The Dirac operator and codimension one collapse

In Chapter 2.2 we introduced the set Mpn`1, d, Cq:“

"

pM, gq PMpn`1, dq:Cď volpMq injpMq

*

of isometry classes of closedpn`1q-dimensional Riemannian manifolds. In Theorem 2.17 we showed that any n-dimensional limit space pB, hq of a sequence in Mpn `1, d, Cq is a Riemannian orbifold with a C^1,α-metric h. In addition, the second derivatives of h exist almost everywhere and }sec^h}L⁸ ď Kpn, d, Cq. As discussed in Chapter 3.2.1 spin structures and Dirac operators are also defined on Riemannian orbifolds. In this section we give an explicit description of the structure and behavior of the Dirac spectrum on any collapsing sequence inMpn`1, d, Cq.

LetpM_i, g_iqiPNbe a sequence of Riemannian spin manifolds inMpn`1, d, Cq converg-ing to an n-dimensional Riemannian orbifold pB, hq. By Corollary 1.29 there is an index I such that for any iěI there is a fibration fi :Mi ÑB defining an S¹-orbifold bundle with structure group in AffpS¹q – S¹ ¸ t˘1u. If B is orientable, then f_i : M_i Ñ B is an S¹-principal bundle. Otherwise, we take the orientation cover Bˆ of B and consider the pullback bundle fi : ˆMi Ñ B. As the structure group of the fibrationˆ fi : Mi Ñ B lies in AffpS¹q this pullback bundle is an S¹-principal bundle. Hence, it often suffices to consider sequences of S¹-principal orbifold bundles. Moreover, for all i ěI there are metrics ˜gi on Mi and ˜hi on B such that the fibrations fi : pMi,˜giq Ñ pB,h˜iq are Rie-mannian submersions, see Corollary 1.29. In addition, we have lim_iÑ8}˜g_i ´g_i}C¹ “ 0 and lim_iÑ8}˜h_i´h}_C¹ “0. Since Dirac eigenvalues are continuous under a C¹-variation of metrics, see Theorem C.4, it suffices to consider sequences ofRiemannian S¹-principal bundles i.e. S¹-principal bundles f : pM, gq Ñ pB, hq such that f is a Riemannian sub-mersion.

For the moment we fix a Riemannian S¹-principal bundle f : pM, gq Ñ pB, hq such thatpM, gqis a spin manifold with a fixed spin structure. As discussed in Chapter 3.2.1 we distinguish between two kinds of spin structures on the total spacepM, gq, the projectable spin structures, if the S¹-action lifts to P_SpinM, and the non projectable spin structures, where a double cover of the S¹-action acts on P_SpinM. If the spin structure on M is projectable then the spin structure onM induces a spin structure onB. Otherwise there is an induced spin^c structure. In the following, we discuss the structure of the spinor bundleΣM following [Amm98a], [Amm98b, Kapitel 7].

The isometricS¹-action on pM, gq induces a Killing vector field K. Furthermore, the length of the S¹-fibers of f : M Ñ B equals 2πl, where l :“ |K|. In particular, we can viewl as a function defined on the base space B. As explained above, there is always an induced isometricS¹-action on PSpinM. Thus, we can define the Lie-derivative of a spinor

4.2. THE DIRAC OPERATOR AND CODIMENSION ONE COLLAPSE 81 ϕin the direction of K as

L_Kϕpxq:“ d ds

ˇ ˇ ˇ ˇs“0

κ_´spϕpκ_spxqqq,

where κ denotes the S¹-action on ΣM and on M respectively. By construction L_K is the differential of the S¹-action on L²pΣMq. It follows from representation theory for compact abelian groups that L_K has the eigenvaluesik, where k PZif the spin structure onM is projectable andk P pZ`¹₂qif the spin structure on M is non projectable. Thus, L²pΣMq decomposes as

L²pΣMq “à

k

V_k,

where V_k is the eigenspace ofL_K with respect to the eigenvalueik.

Since S¹ acts on L²pΣM) as isometries, L_K commutes with the Dirac operator D^M. Therefore, L_K and D^M are simultaneously diagonalizable, i.e. for any eigenspinor ϕ of D^M there is ak such that ϕP V_k. For any fixedk, letλ_j,k be the eigenvalues ofD^M_|V

k such that

. . .ďλ_´1,k ďλ_0,k ă0ďλ_1,k ďλ_2,k ď. . . .

Remark 4.9. It is easy to check that LK is the same as ∇^aff_K in this setting. Hence, V0

is the space of affine parallel spinors.

Let L :“ M ˆS¹ C be the associated line bundle. For any k P Z, resp. k P pZ` ¹₂q, Ammann constructed the isometry

Q_k :L²p^˛ΣB bL^´kq ÑV_k,

in [Amm98b, Lemma-Definition 7.2.3]. Fork “0this isometry coincides with the isometry constructed in Lemma 3.26. Using the isometryQ_kwe can generalize the bounds on Dirac eigenvalues given in Theorem 4.1 to any collapsing sequence in Mpn`1, d, Cq, proving Proposition 0.4.

Proposition 4.10. Let pM_i, g_iqiPNbe a sequence of spin manifolds inMpn`1, d, Cq con-verging to ann-dimensional Riemannian orbifold pB, hq. Suppose that the spin structures on Mi are either all projectable or non projectable. Then we can number the Dirac eigen-values pλ<sub>j,k</sub>piqq<sub>j,k</sub> withj P Zand kP Z(projectable spin structures), resp. k P pZ`¹₂q(non projectable spin structures), such that for any εą0 there is an indexI ą0 such that for all i ě I there are fibrations fi : Mi Ñ B with fibers diffeomorphic to S¹ such that for all j P Z and k PZ (projectable spin structures), resp. k P pZ` ¹₂q (non projectable spin structures),

|λ_j,kpiq| ě sinh ˆ

arsinh ˆ |k|

}l_i}8

´1 2

”n 2

ı¹₂

C_A´ε

˙

´ε

˙ .

Here 2πl_i is the length of the fibers and C_A is a constant depending on n, d and C.

In particular, limiÑ8|λj,kpiq| “ 0 whenever k ‰0 since limiÑ8li “0.

82 CHAPTER 4. THE BEHAVIOR OF DIRAC EIGENVALUES For any i ě I, let ω_i P ΩpM_i,V_iq be the orthogonal projection onto V_i :“ kerpdf_iq, where f_i :M_i ÑB. If, in addition, there is a constant C such that

}dω_i}C^0,1 ďC

for all iěI, then for all j P Z and k P Z (projectable spin structures), resp. k P pZ` ¹₂q (non projectable spin structures),

lim sup

iPN

ˆ

minpPB l_ippq|λ_j,kpiq|

˙ ď |k|.

Proof. By Corollary 1.29 and Proposition 1.31, there is an index I₁ such that for any i ě I₁ there is an S¹-orbifold bundle f_i : M_i Ñ B with affine structure group. If B is orientable then this is anS¹-principal orbifold bundle. If B is non orientable we consider the pullback bundle fˆ_i : ˆM_i Ñ Bˆ over the orientation covering B. Since the structureˆ group of the fibration f_i : M_i Ñ B lies in AffpS¹q – S¹ ¸ t˘1u this pullback is an S¹-principal orbifold bundle. As (non) projectable spin structures pull back to (non) projectable spin structures and as the spectrum σpD^Miq is a subset of σpD^Mˆiq we can assume without loss of generality that the limit spaceB is orientable

Applying Corollary 1.29 to the sequence pMi, giqiPN there are metrics g˜i on Mi and metrics˜h_ionB such that the fibrationf_i :pM_i,g˜_iq Ñ pB,˜h_iqis a RiemannianS¹-principal orbifold bundle for alliěI₁. Moreover,

iÑ8lim }g˜_i´g_i}C¹ “0,

iÑ8lim}˜h_i´h}_C¹ “0.

The change of the Dirac spectra is controlled by

|arsinhpλ^D_j,k^˜ piqq ´arsinhpλ^D_j,kpiqq| ďC}g_i´g˜_i}C¹, (4.10.1) for a positive constant C, see Theorem C.4. Here λ_j,kpiq^D denotes an eigenvalue of D^Mi and λ_j,kpiq^D˜ an eigenvalue of D˜^Mi, where the eigenvalues are numbered as explained in the beginning of this section. At this point we want to remark, that the numbering of the Dirac eigenvalues was derived for Riemannian S¹-principal orbifold bundles only. Hence, this numbering is a priori only defined for the eigenvaluespλ^D_j,k^˜ piqq_j,k of the Dirac operator D˜^Mi on pM_i,˜g_iq. Nevertheless, it follows from Theorem C.4 that there is an induced numbering pλ^D_j,kpiqqj,k of the eigenvalues of the original Dirac operator D^Mi on pM, g_iq such that the inequality (4.10.1) holds.

As shown in [Amm98b, Beweis von Satz 7.2.1], see also [Amm98a], the Dirac operator can be written as

D˜^Mi “ 1 l_iγ

ˆKi

l_i

˙

L_Ki `D^Hi ´1 4γ

ˆKi

l_i

˙

γpl_iF_iq,

where F_i :“ d˜ω_i is the curvature of the unique connection one-form i˜ω_i, whose kernel is orthogonal to the fibers, and D^Hi is described by its action on the eigenspaces V_kpiq of L_Ki,

D^Hi_|Vkpiq:“Q_k,i˝D_k,i˝Q^´1_k,i.

4.2. THE DIRAC OPERATOR AND CODIMENSION ONE COLLAPSE 83 In the above equation, D_k,i is the twisted Dirac operator on Σ_iBbL^´k_i if n is even, and it is the twisted Dirac operator onpΣ^{`</sup><sub>i</sub> B‘Σ<sup>´</sup><sub>i</sub> Bq bL<sup>´k</sup><sub>i</sub> , if nis odd. Here Σ<sup>`}_i B and Σ^´_i B are two copies of the spinor bundle Σ_iB. However, Clifford multiplication by vector fields X P ΓpT Bq acts on Σ^`_i B asγpXq and on Σ^´B as ´γpXq. For more details we refer the reader to Appendix B.

Moreover, a straightforward calculation shows that

}l_iF_i}8 “2}A_i}8, (4.10.2) where A_i denotes the A-tensor of the Riemannian submersion f_i :pM_i,˜g_iq Ñ pB,˜h_iq. By Theorem 2.17 there is a constantKpn, d, Cqsuch that}sec^B}L⁸ ďKpn, d, Cq. Now we fix a positive constant K˜pn, d, Cq ąKpn, d, Cq and a positive constant C˜pnq ą1. It follows from Lemma 1.27 that there is an index I₂ ě I such that for all i ě I₂ the sectional curvature of pB,˜h_iq is bounded by K, i.e.˜ }sec^˜hi}8 ď K˜ and the sectional curvature of pM_i,˜g_iq is bounded by C, i.e.˜ }sec^g˜i}8 ďC.˜

Let pξ1, . . . , ξn, ζ1q be a split orthonormal frame, see Definition 3.1. In particular, pξ₁, . . . , ξ_nq is the horizontal lift of an orthonormal frame pξˇ₁, . . . ,ξˇ_nq in pB,˜h_iq. Now it follows directly from O’Neill’s formula (B.7.3) that

|A_i|² “ ÿ

iăj

|Apξ_i, ξ_jq|²

“ 1 3

ÿ

iăj

sec^˜hipξˇ_i,ξˇ_iq ´sec^˜gipξ_i, ξ_jq ď npn´1q

6

´K˜ `C˜pnq

¯

“:C_Apn, d, Cq². Applying [HM99, Lemma 3.3] and the identity (4.10.2) we obtain

›

›

›

› 1 4γ

ˆK_i l_i

˙

γpl_iF_iq

›

›

›

›8

ď 1 4

”n 2

ı¹₂

}l_iF_i}8 ď 1 2

”n 2

ı¹₂

}A_i}8 ď 1 2

”n 2

ı¹₂ C_A.

Since ¹₄γ

´Ki

li

¯

γpliFiq is symmetric it follows from [Kat76, Chapter 5, Theorem 4.10] that dist

ˆ

σpD˜^Miq, σ ˆ1

l_iγ ˆKi

l_i

˙

L_Ki `D^Hi

˙˙

ď

›

›

›

› 1 4γ

ˆKi

l_i

˙

γpl_iF_iq

›

›

›

›8

ď 1 2

”n 2

ı¹

2 C_A. (4.10.3) Let λ^W_j,kpiq be an eigenvalues of W_i :“ _l¹_iγ

´Ki

li

¯

L_Ki `D^Hi. It was shown in [Amm98a]

that for any εą0 there is anI ěI₂ such that

|λ^W_j,kpiq| ě |k|

}l_i}8

´ε,

for all iěI. Applying the inequalities (4.10.3) and (4.10.1) we obtain the claimed lower bound.

84 CHAPTER 4. THE BEHAVIOR OF DIRAC EIGENVALUES It remains to prove the upper bound. As explained in the beginning of this proof it suffices to consider theS¹-principal bundles f_i :pM_i,˜g_iq Ñ pB,˜h_iq. Leti˜ω_i be the unique connection one-form such thatkerpω˜_iqis orthogonal to the fibers with respect tog˜_i. Since

iÑ8lim }˜g_i´g_i}C¹ “0,

iÑ8lim}˜h_i´h}_C¹ “0,

andω˜_icoincides with the orthogonal projection ontokerpdf_iqit follows from the additional assumptions that the curvaturesF˜_i :“d˜ω_i are all uniformly bounded in C^0,1pB, hq for all iě I. Thus, we can apply Theorem D.8 to deduce that there is subsequence pMi,g˜iqiPN

such that allM_iare diffeomorphic to a fixed manifoldM and such that the sequencepω˜_iqiPN

of connection one-forms converge inC^1,α for anyαP r0,1q. Applying these isomorphisms it suffices to consider the sequence

´

fi :pM,g˜iq Ñ pB,˜hiq

¯

iPN

of S¹-principal bundles.

Next, we recall that for anyk, D_|V^Hi

kpiq “Q_k,i˝D_k,i˝Q^´1_k,i.

As the manifold M does not depend on i the same holfd for the associated line bundle L“MˆS¹C. Moreover, as in Step 4 of the proof of Theorem 4.5, we obtain an isometry

L²p^˛Σ_iB bL^´kq ÑL²p^˛ΣBbL^´kq,

for anyiPN. Applying these isometries and thatlim_iÑ8}˜h_i´h}_C¹ “0and the connection one-forms pω˜_iqiPN converge in C^1,α for any αr0,1q it follows from Theorem C.5 that the spectrum pµ_jpiqq_jPZ of the twisted Dirac operators D_k,i converges in the arsinh-topology to the spectrumpµ_jqjPZ of the twisted Dirac operator D_k,8 on ^˛ΣBbL^´k. As before, we consider the operator

W_i :“ 1 l_iγ

ˆK_i l_i

˙

L_Ki`D^Hi. It is straightforward to check that

W_i² “ ´1

l_i²pL_Kiq²` pD^Hiq²´γ

ˆgradpl_iq l_i²

˙ .

AsWi is a self-adjoint operator it is immediate thatW_i² is a nonnegative self-adjoint oper-ator. A straightforward calculation using the Rayleigh quotient shows that any eigenvalue λ^W_j,kpiq of W_i|Vkpiq satisfies

pλ^W_j,kpiqq² ď |k|²

minpPBlippq² `µ<sub>j</sub>piq<sup>2</sup>` }gradpl_iq}8

minpPBlippq². Multiplying this equation withmin_pPBl_ippq² leads to

min

pPB l_ippq²pλ^W_j,kpiqq² ď |k|²`min

pPB l_ippq²µ_jpiq²`min

pPB l_ippq}gradpl_iq}8

4.2. THE DIRAC OPERATOR AND CODIMENSION ONE COLLAPSE 85 Next, we observe that the T-tensor of the Riemannian submersion f_i :pM,˜g_iq Ñ pB,h˜_iq is given by

T_i ˆK_i

li

,K_i li

˙

“ gradl_i li

.

As the tensors pT_iqiPN are uniformly bounded in i, see for instance Corollary 1.29, lim sup

iÑ8

}gradpl_iq}8 “0.

Furthermore, the spectrum pµ_jpiqq_jPZ of D_k,i converges to the spectrum pµ_jqjPZ of D_k,8 in the arsinh-topology. Combining these observations we obtain that

lim sup

iÑ8

minpPB lippq²pλ^W_j,kpiqq² ď |k|².

As for the lower bound, we conclude the claim by applying the inequalities (4.10.3),

(4.10.1) and taking the limit iÑ 8. l

An immediate consequence of this proposition is that the eigenvalues of the restrictions D^M_|Vⁱ

kpiqtend to˘8in the limitiÑ 8whenever k‰0. In the case of non projectable spin structures this means that all eigenvalues diverge. Whereas in the case of projectable spin structures all eigenvalues diverge except those corresponding to the subspaceV₀piq, which is the space of affine parallel spinors. This coincides with the previous results by Ammann, Theorem 4.1 and Lott, Theorem 4.2, Theorem 4.3. In the following theorem we summarize the complete behavior of Dirac eigenvalues on collapsing sequences in Mpn`1, d, Cq.

Theorem 4.11. Let pM_i, g_iqiPN be a collapsing sequence of Riemannian spin manifolds in Mpn `1, d, Cq. Then there is a C^1,α-Riemannian orbifold pB, hq such that for a subsequence, relabeled as pM_i, g_iqiPN, there are S¹-orbifold bundles f_i :M_i ÑB for which one of the following two cases occur:

Case 1: There is a subsequence, relabeled as pMi, giqiPN such that the spin structures on pM_i, g_iq are non projectable. Then the eigenvalues of the Dirac operator D^Mi can be numbered as pλ_j,kpiqq jPZ

kPpZ`¹₂q

such that for all ε ą0 there is an index I ą0 such that for all iěI,

|λ_j,kpiq| ě sinh ˆ

arsinh ˆ |k|

}li}8

´1 2

”n 2

ı¹₂

C_A´ε

˙

´ε

˙ .

Here, 2πl_i is the length of the fibers and C_A is a constant depending on n, d and C. In particular, lim_iÑ8|λ_j,kpiq| “ 8 for all j PZ and k P pZ`¹₂q sincelim_iÑ8l_i “0.

For any i ě I, let ω_i P ΩpM_i,V_iq be the orthogonal projection onto V_i :“ kerpdf_iq, where f_i :M_i ÑB. If, in addition, there is a constant C such that

}dω_i}C^0,1 ďC for all iěI, then for all j PZ and k P pZ` ¹₂q,

lim sup

iPN

ˆ

minpPB lippq|λj,kpiq|

˙ ď |k|.

86 CHAPTER 4. THE BEHAVIOR OF DIRAC EIGENVALUES Case 2: There is a subsequence, relabeled as pM_i, g_iqiPN, such that the spin structures on pM_i, g_iq are projectable and all of them induce the same spin structure on B, if B is orientable, resp. the same pin^´-structure on B, if B is nonorientable. Then the eigen-values of the Dirac operator D^Mi can be numbered as pλ_j,kpiqqjPZ

kPZ

such that for all ε ą 0 there is an I ą0 such that for all iěI,

|λ_j,kpiq| ěsinh ˆ

arsinh ˆ |k|

}l_i}8

´ 1 2

”n 2

ı¹₂

C_A´ε

˙

´ε

˙ . In particular, lim_iÑ8|λ_j,kpiq| “ 8 for all j PZ and k‰0 since lim_iÑ8l_i “0.

For any i ě I, let ω_i P ΩpM_i,V_iq be the orthogonal projection onto V_i :“ kerpdf_iq, where f_i :M_i ÑB. If, in addition, there is a constant C such that

}dω_i}C^0,1 ďC for all iěI, then for all j P Z and k PZ,

lim sup

iPN

ˆ

minpPB l_ippq|λ_j,kpiq|

˙ ď |k|.

Fork “0, the eigenvaluesλ_j,0piqconverge uniformly with respect to thearsinh-topology to the eigenvalues of the operator

D^B` i

4ω_n^CγpFq, if n is even, ˆ D^B ₄ⁱγpFq

i

4γpFq ´D^B

˙

, if n is odd.

If B is orientable,

• D^B is the Dirac operator of B,

• ω_n^C is the complex volume element of ΣB, i.e. ω^C_n “ ir^n`12 sγpe₁q ¨ ¨ ¨γpe_nq for any orthonormal frame pe₁, . . . , e_nq,

• F is a C^0,α-two-form for α P r0,1q.

If B is non orientable

• D^B is the twisted Dirac operator on the twisted pin^´ bundleΣ^PBbdetpT Bq^C, where detpT Bq^C is the complexified determinant bundle,

• ω_n^C is the complex volume element of Σ^PB bdetpT Bq^C,

• F is a C^0,α-two-form for α P r0,1q.

4.2. THE DIRAC OPERATOR AND CODIMENSION ONE COLLAPSE 87 Proof. The behavior of the divergent eigenvalues as well as the upper bound follow di-rectly from Proposition 4.10. As usual, we switch to invariant metrics and work with the resulting sequence of Riemannian S¹-orbifold bundles f_i : pM_i,g˜_iq Ñ pB,˜h_iq. Under the assumption that all spin structures of pM_i,gq˜ _iPN are projectable, each of them induces a spin structure onB. As there are only finitely many equivalence classes of spin structures onB (see for instance [LM89, Chapter II, Theorem 1.7]), there is a subsequence relabeled aspM_i,g˜_iqiPN, such that the spin structure onpM_i,g˜_iqinduces, up to equivalence, the same spin structure onB for alliPN. We recall that for anyiPN there is a unique imaginary connection one-formi˜ω_i of the RiemannianS¹-orbifold bundlef_i :pM_i,˜g_iq Ñ pB,h˜_iqsuch that kerpω_iq is orthogonal to the fibers with respect tog˜_i. Let F_i :“d˜ω_i be the curvature form of ω˜_i. To show the convergence behavior of the eigenvalues pλ_j,0piqq_jPZ, it suffices to observe that A_i|HiˆHi “ ¹₂l_iF_iifBis orientable. IfBis non orientable we defineF_i :“2A_i. Then the claim follows from Theorem 4.5 and the rules for Clifford multiplication derived

in Appendix B. l

Finally, we want to mention that the same statement as in Proposition 4.8 also hold for general collapsing sequences in Mpn`1, d, Cq. It is easy to check that all arguments given in the proof of Proposition 4.8 also work when B is a Riemannian orbifold.

Appendix A

Infranilmanifolds

In this appendix we recall the basic properties and definitions of infranilmanifolds. For a thorough introduction to infranilmanifolds we refer to [Dek17], [CFG92, Section 3]

and [Lot02c, Section 3].

Let N be a connected and simply-connected nilpotent Lie group. The Lie algebra n of N is nilpotent, i.e. there is a k PN such that the lower central series

n₁ “n, n₂ “ rn,n₁s, n₃ “ rn,n₂s, ...

terminates at nk“0.

On the Lie groupN, there is a canonical flat connection∇^affdefined by the requirement that all left-invariant vector fields are parallel. Let AffpNq denote the subgroup of the diffeomorphism groupDiffpNqthat preserves ∇^aff. It follows thatAffpNqis isomorphic to the semi-productN_L¸AutpNq. Here N_Ldenotes the left-action of N on itself. Note that N_L isomorphic to N via the isomorphism N Ñ N_L, g ÞÑL_g. As usual, AutpNq denotes the automorphism group of N.

An infranilmanifold Z is a quotient ΓzN of a connected and simply-connected nilpo-tent Lie groupN by a cocompact discrete subgroupΓ ofAffpNq. By the generalized first Bieberbach Theorem (see for instance [Dek17, Theorem 3.4]), the subgroup Γ :ˆ “ΓXNL

is of finite index in Γ. In fact, there is a constant Cpkq depending only on k :“ dimpZq such that rΓ : ˆΓs ă Cpkq [Gro78, Main result]. Thus, we have the following diagram of short exact sequences.

1 N_L AffpNq AutpNq 1,

1 Γˆ Γ ppΓq 1.

p

It follows that Z is finitely covered by the nilmanifold Zˆ :“ΓˆzN. The finite deck trans-formation group is given by F :“ ppΓq. Since Γ is a subgroup of AffpNq it follows that the flat connection ∇^aff onN descends to a well-defined flat connection on Zˆ and on Z.

Let n be the Lie algebra of N. The space of affine parallel vector fields on N and Zˆ is isomorphic to n. Thus, the space of affine parallel vector fields on an infranilmanifold Z is isomorphic to the subspacen^F consisting of those elements that are invariant under the induced action of F onn. Obviously,n^F is finite dimensional.

89

90 APPENDIX A. INFRANILMANIFOLDS Let g be a left-invariant metric on N. For a local orthonormal frame pe₁, . . . , e_kq of pN, gq the structural coefficients of the Lie algebra n of N are given by

re_a, e_bs “

k

ÿ

c“1

τ_ab^ce_c.

The Christoffel symbols of∇^aff are trivial and the Christoffel symbols of the Levi-Civita connection can be calculated using the Koszul formula,

Γ^c_ab “ 1 2

`τ_ab^c ´τ_ac^b ´τ_bc^a˘

. (A.0.1)

It follows that∇^aff is identical to the Levi-Civita connection if and only if N is abelian, i.e. N is isometric to the additive group pR^k,`q with the euclidean metric. We fix the following terminology for tensor fields on a Riemannian infranilmanifold pZ, gq.

Notation A.1. Let pZ, gq be a Riemannian infranilmanifold. A tensor field X on Z is calledaffine parallel if it is parallel with respect to the affine connection, i.e. ∇^affX “0.

This is equivalent to say that X lifts to a left-invariant tensor field X˜ on the universal coverN. On the other hand, a parallel tensor fieldX onZ is parallel with respect to the Levi-Civita connection on pZ, gq.

In the remainder of this appendix we consider a closed k-dimensional Riemannian infranilmanifoldspZ, gqwith an affine parallel metricg. Letg_N be the lift ofgtoN. Since g is affine parallel g_N is a left-invariant metric on N. Hence, the oriented orthonormal frame bundle is trivial, i.e.P_SON –NˆSOpkq. Thus, there is a canonical spin structure onN given by

N ˆSpinpkq ÑN ˆSOpkq. (A.1.1)

We recall that the equivalence classes on a spin manifold M are in one-to-one correspon-dence with the cohomology class H¹pM,Z2q, [LM89, Chapter II, Theorem 1.7]. Since N is connected and simply-connected the cohomology group H¹pN,Z2qis trivial. Thus, the spin structure defined by (A.1.1) is, up to equivalence, the only spin structure onN.

As the metric g on Z “ΓzN is affine parallel it follows that Γ is a discrete group of isometries of pN, g_Nq. Hence, the oriented orthonormal bundle ofZ is isomorphic to

P_SOZ –ΓzpN ˆSOpkqq.

At this point we want to remark that there are examples of infranilmanifolds that are not spin, e.g. the Kleinian Bottle. An infranilmanifoldZ is spin if and only ifF ĂSOpkq and if there exists a lift

Spinpkq

Γ F SOpkq.

˜ ρ

ρ

91 The different equivalence classes of spin structures ofZ correspond to different lifts of the map ΓÑSOpkqto ΓÑSpinpkq. Moreover, the group

HompΓ,Z2q – H¹pΓ,Z2q –H¹pZ,Z2q acts freely and transitively on the set of equivalence classes.

It follows that the Spinpkq-principal bundle ofZ is given by P_SpinpZq – ΓzpN ˆSpinpkqq.

Let θ_k : Spinpkq ÑAutpΣ_kq be the canonical complex spinor representation, where Σ_k is a complex vector space with dim_CpΣkq “ 2r^k2s. If there is a given lift ρ˜ : Γ Ñ Spinpkq, then Γ acts onΣ_k via

ΓˆΣ_k ÑΣ_k,

pγ, ϕq ÞÑθ_kpρpγqqpϕq.˜ Thus, the spinor bundle of Z is defined as

ΣZ “P_SpinZ ˆθk Σ_k–ΓzpN ˆΣ_kq.

Next, we recall the affine connection ∇^aff on Z that is induced by the canonical flat connection∇^aff onN for which all left-invariant vector fields are parallel. Since the metric g on Z is affine parallel ∇^aff induces a connection on P_SOZ and on P_SpinZ. For brevity, we continue to write ∇^aff for these induced connections. In this thesis we are mainly interested in the space of affine parallel spinors on an infranilmanifold pZ, gq with an affine parallel metric g and a fixed spin structure, i.e.

P :“ tϕPL²pΣZq:∇^affϕ“0u.

First we observe that the space of affine parallel spinors is isomorphic to Σ^Γ_k “ tν PΣ_k:θ_kpρpγqqpνq “˜ ν, @γ PΓu.

Since Γˆ Ă N_L it is immediate that ρpγq “ Id for all γ P Γ. Thus,ˆ ρp˜ Γqˆ takes values in t˘1u. Here ˘1 denote the two preimages of the identity Id P SOpkq under the double cover Spinpkq Ñ SOpkq. We conclude that Σ^Γ_k “ t0u if there exists a γ P Γˆ such that

˜

ρpγq “ ´1. If ρ˜_|Γˆ “ 1 then Σ^Γ_k “ Σ^F_k, where the latter is the space of all elements in Σ_k that are fixed by the action of the finite group F “ ρpΓq. Since Σ^F_k Ă Σ_k, the space of affine parallel spinors on Z is finite dimensional.

Appendix B

Spinors on Riemannian submersions

This appendix deals with Riemannian submersions f :M ÑB where M is a spin man-ifold. We discuss how the Clifford multiplication of vertical and horizontal vectors acts on the spinors of M. The main goal of this appendix is to derive formulas for the spino-rial connection and the Dirac operator on the total space M expressing the influence of the “vertical” and the “horizontal” geometry. Afterwards we recall O’Neill’s formulas for Riemannian submersions which are constantly used in this thesis.

We start with an elementary discussion of the canonical complex spin representation (see [LM89, Chapter I, §5] for more details). First, we recall that the group Spinpnq is contained in the Clifford algebra Clpnq:“ClpCⁿq of Cⁿ.

If n is even, then there is a unique irreducible representation χ_n : Clpnq Ñ GLpΣ_nq.

In the other case, i.e. n is odd, there are two inequivalent irreducible representations χ^˘_n : Clpnq Ñ GLpΣ_nq. Here Σ_n is a complex vector space of complex dimension 2rⁿ2s. The canonical complex spin representation is defined as

θ_n :“

#

χn|Spinpnq, if n is even, χ^`_n|Spinpnq, if n is odd.

It is important to remark here, that the restrictions χ^{`</sup><sub>n|Spinpnq</sub>and χ<sup>´</sup><sub>n|Spinpnq</sub>are equivalent to each other, although the non restricted representations χ<sup>`}_n and χ^´_n are inequivalent.

If n is even the canonical complex spin representation splits Σ_n “ Σp^`_n ‘Σp^´_n such that the restrictions θ_n|Σp˘

n are inequivalent irreducible representations of Spinpnq. This splitting corresponds to the ˘1 eigenspaces of the complex volume element

ω^C_n :“ir^n`12 sγpe₁q ¨ ¨ ¨γpe_nq.

Here pe₁, . . . , e_nq is the standard basis of Rⁿ and γ : Rⁿ Ñ GLpΣ_nq denotes Clifford multiplication, i.e. γpvqγpwq `γpwqγpvq “ ´2xv, wy, where x., .y is the standard scalar product on Rⁿ. The map

Σ_n“Σp^{`</sup><sub>n</sub> ‘Σp<sup>´</sup><sub>n</sub> ÑΣ<sub>n</sub> “Σp<sup>`}_n ‘Σp^´_n, ψ “ψ<sup>`</sup>`ψ^´ ÞÑψ¯“ψ^`´ψ^´ is calledcomplex conjugation.

93

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