thenumberσ (T2,g)is aconformalinvariantof(T2,g)whichwillbecalled cosystole. 1 The most difficult step in the proof of this theorem is to find the estimate (4). This step willbeperformedinTheorem9.1. Forprovingtheabovetheorem,wewillusetheexplicit formula for the spectra of flat tori (Proposition 7.2, [Fri84]). Another important tool for theproofisthefollowingproposition. PROPOSITION5.2. Let M beacompactmanifoldwithtwo conformalmetricsg˜andg = e2ug˜. Let D and D be the correspondingDirac operatorswith respect to a common spin structure. We denote the eigenvalues of D2 by µ µ ... and the ones of D2 by 1 2 f ≤ ≤ µ µ .... 1 2 ≤ ≤ f Then e e µ min e2u(m) µ µ maxe2u(m) i = 1,2,.... i i i m∈M ≤ ≤ m∈M ∀ e This proposition is based on Hitchin’s transformation formula for spinors [Hit74] (see section 5fora proof). In section 6 we will define a norm on H1(T2,Z ), the L2-norm. This norm allows us to 2 deriveexplicitlowerboundsforthefirsteigenvalueofD2onT2. Thislowerboundisnon- trivialifthespinstructureisnon-trivial. Thecosystoleσ (T2,g)canalsobeexpressedin 1 terms oftheL2-norm σ (T2,g) := inf α α H1(T2,Z ), α = 0 . 1 k kL2 | ∈ 2 6 n o (see section6, inparticularProposition6.1(a)). THEOREM 2.2. Let(T2,g)beaRiemannian2-toruswithspinhomomorphismχ. Assume that K < 4π. Then thefirsteigenvalueµ ofD2 satisfies k gkL1(T2,g) 1 4π2 χ 2 µ Area k kL2 , 1 g ≥ exp 2 ( K , K Area 1−(1/p),p,σ (T2,g)−2 S k gkL1(T2,g) k gkLp(T2,g) g 1 (cid:16) (cid:17) The equalityisattainedifandonlyifg isflat. 6 From this theorem we will obtain two corollaries estimating µ in terms of the systole 1 sys , thespinningsystolespin-sys and thenon-spinningsystolenonspin-sys . 1 1 1 sys (T2,g) := inf length(γ) γ isanon-contractibleloop. 1 { | } spin-sys (T2,g,χ) := inf length(γ) γ isaloopwith χ([γ]) = 1. 1 { | − } nonspin-sys (T2,g,χ) := inf length(γ) γ isanon-contractibleloopwithχ([γ]) = 1 1 { | and [γ]isaprimitiveelementinH (T2,Z). 1 } An element α H (T2,Z) is called primitive if there are no k N, k 2, β 1 ∈ ∈ ≥ ∈ H (T2,Z) withα = k β. 1 · COROLLARY 2.3. Let (T2,g) be a Riemannian 2-torus with non-trivial spin homomor- phismχ. Assumethat K < 4π. Then thefirsteigenvalueµ ofD2 satisfies k gkL1(T2,g) 1 π2nonspin-sys (T2,g,χ)2 µ Area 2 1 . 1 g ≥ exp 2 ( K , K Area 1−(1/p),p, Areag S k gkL1(T2,g) k gkLp(T2,g) g sys1(T2,g)2 (cid:16) (cid:17) The equalityisattainedifandonlyifg isflat. COROLLARY 2.4. Let (T2,g) be a Riemannian 2-torus with non-trivial spin homomor- phismχ. Assumethat K < 4π. Then thefirsteigenvalueµ ofD2 satisfies k gkL1(T2,g) 1 π2 µ spin-sys (T2,g,χ)2 . 1 1 ≥ exp 4 ( K , K Area 1−(1/p),p, Areag S k gkL1(T2,g) k gkLp(T2,g) g sys1(T2,g)2 (cid:16) (cid:17) The equalityisattainedifandonlyif (a) g isflat,i.e. (T2,g)isisometrictoR2/Γfora suitablelatticeΓ,and (b) therearegeneratorsγ ,γ forΓ statisfyingγ γ ,χ(γ ) = 1and χ(γ ) = 1. 1 2 1 2 1 2 ⊥ − UsingProposition6.1andtheinequalitiesfromsection10thetwocorollariesimmediately followfrom Theorem2.2. We now turn to theLaplace operatorand to theDirac operatorassociated to a trivialspin structure. We recall a well-known proposition that is the analogue of Proposition 5.2 for theLaplacian onsurfaces (section5). PROPOSITION5.1. LetM beacompact2-dimensionalmanifoldwithtwoconformalmet- rics g˜ and g = e2ug˜. The eigenvalues of the Laplacian on functions corresponding to g and g˜willbedenoted as0 = λ < λ λ ...and 0 = λ˜ < λ˜ λ˜ ...respectively. 0 1 2 0 1 2 ≤ ≤ Then λ min e2u(m) λ λ maxe2u(m) i = 1,2,.... i i i m∈M ≤ ≤ m∈M ∀ e 7 TogetherwithProposition7.1 and Theorem9.1 weobtain THEOREM 2.5. Let (T2,g) be a torus conformally equivalent to R2/Γ, vol(R2/Γ) = 1. Let Γ∗ be the latticedual to Γ. Let 0 ℓ ℓ ℓ ... be the sequence of lengths of 0 1 2 ≤ ≤ ≤ ≤ Γ∗, and let (λ i = 0,1,2,...) be thespectrum of theLaplacian on functionson (T2,g), i | then e−2oscu4π2ℓ2 λ Area e2oscu4π2ℓ2. i ≤ i g ≤ i If K < 4π,then k gkL1(T2,g) oscu K , K Area 1−(1/p),p,σ (T2,g)−2 . ≤ S k gkL1(T2,g) k gkLp(T2,g) g 1 (cid:16) (cid:17) NotethatthistheoremalsoprovidesboundsfortheLaplacian onforms: By Poincare´ du- alitythespectrumon2-formsisthesameasthespectrumonfunctions,andtheLaplacian on 1-formsalso hasthesamenon-zero eigenvalues,but each withmultiplicitytwo. Thetheorem implies,inparticular, alowerboundon thefirst positiveeigenvalue. THEOREM 2.6. Let (T2,g) be a Riemannian 2-torus. Assume that K < 4π. k gkL1(T2,g) Then thefirstpositiveeigenvalueλ oftheLaplacianonfunctionssatisfies 1 4π2σ (T2,g)2 1 λ Area . 1 g ≥ exp 2 ( K , K Area 1−(1/p),p,σ (T2,g)−2 S k gkL1(T2,g) k gkLp(T2,g) g 1 (cid:16) (cid:17) The equalityisattainedifandonlyifg isflat. COROLLARY 2.7. Let(T2,g)beaRiemannian2-torus. Assumethat K < 4π. k gkL1(T2,g) Then thefirstpositiveeigenvalueλ oftheLaplacianonfunctionssatisfies 1 4π2sys (T2,g)2 λ Area 2 1 . 1 g ≥ exp 2 ( K , K Area 1−(1/p),p, Areag S k gkL1(T2,g) k gkLp(T2,g) g sys1(T2,g)2 (cid:16) (cid:17) The equalityisattainedifandonlyifg isflat. Remark. Theorem2.6andCorollary2.7alsoholdforthefirstpositveeigenvalueofD2, ifthespinstructureistrivial. Theorem2.5holdsforthespectrumofD2,ifwedoublethe multiplicities. Thestructureofthepaperisasfollows: Inthefollowingsections(sections3–11)wewill prove our main results. In section 12, we will apply the inequalities in Proposition 9.1 in order to obtain a lower bound on the Willmore functional. Finally, in section 13 we assumethatamanifoldofabitrarydimensionn 2carriestwospinstructures. Wederive ≥ an upperboundforthespectraofthecorrespondingDiracoperators. 8 3 Overview We want to obtain upper and lower bounds for the eigenvalues of the Dirac operator and theLaplaceoperatoron aRiemannian2-torus(T2,g). TheCliffordactionofthevolumeelementonspinorsanticommuteswiththeDiracopera- torD. Thus,thespectrumofD issymmetricandisuniquelydeterminedbythespectrum of its square D2. Therefore we will study the spectrum of D2 instead of the spectrum ofD. In theliteratureD2 isoften called theDirac Laplacian. In order to prove bounds on eigenvalues we use the uniformization theorem which tells usthatwecan writeg asg = e2ug withareal-valued functionuandaflat metricg . For 0 0 flattorithespectrumoftheLaplacianandtheDiracoperatorisknown: thespectracanbe calculated in termsofthedual latticecorrespondingto (T2,g ). 0 We obtainboundsthroughthefollowingsteps. (a) Comparisonofthespectrumof(T2,g)andthespectrumof(T2,g )(Propositions5.1 0 and 5.2). (b) Introduction of certain spin-conformal invariants that contain information about the dual latticecorrespondingto(T2,g ) (section 6). 0 (c) Theknowledgeofspectraofflat tori(section7). (d) A boundon oscu = maxu minu (section9). − (e) Derivation, in section 10, of certain inequalities that are in a sense inverse to the inequalitiesinProposition6.1 and containacurvatureterm. In section11, wecombinetheinequalitiesand derivethemainresults. 4 Spin structures The eigenvalues of D depend on the spin structures and we want to find estimates de- pending on the spin structure. In this section we recall some important facts about spin structures. Good references about spin structures are [LM89], [BG92] and [Swi93, sec- tionII].WewilldefinespinstructureswithoutfixingaRiemannianmetric. Thisdefinition willallowustoidentifyspinstructuresondiffeomorphicbutnotisometricmanifolds(see Proposition5.2). Let M be an oriented manifold of dimension n 2. The bundle GL+(M) of oriented ≥ bases over M is a principal GL+(n,R)-bundle. The fundamental group of GL+(n,R) is 9 Z for n = 2 and Z for n 3. Therefore GL+(n,R) has a unique connected double 2 ≥ coveringΘ : GL+(n,R) GL+(n,R). → Definition. Agspin structure on M is a pair (GL+(M),ϑ) where GL+(M) is a principal GL+(n,R)-bundleoverM andϑ isadoublecoveringGL+(M) GL+(M) such that g →g g GL+(M) GL+(n,R) GL+g(M) × → g g g ց ϑ Θ ϑ M (5) ↓ × ↓ ր GL+(M) GL+(n,R) GL+(M) × → commutes. Thehorizontalarrows are givenbythegroupaction. There is a spin structure on M if and only if the second Stiefel-Whitney class w (TM) 2 vanishes. Such manifoldsare called spin. Fromnowon weassumethat M is spin. + Two spin structures (GL+(M),ϑ) and (GL (M),ϑ ) are identified if there is a fiber 1 1 preservingisomorphismofprincipalGL+(n,R)-bundlesα : GL+(M) GL+(M) with g g → 1 ϑ = ϑ α. 1 ◦ g g g Thesetofallspinstructures(GL+(M),ϑ)overM willbedenotedbySpin(M). Theset Spin(M) has the structure of an affine space associated to the vector space H1(M,Z ), 2 g i.e. H1(M,Z ) acts freely and transitively on Spin(M). We will describe this action: 2 Elements in H1(M,Z ) can be viewed as principal Z -bundles over M [LM89, Ap- 2 2 pendix A]. Let π : P M be the Z -bundle defined by χ H1(M,Z ). Let χ 2 2 → ∈ (GL+(M),ϑ) be a spin structure. The group Z acts by deck-transformation both on 2 GL+(M) andP . Wedefine g χ g GL+(M) := (GL+(M) P )/Z 1 ×M χ 2 where Z acts diagonallyognthefiberwisegproductofthebundles. Themap 2 ϑ π : GL+(M) P GL+(M) (A,α) ϑ(A) M M χ × × → 7→ is invariant under the Z -gaction and therefore defines a map ϑ : GL+(M) GL+(M) 2 1 1 → + compatiblewith(5). Theactionofχmaps(GL+(M),ϑ)tothespinstructure(GL (M),ϑ ). g 1 1 Thisaction isfree and transitive[LM89,II 1]. §g g NowwefixaRiemannianmetricgonM. ThisreducesourstructuregroupfromGL+(n,R) to SO(n). The bundle of positively oriented orthonormal bases SO(M,g) is a principal SO(n)-bundle. The spin group is defined by Spin(n) := Θ−1(SO(n)) and is the unique connected double covering of SO(n). A metric spin structure is a pair (Spin(M,g),ϑ) 10