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Differential Geometry and its Applications 22 (2005) 229–252 www.elsevier.com/locate/difgeo

Spinc-structures and Dirac operators on contact manifolds

Robert Petit

Laboratoire de Mathématiques Jean Leray, UMR 6629 CNRS, Université de Nantes, 2, rue de la Houssinière BP 92208, 44322 Nantes, France Received 24 June 2003; received in revised form 29 January 2004

Communicated by D.V. Alekseevsky, O. Kowalski

Abstract Any contact metric manifold has a Spinc-structure. Thus, we study on any Spinc-spinor bundle of a contact metric manifold, Dirac type operators associated to the generalized Tanaka–Webster connection. Bochner–Lichnerowicz type formulas are derived in this setting and vanishing theorems are obtained.  2005 Elsevier B.V. All rights reserved.

MSC: 32V05; 53C20; 53C25; 53C27; 53D10

Keywords: Contact metric manifold; Spinc-structure; Tanaka–Webster connection; Dirac operator; Vanishing theorems

1. Introduction

In the last years, Spinc-structures have played an important part in the geometry and the topology of manifolds, and especially in dimension four by means of the Seiberg–Witten theory. Among all the manifolds endowed with a Spinc-structure, a central part is played by the almost hermitian manifolds. Ac- tually, any almost hermitian manifold has a canonical Spinc-structure determined by its almost hermitian structure. A contact metric structure on a contact manifold is the data of an almost complex structure on the contact distribution together with a metric compatible both with the almost complex structure and the contact form. A contact manifold endowed with a contact metric structure is refered as a contact metric manifold. Any contact metric manifold has a canonical Spinc-structure determined by its contact

E-mail address: [email protected] (R. Petit).

0926-2245/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.difgeo.2005.01.003 230 R. Petit / Differential Geometry and its Applications 22 (2005) 229–252 metric structure. On the other hand, recall that a basic tool in the study of contact metric manifolds is the generalized Tanaka–Webster connection [18]. This connection preserves both the contact form and the compatible metric (and also preserves the almost complex structure in the setting of contact metric CR manifolds so-called strictly pseudoconvex CR manifolds). Therefore, given a Spinc-structure on a contact metric manifold, then we can define a spinorial connection on the associated spinor bundle by means of the generalized Tanaka–Webster connection together with a connection on the determinant line bundle. A Dirac operator is canonically associated to a such connection. We also define, by restriction to the contact distribution, a sub-Riemannian analogue of this operator called in the following the Kohn–Dirac operator. One of purposes of this article is to derive some Bochner–Lichnerowicz formulas for such operators. The main motivation being to obtain vanishing theorems for the harmonic and subharmonic spinors (i.e., spinors respectively in the kernel of the Dirac operator and the Kohn–Dirac operator). The plan of this article is the following. In Section 2, we recall basic facts concerning the contact metric manifolds and the generalized Tanaka–Webster connection. The definitions of curvatures, and specially of the Tanaka–Webster scalar curvature, are given, as also Bianchi identities. In Section 3,we discuss the canonical Spinc-structure of a contact metric manifold and Dirac type operators associated to the generalized Tanaka–Webster connection. In the particular setting of strictly pseudoconvex CR man- ifolds, we note that the Kohn–Dirac operator is a square root of the Kohn Laplacian. The Section 4 is devoted to obtain Bochner–Lichnerowicz type formulas for the Kohn–Dirac operator (Proposition 4.2). In particular, we derive a Spinc-version of the Weitzenbock–Tanaka formula in [17]. As a main applica- tion, we obtain the following vanishing theorems for the harmonic and subharmonic spinors: there is no harmonic spinor on a compact spin contact metric manifold if its Tanaka–Webster torsion vanishes and its Tanaka–Webster scalar curvature is nonnegative and positive at some point (Theorem 4.2). Any subharmonic spinor ψ on a compact spin strictly pseudoconvex CR manifold with pseudo- Hermitian Ricci tensor nonnegative and pseudo-Hermitian scalar curvature positive at some point can be decomposed as ψ = ψ1 + ψ2,whereψ1 and ψ2 are holomorphic sections of a square root of the canonical bundle (Corollary 4.2). In Section 5, we explain the Bochner–Lichnerowicz formula (10) of Proposition 4.2 on a spin contact metric manifold in terms of infinitesimal variations of the Kohn–Dirac operator with respect to some variations of the metric associated to the contact metric structure (Proposition 5.1). To conclude, Nico- laescu considers in [13] Dirac operators on contact metric manifolds associated to contact connections different from the generalized Tanaka–Webster connection. These connections preserve the almost com- plex structure on the contact distribution even in the almost CR case. However, in comparison with the generalized Tanaka–Webster connection, the torsion terms are in general more complicated.

2. Contact metric manifolds

A contact form on a smooth manifold M of dimension m = 2d + 1 is a 1-form θ satisfying θ ∧ (dθ)d = 0everywhereonM.Ifθ is a contact form on M, the hyperplan subbundle H of TM given by H = Ker θ is called a contact structure. The Reeb field associated to θ is the unique vector field ξ on M satisfying θ(ξ)= 1anddθ(ξ,.) = 0. By a contact manifold (M, θ) we mean a manifold M endowed with a fixed contact form θ. R. Petit / Differential Geometry and its Applications 22 (2005) 229–252 231

Let (M, θ) be a contact manifold, then the pair (H, dθH ) is a symplectic vector bundle. We recall that an almost complex structure J on a symplectic vector bundle V is compatible with a symplectic form ω if, ω(X,JY) =−ω(JX,Y), for any X, Y ∈ V , and, ω(X,JX) > 0, for any X ∈ V/{0}.Wefixanalmost complex structure J on H compatible with dθH . Hence, gθ,H given by gθ,H(X, Y ) = dθ(X,JY) defines a Hermitian metric on H . We extend J on TM by Jξ = 0. This allows to extend gθ,H to a Riemannian metric gθ on TM (called the Webster metric) by setting

gθ (X, Y ) = dθ(X,JY)+ θ(X)θ(Y). Note that the following relations hold: 2 gθ (ξ, X) = θ(X), J =−Id + θ ⊗ ξ, gθ (J X, Y ) = dθ(X, Y ), X, Y ∈ TM.

The metric gθ is said to be associated to θ and we denote by M(θ) the set of metrics associated to θ.We call (θ,ξ,J,gθ ) a contact metric structure and (M,θ,ξ,J,gθ ) a contact metric manifold (cf. Blair [2]). p Let (M,θ,ξ,J,gθ ) be a contact metric manifold. Remember that, for any α ∈ Ω (M),wehave, setting αH = α ◦ Π where Π : TM → H is the canonical projection and αξ = θ ∧ i(ξ)α, the splitting ∗ α = αH + αξ (cf. [15,16]). It follows the decomposition of Ω (M) as ∗ = ∗ ⊕ ∧ ∗ Ω (M) ΩH (M) θ ΩH (M), ∗ where ΩH (M) is the bundle of horizontal forms. Also, we recall the decomposition of any horizontal = + := 1 ± ◦ 2-tensor µH into µH µH+ µH− ,whereµH± 2 (µH µH J)are respectively the J -invariant part and the J -anti-invariant part of µH . In the following, the torsion and the curvature of a connection ∇ are respectively defined by T(X,Y)= [X, Y ]−∇XY +∇Y X and R(X,Y) =[∇Y , ∇X]−∇[Y,X].

Proposition 2.1 (Generalized Tanaka–Webster connection, cf. [17,18,20]).Let(M,θ,ξ,J,gθ ) be con- tact metric manifold, then there exists a unique affine connection ∇ on TM with torsion T (called the generalized Tanaka–Webster connection) such that:

(a) ∇θ = 0, ∇ξ = 0, (b) ∇gθ = 0, =− ⊗ = 1 ◦ L (c) TH dθ ξ and i(ξ)T 2 (J ξ J), ∇ = 1 ∈ (d) gθ (( XJ )(Y ), Z) 2 dθ(X,NH (Y, Z)) for any X, Y, Z TM,

2 where (Lξ J )(X) =[ξ,JX]−J [ξ,X] and N(Y,Z) = J [Y,Z]+[JY,JZ]−J [Y,JZ]−J [JY,Z]+ dθ(Y, Z)ξ.

The endomorphism τ := i(ξ)T is called the generalized Tanaka–Webster torsion. Note that τ is gθ - symmetric with trace-free and satisfies τ(JX)=−J(τ(X)). The curvature R of ∇ satisfies the following Bianchi identities (cf. [5,17]):

RH (X, Y )Z + RH (Z, X)Y + RH (Y, Z)X = ωθ (X, Y )τ(Z) + ωθ (Z, X)τ(Y ) + ωθ (Y, Z)τ(X), (1) + = ∇ − ∇ R(X,ξ)Z
R(ξ,Z)X
( Xτ)(Z) ( Zτ)(X),

 g R (X, Y )Z, W − g R (Z, W)X, Y = ω (Y, Z)g τ(X),W + ω (X, W)g τ(Y),Z θ H θ H θ θ
 θ θ
 − ωθ (X, Z)gθ τ(Y),W − ωθ (Y, W)gθ τ(X),Z , 232 R. Petit / Differential Geometry and its Applications 22 (2005) 229–252


 gθ R(X,ξ)Z,W = gθ (∇W τ)(X), Z − gθ (∇Zτ)(X), W , where ωθ := dθ, (∇Xτ)(Z) =∇Xτ(Z)− τ(∇XZ) and X, Y, Z, W ∈ H . The generalized Tanaka–Webster Ricci endomorphism Ric is given by = Ric(X) tracegθ R(.,X). = =− ∇ Notes that Ric(ξ) δτ with δτ tracegθ ( .τ)(.). = The generalized Tanaka–Webster scalar curvature is s tracegθ (Ric). A contact metric manifold (M,θ,ξ,J,gθ ) for which J is integrable (i.e., ∇J = 0) is called a strictly pseudoconvex CR manifold. The following identities hold for a strictly pseudoconvex CR manifold. For any X, Y, Z ∈ H ,
 = − RicH− (X, Y ) (d 1)ωθ τ(X),Y , (2) + + RH− (X, Y )Z RH− (Z, X)Y RH− (Y, Z)X

= ωθ (X, Y )τ(Z) + ωθ (Z, X)τ(Y ) + ωθ (Y, Z)τ(X). (3)

If (M,θ,ξ,J,gθ ) is a strictly pseudoconvex CR manifold, the symmetric 2-tensor RicH+ is called the = pseudo-Hermitian Ricci tensor, the 2-form ρH (X, Y ) RicH+ (X, J Y ) is called the pseudo-Hermitian

Ricci form, and tracegθ,H RicH+ is called the pseudo-Hermitian scalar curvature. Note that the pseudo- Hermitian scalar curvature coincides with the generalized Tanaka–Webster scalar curvature.

3. Spinc-structures and Dirac operators on contact metric manifolds

In this section we will describe the special features of the Spin (Spinc)-structures on a contact metric manifold and their associated spinor bundles. For the definitions and more details about these notions we refer to [7].

3.1. Spinc-structures on contact metric manifolds

C Let (M,θ,ξ,J,gθ ) be a contact metric manifold and VC the subbundle of T M given by VC = Cξ. Then, we have

C 1,0 0,1 T M = T M ⊕ T M ⊕ VC, where T 1,0M (resp. T√0,1M) is the subbundle√ given by the eigenspace of the complex extension of J on H C to the eigenvalue −1 (resp. − −1).Wehaveanisomorphism  ∧mT CM = ∧pT 1,0M ⊗∧q T 0,1M ⊗∧r ξ , p+q+r=m where we observe that ∧r ξ =0ifr>1. We set ∧p,q := ∧p 1,0 ∗ ⊗∧q 0,1 ∗ ∧p,0 := ∧p,0 ⊕∧p−1,0 ∧ H (M) T M T M , (M) H (M) H (M) θ. := ∧d+1,0 ∧d,0 −1 The bundle K (M) H (M) and its dual, denoted K , are respectively called the canonical p,q = ∧p,q and the anticanonical bundle. In the following, we set ΩH (M) Γ( H (M)). R. Petit / Differential Geometry and its Applications 22 (2005) 229–252 233

A contact metric structure on M2d+1 produces a reduction of the SO(2d + 1)-structure of TM to the subgroup U(d)-structure, where U(d) is viewed canonically as a subgroup of SO(2d + 1). Arguing exactly as in Corollary 3.4.5 of [9], we deduce the following result:

c Proposition 3.1. Any contact metric manifold (M,θ,ξ,J,gθ ) admits a Spin -structure whose determi- nant line bundle is K−1.

c ∧0,∗ ∗ Proposition 3.2. The spinor bundle ΣM can be identified with the bundle H (M) of the (0, ) forms and the Clifford multiplication by a vector field X on a (0,q)form α is given by √
∗  √ = 0,1 ∧ − 0,1 + − q+1 − X.α 2 (XH ) α i(XH )α ( 1) 1 θ(X)α, where XH denotes the horizontal part of X and i is the contraction operator.

The proof of this proposition is based on the following lemma (as in Corollary 3.4.6 of [9]).

Lemma 3.1. For the above defined multiplication, we have

X.X.α =−gθ (X, X)α and √ − d+1 = ( 1) vgθ .α α, = 1 ∧ d where vgθ is the canonical volume element (i.e., vgθ d! θ (dθ) ).

Proof. By the above formula, we have √
∗  √
 √ = 0,1 ∧ − 0,1 + − q+1 − X.X.α 2X. (XH ) α 2X. i(XH )α ( 1) 1 θ(X)X.α
∗  √ √
∗  =− 0,1 0,1 ∧ + − q+2 − 0,1 ∧ 2i(XH ) (XH ) α ( 1) 1 2 θ(X) (XH ) α ∗ √ √ − 0,1 ∧ 0,1 + − q+1 − 0,1 2(XH ) i(XH )α ( 1) 1 2 θ(X)i(XH )α √ √
∗  √ √ + − q+1 − 0,1 ∧ − − q+1 − 0,1 − 2 ( 1) 1 2 θ(X) (XH ) α ( 1) 1 2 θ(X)i(XH )α θ (X)α =− | 0,1|2 − 2 2 XH α θ (X)α. | 0,1|2 = 1 | |2 =− ∈ 0,1 = Since√ XH 2 XH ,wehaveX.X.α gθ (X, X)α.Now,foranyX H ,wehaveJX − −1X0,1. Since the duality is antilinear, we deduce that √ √
∗  JX.α= 2 −1 (X0,1) ∧ α + i(X0,1)α . Hence, we have √ √
∗  X.J X.α = 2 −1X. (X0,1) ∧ α + i(X0,1)α √
2 ∗  = −1 −2|X0,1| α + 4(X0,1) ∧ i(X0,1)α √
∗  =− −1 |X|2α − 4(X0,1) ∧ i(X0,1)α . 234 R. Petit / Differential Geometry and its Applications 22 (2005) 229–252

Now, let {ε1,Jε1,...,εd ,Jεd ,ξ} a local orthonormal frame, where {ε1,Jε1,...,εd ,Jεd } is a local or- thonormal frame of H , then we have √
 =− − − ∗ ∧ εi.J εi.α 1 α 4Zi i(Zi)α , √ ∗ ∗ ∗ with Z = 1 (ε + −1Jε ).Letα = Z ∧ Z ∧···∧Z .Ifi ={j ,j ,...,j }, then (cf. Corollary i 2 i i j1 j2 jq 1 2 q 3.4.5 of [9]) √ ∗ ∧ = =− − Zi i(Zi)α 0andεi.J εi.α 1 α, whereas if i ∈{j1,j2,...,jq }, then ∗ 1 √ Z ∧ i(Z )α = α and ε .J ε .α = −1 α. i i 2 i i = Now, since vgθ ξ.ε1.J ε1 ...εd .J εd , then √ − √ √ √ + = − − d q − q − q+1 − = − d 1 − d+1 vgθ .α ( 1) ( 1) ( 1) 1 α ( 1) ( 1) α. √ − d+1 = 2 Hence, ( 1) vgθ .α α.

Proof of Proposition 3.2. The above action satisfies the Clifford relation and consequently extends by C ∧0,∗ = d complex linear endomorphisms to an action of l(M). Since, at each point, dim( H (M)) 2 , hence ∗ √ d+1 ∗ ∧0, (M) Cl(M) ( − ) v = ∧0, (M) H is an irreductible module for .Now,since 1 gθ Id on H , we deduce c =∧0,∗ = ∧0,q 2 that ΣM H (M) q H (M).

In the following we call the above Spinc-structure the canonical Spinc-structure of the contact metric c manifold (M,θ,ξ,J,gθ ). Now, consider any other Spin -structure on (M, gθ ) with determinant line bundle L. Then, the associated principal SpinC-bundle differs from the canonical principal SpinC-bundle by tensoring with some U(1)-principal bundle. If L is the complex line bundle associated to this U(1)- c =∧0,∗ ⊗ L L2 = ⊗ principal bundle, then the spinor bundle is ΣM H (M) . Note that K L. c c For√ any spinor√ bundle ΣM associated to a Spin -structure on (M, gθ ), the Clifford multiplications by −1ξ and −1ωθ respectively induce the decompositions (cf. [1,10,13]): ΣMc = Σ +Mc ⊕ Σ −Mc, and d c c ΣM = Σ(d−2q)M , q=0 √ √ ± c c where Σ M is the eigenspace of −1 ξ to the eigenvalue ±1andΣkM is the eigenspace of −1 ωθ ± c ∧0,even/odd ⊗ L c  to the eigenvalue k.ByProposition 3.2,wehaveΣ M H (M) and Σ(d−2q)M ∧0,q ⊗ L H (M) .

3.2. Spinorial connections and Dirac type operators on contact metric manifolds

c Let (M, gθ ) be a contact metric manifold endowed with a Spin -structure. Each unitary connection A on L together with the generalized Tanaka–Webster connection ∇ induce a spinorial connection ∇A R. Petit / Differential Geometry and its Applications 22 (2005) 229–252 235

c D DA on ΣM . The Dirac operator associated will be denoted by A. The Kohn–Dirac operator H is the differential operator defined by:  DA = ε .∇A, H i εi i where {εi} is a local orthonormal frame of H .

D DA Proposition 3.3. The operators A and H satisfy the identities: 1 − {D ,λ }=∇A, (4) 2 A ξ ξ 1 λ ◦[D ,λ ]=DA , (5) 2 ξ A ξ H 1 λ ◦[D2 ,λ ]={DA ,λ ◦∇A}, (6) 2 ξ A ξ H ξ ξ 1 2 2 − λ ◦{D2 ,λ }=DA −∇A , (7) 2 ξ A ξ H ξ,ξ c where λ. is the endomorphism of Γ(ΣM ) given by the Clifford product by a form or a vector field and [ , ] (resp. { , }) is the commutator (resp. anticommutator).

Proof. We have D = DA + ◦∇A A H λξ ξ , DA ◦∇A ∇A with H (resp. λξ ξ ) anticommutes (resp. commutes) with λξ (since λξ is -parallel). Hence, {D }={ ◦∇A }=− ∇A A,λξ λξ ξ ,λξ 2 ξ , and [D ]=[DA ]=− ◦ DA A,λξ H ,λξ 2λξ H . Now, we have D2 = DA + ◦∇A ◦ DA + ◦∇A = DA2 −∇A2 +{DA ◦∇A} A ( H λξ ξ ) ( H λξ ξ ) H ξ,ξ H ,λξ ξ , DA2 −∇A2 {DA ◦∇A} with H ξ,ξ (resp. H ,λξ ξ ) commutes (resp. anticommutes) with λξ . The last two equations follow directly. 2

A c Let ∇ and DA be the spinorial connection and the Dirac operator on ΣM induced by the Levi-Civita connection on TM together with a unitary connection A on L.Wehave

Proposition 3.4. For any ψ ∈ Γ(ΣMc), we have

A 1 1 1 ∇Aψ − ∇ ψ = J X.ξ.ψ − θ(X)ω .ψ − τ(X).ξ.ψ, X X 4 4 θ 2 1 D ψ − D ψ = ξ.ω .ψ. A A 4 θ 236 R. Petit / Differential Geometry and its Applications 22 (2005) 229–252

Moreover, if M is spin, we have

∇ξ ψ = Lξ ψ, where Lξ is the metric Lie derivative associated to the Webster metric (cf. [3]).

Proof. The calculation is local and locally the Spinc-spinor bundle can be decomposed as ΣMc = ΣM⊗ L1/2,whereΣM is the local Spin-spinor bundle and L1/2 is a local square root of L. Also it is sufficient to verify the formula for a local section of ΣM. The difference between the generalized Tanaka–Webster connection ∇ and the Levi-Civita connection ∇ is given by (cf. [4]):   1 1 ∇−∇=− θ  J + ω − A ⊗ ξ + τ ⊗ θ, 2 2 θ θ where  denotes the symmetric product (i.e., for any X, Y ∈ TM, (θ J )(X, Y ) = θ(X)JY +θ(Y)JX). Now, let {ei} a local gθ -orthonormal frame and {ψα} a local spinorial frame, then 1  ∇ ψ − ∇ ψ =− g (∇ e − ∇ e ,e )e .e .ψ . X α X α 2 θ X j X j i i j α 1
i

In a local orthonormal frame {ε1,...,ε2d ,ξ},where{εi}, i
2d, is a local orthonormal frame of H ,we have 1 
 1  ∇ ψ − ∇ ψ =− g (∇ − ∇ )ε ,ε ε .ε .ψ + g (∇ ξ,ε )ε .ξ.ψ X α X α 2 θ X X j i i j α 2 θ X i i α 1
i

Remark 3.1. As a consequence of the previous proposition, we obtain that if M is compact, then the Dirac c operator DA is formally self-adjoint for the natural inner product on Γ(ΣM ). A such Dirac operator is called a nice geometric Dirac operator by Nicolaescu [13]. R. Petit / Differential Geometry and its Applications 22 (2005) 229–252 237

Now, suppose that M is a strictly pseudoconvex CR manifold endowed with a Spinc-structure. Let c =∧0,∗ ⊗ L L ΣM H (M) be the spinor bundle (with as above). To each unitary connection A on L, 2 correspond a unitary connection A on L. The correspondence is given by A = Ac ⊗ A,whereAc is the Webster connection on K (cf. [1]). The associated covariant derivative on L is denoted by ∇A.Let 0,∗ ; L = ∧0,∗ ⊗ L ∇Aq 0,q ; L → 0,q ; L ΩH (M ) Γ( H (M) ), we define, W : ΩH (M ) ΩH (M ) by the usual rule ∇Aq ⊗ = ∇q ⊗ + ⊗∇A W (α z) ( W α) z α W z, ∇q 0,q ∈ L where W α is the natural extension of the generalized Tanaka–Webster connection to ΩH (M), z Γ( ) ∈ C and W T M. ∗ q q A 0,q ; L → 0,q+1 ; L A 0,q ; L → 0,q−1 ; L Let ∂H : ΩH (M ) ΩH (M ) (resp. ∂H : ΩH (M ) ΩH (M ))givenby

d ∗ d Aq ∗ Aq Aq Aq ∂ = Z ∧∇ (resp. ∂ =− i(Zi)∇ ). H i Zi H Zi i=1 i=1 0,∗ ; L It follows from Proposition 3.2, that we have on ΩH (M )

d d ∗  √  Aq Aq ∇A = ∇Aq DA = + W W , H 2 (∂H ∂H ), q=0 q=0 and d √ ◦∇A = − q+1 − ∇Aq λξ ξ ( 1) 1 ξ . q=0

A c Remark 3.2. For the canonical Spin -structure, the operator ∂H coincides with the usual operator ∂H 2  (q+1)∗ q DAac = d 2q 2q 2q = + and H q=0 H ,where H is the Kohn Laplacian on (0,q)-forms (i.e., H 2(∂H ∂H (q−1) q∗ ∂H ∂H )).

4. Lichnerowicz type formulas and vanishing theorems on Spinc contact metric manifolds

4.1. Lichnerowicz type formulas

Let (M, gθ , ∇) be an (2d + 1)-dimensional contact metric manifold endowed with the Webster metric and with the generalized Tanaka–Webster connection and E be a Riemannian vector bundle over M. Denote by Ω∗(M; E) the bundle of E-valued forms on M. Remember (cf. [14,19]) that for any α ∈ Ωp(M; E), the covariant derivative and the divergence of α are respectively given by: p ∇ =∇E − ∇ ( Xα)(X1,...,Xp) X α(X1,...,Xp) α(X1,..., XXi,...,Xp), i=1 =− ∇ (δα)(X1,...,Xp−1) tracegθ ( . α)( . ,X1,...,Xp−1). 238 R. Petit / Differential Geometry and its Applications 22 (2005) 229–252

∈ p ; For any αH ΩH (M E),wedefine =− ∇ (δH αH )(X1,...,Xp−1) tracegθ,H ( . αH )( . , X1,...,Xp−1), and for p  2 1 (∧ α )(X ,...,X − ) = trace ω (., .)α (., .,X ,...,X − ), H H 1 p 2 2 gθ,H θ H 1 p 2 with Xi ∈ H . Let S be a Dirac bundle over M. For any Q ∈ Ω2(M;∧2(M)) and α ∈ Ω1(M; End(S)),wedefine Q(α) ∈ Ω1(M; End(S)) by  1 
 Q(α)(X) = Q(e ,X).α(e ) = Q(e ,X) (e ,e )e .e .α(e ), i i 2 i j k j k i i i,j,k where {ei} is a local gθ -orthonormal frame of TM. In the following, the connections (resp. curvatures) on Cl(M) or S will be denoted by ∇ (resp. R).

Proposition 4.1 (Mok–Siu–Yeung type formulas). (i) If δH QH = 0, then we have
 ∇ = RQ − ◦∇ δH QH ( ) H λ(∧H QH ) ξ , (8) RQ { } where H is the endomorphism given in a local orthonormal frame εi of H by 1  RQ =− Q (ε ,ε ).R (ε ,ε ). H 2 H i j H i j i,j

(ii) If δQξ = 0, then we have
 ∇ = RQ + DQ δ Qξ ( ) ξ τ , (9) RQ DQ where ξ (resp. τ ) is the endomorphism (resp. differential operator) given by   RQ =− DQ = ∇ ξ Q(ξ, εi).R(ξ, εi)(resp. τ Q(ξ, εi). τ(εi )). i i

Proof. Let {ε1,...,ε2d ,ξ} be a local orthonormal frame, where {εi}, i
2d, is a local orthonormal frame of H . For any X ∈ H ,wehave
  ∇ = ∇ QH ( ) (X) QH (εi,X). εi . i We obtain
 
 


  ∇ =− ∇ ∇ =− ∇ ∇ − ∇ ∇ δH QH ( ) εj QH ( ) (εj ) εj QH ( ) (εj ) QH ( ) ( εj εj ) j j 

  =− ∇ ∇ − ∇ ∇ εj QH (εi,εj ). εi QH (εi, εj εj ). εi i,j R. Petit / Differential Geometry and its Applications 22 (2005) 229–252 239 

  =− ∇ ∇ + ∇ ∇ − ∇ ∇ εj QH (εi,εj ) . εi QH (εi,εj ). εj εi QH (εi, εj εj ). εi i,j
 =− ∇ ∇ + ∇ ∇ + ∇ ∇ ( εj QH )(εi,εj ). εi QH (εi,εj ). εj εi QH ( εj εi,εj ). εi i,j  =− (∇ Q )(ε ,ε ).∇ − Q (ε ,ε ).(∇ ∇ −∇∇ ) εj H i j εi H i j εj εi εj εi i,j  i,j =− (δ Q )(ε ).∇ − Q (ε ,ε ).∇2 H H i εi H i j εj ,εi i i,j  1  =− (δ Q )(ε ).∇ − Q (ε ,ε ).(∇2 −∇2 ). H H i εi 2 H i j εj ,εi εi ,εj i i,j Now, for any X, Y ∈ H ,wehave ∇2 −∇2 = −∇ = + ∇ Y,X X,Y RH (X, Y ) TH (X,Y ) RH (X, Y ) ωθ (X, Y ) ξ .

The assumption δH QH = 0 together with the previous formula yield
 1  1  δ Q (∇) =− Q (ε ,ε ).R (ε ,ε ) − ω (ε ,ε )Q (ε ,ε ).∇ H H 2 H i j H i j 2 θ i j H i j ξ i,j i,j = RQ − ∧ ∇ H ( H QH ). ξ . For any X ∈ TM,wehave
   ∇ = ∇ + ∇ =− ∇ + ∇ Qξ ( ) (X) Qξ (εi,X). εi Qξ (ξ, X). ξ θ(X) Q(ξ, εi). εi Q(ξ, X). ξ . i i We obtain
 

 ∇ =− ∇ ∇ − ∇ ∇ δ Qξ ( ) εj Qξ ( ) (εj ) ξ Qξ ( ) (ξ) j 


 
 =− ∇ ∇ − ∇ ∇ −∇ ∇ εj Qξ ( ) (εj ) Qξ ( ) ( εj εj ) ξ Qξ ( ) (ξ) j 

  
 =− ∇ ∇ − ∇ ∇ + ∇ ∇ εj Q(ξ, εj ). ξ Q(ξ, εj εj ). ξ ξ Q(ξ, εj ). εj j j 

  =− ∇ ∇ + ∇ ∇ − ∇ ∇ εj Q(ξ, εj ) . ξ Q(ξ, εj ). εj ξ Q(ξ, εj εj ). ξ j 

  + ∇ ∇ + ∇ ∇ ξ Q(ξ, εj ) . εj Q(ξ, εj ). ξ εj j 
 
 =− ∇ ∇ + ∇ ∇ εj i(ξ)Q (εj ). ξ ξ i(ξ)Q (εj ). εj j j  + ∇ ∇ −∇ − ∇ ∇ Q(ξ, εj ).( ξ εj ∇ξ εj ) Q(ξ, εj ). εj ξ j  j  =−(δQ )(ξ).∇ − (δQ )(ε ).∇ − Q(ξ, ε ).(∇2 −∇2 ). ξ ξ ξ j εj j εj ,ξ ξ,εj j j 240 R. Petit / Differential Geometry and its Applications 22 (2005) 229–252

∇2 −∇2 = −∇ ∈ = Using the relation X,ξ ξ,X R(ξ,X) τ(X),(X H ), and δQξ 0, we obtain
   ∇ =− + ∇ = RQ + DQ 2 δ Qξ ( ) Q(ξ, εi).R(ξ, εi) Q(ξ, εi). τ(εi ) ξ τ . i i

Remark 4.1. The classical example of ∧2(M)-valued 2-form on a Riemannian manifold M is the cur- vature tensor. Moreover, on a locally symmetric space the curvature tensor is parallel for the Levi-Civita connection. In [11] and [21], Mok, Siu and Yeung derive similar formulas to (8) for harmonic maps and forms defined on symmetric spaces. As applications they have obtained rigidity theorems for the sym- metric spaces. Recently, by a Riemannian analogue of (8), Friedrich and Kirchberg [6] have obtained estimates on the first eigenvalue of the Dirac operator on Riemannian manifolds with harmonic curvature tensor.

A horizontal 2-form αH such that ∧H αH = 0 is called a primitive 2-form. For any horizontal 2-form

αH , we associate a primitive 2-form denoted αH0 by setting 1 α = α − ω ⊗ (∧ α ). H0 H d θ H H DA DA ∇A ∗∇A Now, let H,J , H,τ and ( H ) H be the differential operators respectively given by    ∗ 2 DA = ε .∇A , DA = ε .∇A ,(∇A ) ∇A =− ∇A , H,J i Jεi H,τ i τ(εi ) H H εi ,εi i i i { } ∇A ∗∇A ∇A ∗∇A where εi is a local orthonormal frame of H , and, ( 1,0) 1,0 (resp. ( 0,1) 0,1) the differential operator given by   ∗ 2 ∗ 2 (∇A ) ∇A =− ∇A (resp. (∇A ) ∇A =− ∇A ), 1,0 1,0 Zi ,Zi 0,1 0,1 Zi ,Zi i i √ = 1 − − { } with Zi 2 (ei 1Jei) where e1,Je1,...,ed ,Jed is a local orthonormal frame of H . Proposition 4.2 (Lichnerowicz type formulas).LetΩA be the curvature 2-form on L associated to a connection A. Then the following identities hold: √ 1 −1 {DA ◦∇A}= ◦ DA − + H ,λξ ξ λξ H,τ λξ.δτ λθ∧i(ξ)ΩA , (10) 2 √ 2 2 ∗ 1 −1 DA = ∇A ∇A − ◦∇A + + ( ) λωθ s λΩA . (11) H H H ξ 4 2 H Moreover, in the strictly pseudoconvex case, we have  √   √  2 −1 ∗ −1 ∗ DA = 2 1 + λ (∇A ) ∇A + 2 1 − λ (∇A ) ∇A H d ωθ 0,1 0,1 d ωθ 1,0 1,0   √ − + 1 − 2 + 1 s λωθ .ρH λΩA , (12) 4 d 2 H0 √ A2 A2 D − D =− −1 λ A . (13) H,J H ΩH− R. Petit / Differential Geometry and its Applications 22 (2005) 229–252 241

Remark 4.2. Similar identities to (10) and (11) have been obtained in dimension 3 by Mrowka, Ozsvath, Yu [10] and Nicolaescu [12,13]. Note that identity (12) is a spinorial analogue of the Weitzenbock– 0,q Tanaka formula on ΩH (M) in [17].

Proof of Proposition 4.2. All these Lichnerowicz type formulas on ΣMc come from formulas (8) and (9) for some particular curvature tensors on M. Consider the parallel ∧2(M)-valued 2-form Qcan defined by Qcan(X, Y ) = X∗ ∧ Y ∗. By taking Q = Qcan in (8) and (9), we obtain the formulas: DA2 − ∇A ∗∇A =− ◦∇A + RA H ( H ) H λωθ ξ H and {DA ◦∇A}= ◦ DA + RA H ,λξ ξ λξ H,τ ξ , RA RA where H and ξ are the endomorphisms locally given by: 1   RA =− ε .ε .RA (ε ,ε ), RA =− ξ.ε .RA(ξ, ε ). H 2 i j H i j ξ i i i,j i Now, we have to calculate the curvature terms. Since the calculation is local, we may assume ΣMc = ΣM ⊗ L1/2,whereΣM is the local spinor bundle and L1/2 is a local square root of L. We furnish ΣMc with the tensor product metric and tensor product connection. Hence, we have for any ψ ∈ Γ(ΣM)and any σ ∈ Γ(L1/2),

 1/2 RA(X, Y )(ψ ⊗ σ)= R(X,Y)ψ ⊗ σ + ψ ⊗ RL ,A(X, Y )σ √
 −1 = R(X,Y)ψ ⊗ σ − ΩA(X, Y )ψ ⊗ σ, 2 where R is the curvature of the local spinor bundle. We deduce that  √  1 
 −1 RA (ψ ⊗ σ)=− ε .ε . R (ε ,ε )ψ ⊗ σ − ΩA (ε ,ε )ψ ⊗ σ H 2 i j H i j 2 H i j i,j √ −1 = (R ψ)⊗ σ + (ΩA.ψ) ⊗ σ, H 2 H and  √  
 −1 RA(ψ ⊗ σ)=− ξ.ε . R(ξ,ε )ψ ⊗ σ − ΩA(ξ, ε )ψ ⊗ σ ξ i i 2 i i √ −1
 = (R ψ)⊗ σ + ξ.i(ξ)ΩA.ψ ⊗ σ. ξ 2 Now, R is given by 1 
 R(X,Y) = g R(X,Y)ε ,ε ε .ε . (14) 4 θ i j i j 1
i,j
2d 242 R. Petit / Differential Geometry and its Applications 22 (2005) 229–252

Using (14),wehave 1 
 RH =− gθ RH (εi,εj )εk,εl εi.εj .εk.εl 8

1 i,j,k,l 2d 1  
 
 =− gθ RH (εi,εj )εk,εl εi.εj .εk + 2 gθ RH (εi,εj )εi,εl εj .εl 8


= =


1 l 2d 1 i j k 2d 1 i,j 2d 1  
 =− gθ RH (εi,εj )εk + RH (εk,εi)εj + RH (εj ,εk)εi,εl εi.εj .εk .εl 24


= =
1 l 2 d 1 i j k 2d 1  
 − gθ RH (εi,εj )εi,εl εj .εl 4



1 l 2d 1 i,j 2d 1  
=− gθ ωθ (εi,εj )τ(εk) + ωθ (εk,εi)τ(εj ) 24


= =
1 l 2d 1 i j k 2d  1  + ω (ε ,ε )τ(ε ), ε ε .ε .ε .ε − ε .R (ε ,ε )ε θ j k i l i j k l 4 j H i j i 1
i,j
2d 1 
 1 
 =− g ω (ε ,ε )τ(ε ), ε ε .ε .ε .ε + g ω (ε ,ε )τ(ε ), ε ε .ε 8 θ θ i j k l i j k l 4 θ θ i j i l j l 1
i,j,k,l
2d 1
i,j,l
2d 1  − ε . Ric(ε ) 4 j j 1
j
2d 1  1  1  =− ω . ε .τ(ε ) + ε .J τ(ε ) − ε . Ric(ε ). 4 θ i i 4 i i 4 j j 1
i
2d 1
i
2d 1
j
2d Since τ and J ◦ τ are symmetric endomorphisms, then the two first terms in the last expression are zero. Restricted to H , the endomorphism Ric is symmetric. We deduce that 1  1 
 1 − ε . Ric(ε ) = g Ric(ε ), ε = s. 4 j j 4 θ j j 4 1
j
2d 1
j
2d

Now, we have to calculate the term Rξ . 1 
 Rξ =− gθ R(ξ,εi)εj ,εk ξ.εi.εj .εk 4

1 i,j,k 2d 1 
 
 =− g R(ξ,ε )ε ,ε ξ.ε .ε .ε − 2 g R(ξ,ε )ε ,ε ξ.ε 4 θ i j k i j k θ j j k k 1
i=j=k
2d 1
j,k
2d 1 


 =− g R(ξ,ε )ε ,ε + g R(ξ,ε )ε ,ε + g R(ξ,ε )ε ,ε ξ.ε .ε .ε 12 θ i j k θ k i j θ j k i i j k 1
i=j=k
2d 1  − ξ.R(ε ,ξ)ε 2 j j 1
j
2d R. Petit / Differential Geometry and its Applications 22 (2005) 229–252 243

1 

 =− g R(ξ,ε )ε + R(ε ,ξ)ε ,ε − g R(ε ,ξ)ε ,ε ξ.ε .ε .ε 12 θ i j j i k θ k i j i j k 1
i=j=k
2d 1 − ξ.Ric(ξ) 2 1 

 =− g (∇ τ)(ε ) − (∇ τ)(ε ), ε − g (∇ τ)(ε ), ε 12 θ εj i εi j k θ εj k i 1
i=j=k
2d
 1 + g (∇ τ)(ε ), ε ξ.ε .ε .ε − ξ.δτ. θ εi k j i j k 2

Since gθ ((∇Xτ)(Y ), Z) = gθ ((∇Xτ)(Z), Y ), then the first term in the last expression is zero. Hence the result. Now, suppose that M is strictly pseudoconvex (i.e., ∇J =∇ωθ = 0) and consider the 2-forms

can can 1 can can 1 can can can Q = Q − ωθ ⊗ (∧H Q ) and Q = (Q − Q ◦ J)= (Q )0. H0 H d H H− 2 H H H− Qcan and Qcan are parallel and primitive. Formula (8) for Qcan and Qcan gives respectively H0 H− H0 H−

2
∗  DA − (∇A ) ∇A = RA H H H 0 H0 and DA2 − DA2 =− RA H,J H 2 H− , where  
  ∗ 2 1 2 (∇A ) ∇A =− ∇A − ω .∇A H H 0 εi ,εi d θ εi ,J εi i and where RA and RA are the endomorphisms locally given by: H0 H−   A 1 A A 1 A R =− εi.εj .R (εi,εj ), R =− εi.εj .R (εi,εj ). H0 2 H0 H− 2 H− i,j i,j √ { } = 1 − − Let e1,Je1,...,ed ,Jed be a local orthonormal frame of H and Zi 2 (ei 1Jei). A direct calcu- lation yields  √   √  −1 ∗ −1 ∗ 1 + ω (∇A ) ∇A + 1 − ω (∇A ) ∇A d θ 0,1 0,1 d θ 1,0 1,0   d

 1 2 2 1 2 2 1 ∗ =− ∇A +∇A − ω . ∇A −∇A = (∇A ) ∇A . 2 ei ,ei Jei ,J ei d θ ei ,J ei Jei ,ei 2 H H 0 i=1 Now, we have √ −1 RA (ψ ⊗ σ)= (R ψ)⊗ σ + (ΩA .ψ) ⊗ σ H0 H0 H0  2  √ − 1 1 1 A = sψ + ωθ .(∧H RH )ψ ⊗ σ + (Ω .ψ) ⊗ σ, 4 d 2 H0 244 R. Petit / Differential Geometry and its Applications 22 (2005) 229–252 and √ −1 RA (ψ ⊗ σ)= (R ψ)⊗ σ + (ΩA .ψ) ⊗ σ. H− H− 2 H− We have 1 
 1 
 ∧ R = g R (ε ,Jε )ε ,ε ε .ε = g (∧ R )(ε ), ε ε .ε . H H 8 θ H i i j k j k 4 θ H H j k j k 1
i,j,k
2d 1
j,k
2d For any X ∈ H ,wehaveusing(1),

RH (εi,Jεi)X + RH (X, εi)J εi + RH (J εi,X)εi = τ(X)+ ωθ (X, εi)τ(J εi) − gθ (X, εi)τ(εi). Hence, we obtain

(∧H RH )(X) − J Ric(X) = (d − 1)τ(X). Using (2), we obtain that, for Y ∈ H


 ∧ =− + − =− =− gθ ( H RH )(X), Y gθ Ric(X), J Y (d 1)gθ τ(X),Y RicH+ (X, J Y ) ρH (X, Y ). ∧ =−1 We deduce that H RH 2 ρH . Using (3),wehave 
 R =−1 H− gθ RH− (εi,εj )εk,εl εi.εj .εk.εl 8

1 i,j,k,l 2d  
 =− 1 + + gθ RH− (εi,εj )εk RH− (εk,εi)εj RH− (εj ,εk)εi,εl εi.εj .εk .εl 24


= =
1 l 2d 1 i j k 2d  
 − 1 gθ RH− (εi,εj )εi,εl εj .εl 4



1 l 2 d 1 i,j 2d 1  
= gθ ωθ (εi,εj )τ(εk) + ωθ (εk,εi)τ(εj ) 24


= =
1 l 2d 1 i j k 2d    + − 1 ωθ (εj ,εk)τ(εi), εl εi.εj .εk .εl gθ (RH− (εi,εj )εi,εl)εj .εl 4



1 l 2 d 1 i,j 2d 1  
 =− g R (ε ,ε )ε ,ε ε .ε . 4 θ H− i j i l j l 1
l
2d 1
i,j
2d Now, for X, Y ∈ H ,wehaveusing(2) 

 = = − gθ RH− (εi,X)εi,Y RicH− (X, Y ) (d 1)ωθ τ(X),Y . (15) 1
i
2d Formula (15) yields   d − 1  R =− ε .J τ(ε ) = 0. 2 H− 4 j j 1
j
2d R. Petit / Differential Geometry and its Applications 22 (2005) 229–252 245

4.2. Vanishing theorems

∈ c D = DA = Definition 4.1. A spinor field ψ Γ(ΣM ) such that Aψ 0 (resp. H ψ 0) will be called a har- monic spinor (resp. subharmonic spinor).

Definition 4.2 [2]. A contact metric manifold for which the Tanaka–Webster torsion vanishes is called a k-contact metric manifold.

Theorem 4.1. Let M be a compact k-contact metric manifold of dimension m  3 endowed with a Spinc- structure. We suppose that the determinant line bundle L is endowed with a unitary connection A for A 1  | |+···+| | which the curvature form Ω is horizontal. If 2 s λ1 λd at each point, where λ1,...,λd are the eigenvalues of ΩA, then any harmonic spinor is parallel. If the previous inequality is strict at some point, then there is no harmonic spinor.

Proof. Since M has Tanaka–Webster torsion vanishing and that i(ξ)ΩA = 0, Formula (10) yields {DA ◦∇A}= H ,λξ ξ 0. We deduce by (6), (7) that

D2 = DA2 −∇A2 A H ξ,ξ. Let ψ ∈ Γ(ΣMc), we obtain by integrating

|D |2 = |DA |2 +|∇A |2 Aψ vgθ H ψ ξ ψ vgθ . M M DA =∇A = If ψ is a harmonic spinor, then we deduce from the last equation that H ψ ξ ψ 0. Hence, we have using (11): √ 1 −1 |∇A ψ|2 + s|ψ|2 + ΩA .ψ, ψ v = 0. (16) H 4 2 H gθ M

Now, we can find a local orthonormal frame {ei} of H such that  A = ∧ ΩH λie2i−1 e2i. 1
i
d  We deduce that | ΩA .ψ, ψ |
|ΩA.ψ||ψ|
( |λ |)|ψ|2 and, using the assumption on s, that √ H H 1
i
d i 1 | |2 + −1 A  2 4 s ψ 2 ΩH .ψ, ψ 0. The result follows immediately from (16).

If M is spin, then L is trivial and flat. We deduce the following:

Theorem 4.2. Let M be a compact spin k-contact metric manifold of dimension m  3. Suppose that the Tanaka–Webster scalar curvature of M is nonnegative and positive at some point, then there is no harmonic spinor.

In the following of this section we suppose that M is a strictly pseudoconvex CR manifold. 246 R. Petit / Differential Geometry and its Applications 22 (2005) 229–252

Definition 4.3. A strictly pseudoconvex CR manifold for which the Tanaka–Webster torsion vanishes is called a Sasakian manifold.

Corollary 4.1. Let M be a compact Sasakian manifold of dimension m  3 with pseudo-Hermitian Ricci tensor nonnegative. Then any harmonic spinor for the canonical Spinc-structure is parallel.

−1 Proof. Let Aac be the Webster connection on K ,thenwehave(cf.[1]): 1


 ΩAac (X, Y ) =− ω X, Ric(Y ) + ω Ric(X), Y = ρ (X, Y ). H 2 θ θ H Moreover, M is a Sasakian manifold, hence we have ΩAac (ξ, X) = 0. Now, we can find a local orthonor- mal frame {ε1,Jε1,...,εd ,Jεd } of H such that  ρH = λiεi ∧ Jεi, 1
i
d =− | |+ with λi RicH+ (εi,εi). Since the pseudo-Hermitian Ricci tensor is nonnegative, we deduce that λ1 ···+| |= 1 2 λd 2 s. The result follows from Theorem 4.1.

Remember [17,19] that a Hermitian vector bundle E over a strictly pseudoconvex CR manifold M ∇ (0,2) = endowed with a Hermitian connection is holomorphic if and only if RH 0whereR is the curva- ∇ E =∇ = ture of . A section σ of a holomorphic bundle E is said to be holomorphic if (∂H σ)(Z) Zσ 0 for any Z ∈ T 0,1M. Note that the canonical and anticanonical bundles defined above are examples of holomorphic (line) bundles.

Theorem 4.3. Let M be a compact strictly pseudoconvex CR manifold of dimension m  3 with pseudo- Hermitian Ricci tensor nonnegative and pseudo-Hermitian scalar curvature positive at some point. We suppose M endowed with a Spinc-structure for which the determinant line bundle L is endowed with a holomorphic, unitary connection A such that (ΩA ) = 0. Then, for any subharmonic spinor ψ, we have H0 + −1 ψ = ψ1 + ψ2,whereψ1 is a holomorphic section of L and ψ2 is a holomorphic section of K ⊗ L .

Proof. Since A is a holomorphic connection on L and (ΩA ) = 0, we have ΩA = 0. Let ψ be a H0 + H0 subharmonic spinor on M, we obtain by integrating equation (12):  √    √   −1 −1 1 + ω .∇A ψ,∇A ψ + 1 − ω .∇A ψ,∇A ψ d θ 0,1 0,1 d θ 1,0 1,0 M   1 2 + s|ψ|2 − ω .ρ .ψ, ψ v = 0. (17) 8 d θ H gθ

Now, let {ε1,Jε1,...,εd ,Jεd } be a local orthonormal frame of H such that   ωθ = εi ∧ Jεi and ρH = λiεi ∧ Jεi, 1
i
d 1
i
d =−
∈ with λi RicH+ (εi,εi). If the pseudo-Hermitian Ricci tensor is nonnegative, then λi 0fori { } | |
| | | |= s | | | |
| | 1,...,d , and we obtain that ρH .ψ ( 1
i
d λi ) ψ 2 ψ . Using the inequality ωθ .ψ d ψ , R. Petit / Differential Geometry and its Applications 22 (2005) 229–252 247 we have     sd  ω .ρ .ψ, ψ  =  ρ .ψ, ω .ψ 
|ρ .ψ||ω .ψ|
|ψ|2. (18) θ H H θ H θ 2  ∈R | |2 − 2  = d d−2q Since ωθ .ρH .ψ, ψ , we deduce that s ψ d ωθ .ρH .ψ, ψ 0. Now, we have ψ q=0 ψ d−2q ∈ c ∧0,q ⊗ L with ψ Σ(d−2q)M H (M) . We deduce that  √     √  −1 d q −1 d q 1 + ω ψ = 2 1 − ψ d−2q and 1 − ω ψ = 2 ψ d−2q . d θ d d θ d q=0 q=0 Hence (17) becomes    d q   q   1 − ∇A (ψ d−2q )2 + ∇A (ψ d−2q )2 d 0,1 d 1,0 q=0 M   1 2 + s|ψ|2 − ω .ρ .ψ, ψ v = 0. (19) 16 d θ H gθ | |2 − 2  ∇A d−2q = ∈{ − } Since s ψ d ωθ .ρH .ψ, ψ 0, we obtain from (19) that, H ψ 0 for any q 1,...,d 1 , ∇A(ψ d ) = 0and∇A(ψ −d ) = 0, for any Z ∈ T 1,0M.SinceL is holomorphic, the two last equations Z Z d c −d c imply that ψ is a holomorphic section of Σd M  L and ψ is a holomorphic section of Σ(−d)M  ⊗ L−1 = sd | |2 K .Now(19) also gives ωθ .ρH .ψ, ψ 2 ψ . Consequently, the Schwarz inequality in (18) = ∈ ∞ | |2 =−sd | |2 is an equality and we obtain that ρH .ψ fωθ .ψ, f C (M). We deduce that f ωθ .ψ 2 ψ . Since |ρ .ψ||ω .ψ|= sd |ψ|2 and |ρ .ψ|
s |ψ|,wehave|ω .ψ|  d|ψ|. Hence |ω .ψ|=d|ψ| and H θ 2 H 2 θ θ =− s | |2 = d − 2| d−2q |2 f 2d .Now,wehave ωθ .ψ q=0 (d 2q) ψ and       s sd d sd 2q 2 d q − |ω .ψ|2 + |ψ|2 = 1 − 1 − |ψ d−2q |2 = 2s q 1 − |ψ d−2q |2 = 0. 2d θ 2 2 d d q=0 q=0 At a point where s>0, we deduce that ψ d−2q = 0 for any q ∈{1,...,d − 1}. Now, for any ∈{ − } ∇A d−2q = | d−2q | q 1,...,d 1 ,wehave H ψ 0, so ψ is constant on M. Consequently, for any q ∈{1,...,d− 1}, ψ d−2q = 0everywhereonM and ψ = ψ d + ψ −d . 2

Corollary 4.2. Let M be a compact spin strictly pseudoconvex CR manifold of dimension m  3 with pseudo-Hermitian Ricci tensor nonnegative and pseudo-Hermitian scalar curvature positive at some point. Then, for any subharmonic spinor ψ, we have ψ = ψ1 + ψ2,whereψ1 and ψ2 are holomorphic sections of K1/2.

∞ Definition 4.4. A strictly pseudoconvex CR manifold for which ρH = fωθ , f ∈ C (M), is called =− s pseudo-Einstein (f 2d ).

Recall that the condition pseudo-Einstein does not imply that the pseudo-Hermitian scalar curvature is constant. If M is pseudo-Einstein, then ΩAac = 0. We deduce the following corollary: H0 248 R. Petit / Differential Geometry and its Applications 22 (2005) 229–252

Corollary 4.3. Let M be a compact strictly pseudoconvex CR manifold of dimension m  3 endowed with its canonical Spinc-structure. Suppose that M is pseudo-Einstein with pseudo-Hermitian scalar curvature nonnegative and positive at some point. Then, for any subharmonic spinor ψ, we have ψ = ψ1 + ψ2,whereψ1 is a (CR-) holomorphic function on M and ψ2 is a holomorphic section of K.

5. Changes of metrics on a spin contact metric manifold and Dirac operator

Following the works of Bourguignon–Gauduchon [3] and Maier [8] concerning the infinitesimal vari- ation of the Dirac operator with respect to variations of metrics, we are interesting, in this section, to the infinitesimal variation of the Kohn–Dirac operator on a spin contact manifold with respect to variations of a metric associated to a contact form. Let (M,θ,ξ) be a contact manifold and let M(θ) be the set of metrics associated to θ.For ∈ M = gθ ,hθ (θ), we consider Ghθ ,gθ the gθ -symmetric endomorphism of TxM given by hθ (X, Y ) − g (G (X), Y ). Note that G is positive and that G (ξ) = ξ.Now,letb = G 1/2 , then θ hθ ,gθ hθ ,gθ hθ ,gθ hθ ,gθ hθ ,gθ b is a smooth SO + -equivariant map from P (M, g ) to P (M, h ).IfM is spin, then hθ ,gθ 2d 1 SO2d+1 θ SO2d+1 θ there exists a smooth Spin -equivariant map β : P (M, g ) → P (M, h ) which cov- 2d+1 hθ ,gθ Spin2d+1 θ Spin2d+1 θ ers b .LetΣ M and Σ M be the associated spinor bundles, then β extends to an isometry hθ ,gθ gθ hθ hθ ,gθ β : Σ M → Σ M and we have hθ ,gθ gθ hθ β (X.ψ) = b (X).β (ψ). hθ ,gθ hθ ,gθ hθ ,gθ

hθ gθ We denote by ∇ (resp. ∇ ) the Tanaka–Webster connection for hθ (resp. gθ )onTM and Σh M (resp. − θ Σ M). The connection on Σ M given by β 1 ◦∇hθ ◦ β (induced by the g -metric connection gθ gθ hθ ,gθ hθ ,gθ θ − h ,g b 1 ◦∇hθ ◦ b on TM) will be denoted by ∇hθ ,gθ .Now,letD θ θ the differential operator on hθ ,gθ hθ ,gθ H Γ(Σgθ M) defined by Dhθ ,gθ = β−1 ◦ Dhθ ◦ β . H hθ ,gθ H hθ ,gθ

We have, for any ψ ∈ Γ(Σg M),  θ Dhθ ,gθ ψ = ε .∇hθ ,gθ ψ H i b (εi ) hθ ,gθ i 

  1 − − = ε . b 1 ◦∇gθ b + b 1 ◦ Ahθ ,gθ b (ε ) ◦ b .ψ i hθ ,gθ b (εi ) hθ ,gθ hθ ,gθ hθ ,gθ i hθ ,gθ 2 hθ ,gθ i + ε .∇gθ ψ, i b (εi ) hθ ,gθ i

hθ ,gθ where, {εi}i
2d is a local gθ -orthonormal frame of H , . the Clifford multiplication for gθ ,andA (X) = ∇hθ −∇gθ X X . Let (M,θ,ξ,J,gθ ) be a contact metric manifold and let τ be the Tanaka–Webster torsion of ∇.For any t ∈ R,wedefine
 t = tτ = ◦ tτ gθ (X, Y ) gθ e (X), Y and Jt J e . t ∈ M t It follows directly from Proposition 4.1 of [18] that gθ (θ) and so (M,θ,ξ,Jt ,gθ ) is a contact metric manifold. Note that v t = v . gθ gθ R. Petit / Differential Geometry and its Applications 22 (2005) 229–252 249

∇t t t Lemma 5.1. Let be the Tanaka–Webster connection of gθ and let τ be the Tanaka–Webster torsion of ∇t , then we have 1 τ t = τ − (e−tτ ◦∇ etτ). 2 ξ Proof. We have 1 1

 τ t (X) = (J ◦ L J )(X) = Jetτ ξ,Jetτ(X) − Jetτ[ξ,X] . 2 t ξ t 2 Since J ◦ etτ = e−tτ ◦ J ,wehave 1

 τ t (X) = [ξ,X]+Jetτ ξ,e−tτ(J X) . 2

Using the relation τ(X)=[ξ,X]−∇ξ X, we obtain that
 −tτ −tτ −tτ ξ,e (X) = e [ξ,X] + (∇ξ e )(X). −tτ tτ Now, since (∇ξ e )(J X) = J(∇ξ e )(X), we deduce that
 −tτ −tτ tτ ξ,e (J X) = e [ξ,JX] + J(∇ξ e )(X). It follows that 1

 τ t (X) = [ξ,X]+J [ξ,JX]+Jetτ J(∇ etτ)(X) 2 ξ 1


 = J [ξ,JX]−J [ξ,X] − e−tτ (∇ etτ)(X) 2 ξ 1 = τ(X)− (e−tτ ◦∇ etτ)(X). 2 2 ξ =∇t −∇ Lemma 5.2. Let At (X) X X, then we have
 1



 gt A (X)Y, Z = g (∇ etτ)(Y ), Z + g (∇ etτ)(X), Z − g (∇ etτ)(X), Y . (20) θ t 2 θ X θ YH θ ZH Proof. We have (∇t gt )(Y, Z) = Xgt (Y, Z) − gt (∇t Y,Z) − gt (Y, ∇t Z) X θ θ θ X θ X

 = Xgt (Y, Z) − gt (∇ Y,Z) − gt (Y, ∇ Z) − gt A (X)Y, Z − gt Y,A (X)Z θ θ X
θ  X
θ t  θ t = ∇ t − t − t ( Xgθ )(Y, Z) gθ At (X)Y, Z gθ Y,At (X)Z . ∇t t Since is gθ -metric, we deduce that

 t + t = ∇ t gθ At (X)Y, Z gθ Y,At (X)Z ( Xgθ )(Y, Z). By permutations, we obtain


 t + + t − + t − gθ At (X)Y At (Y )X, Z gθ At (X)Z At (Z)X, Y gθ At (Y )Z At (Z)Y, X = ∇ t + ∇ t − ∇ t ( Xgθ )(Y, Z) ( Y gθ )(X, Z) ( Zgθ )(X, Y ). (21) 250 R. Petit / Differential Geometry and its Applications 22 (2005) 229–252

− =∇t −∇t − ∇ −∇ = t − Now, we have At (Y )X At (X)Y Y X XY ( Y X XY) T (X, Y ) T(X,Y).Using Lemma 5.1, we obtain 1 1 T t =−dθ ⊗ ξ + θ ∧ τ t =−dθ ⊗ ξ + θ ∧ τ − θ ∧ (e−tτ ◦∇ etτ) = T − θ ∧ (e−tτ ◦∇ etτ). 2 ξ 2 ξ − =−1 ∧ −tτ ◦∇ tτ We deduce that At (Y )X At (X)Y 2 (θ (e ξ e ))(X, Y ). Hence (21) becomes
 t = ∇ t + ∇ t − ∇ t 2gθ At (X)Y, Z ( Xgθ )(Y, Z) ( Y gθ )(X, Z) ( Zgθ )(X, Y ) 1

  1

  − gt θ ∧ (e−tτ ◦∇ etτ) (Y, Z), X − gt θ ∧ (e−tτ ◦∇ etτ) (X, Z), Y 2 θ ξ 2 θ ξ 1

  + gt θ ∧ (e−tτ ◦∇ etτ) (X, Y ), Z 2 θ ξ = ∇ t + ∇ t − ∇ t ( Xgθ )(Y, Z) ( Y gθ )(X, Z) ( Zgθ )(X, Y ) 1


 
 + θ(X) g ∇ etτ (Y ), Z − g (∇ etτ)(Z), Y 2 θ ξ θ ξ 1


 + θ(Z) g (∇ etτ)(Y ), X + g (∇ etτ)(X), Y 2 θ ξ θ ξ 1


 − θ(Y) g (∇ etτ)(X), Z + g (∇ etτ)(Z), X . 2 θ ξ θ ξ ∇ t = ∇ tτ ∇ tτ = ∇ tτ Since ( Xgθ )(Y, Z) gθ (( Xe )(Y ), Z) and gθ (( ξ e )(X), Y ) gθ (( ξ e )(Y ), X) we obtain



 2gt A (X)Y, Z = g (∇ etτ)(Y ), Z + g (∇ etτ)(X), Z − g (∇ etτ)(X), Y θ t θ X
θ  Y
θ Z  + θ(Z)g (∇ etτ)(X), Y − θ(Y)g (∇ etτ)(X), Z
θ ξ 
θ ξ  tτ tτ = g (∇ e )(Y ), Z + g (∇ − e )(X), Z θ
X θ (Y θ(Y)ξ) tτ − gθ (∇(Z−θ(Z)ξ)e )(X), Y . 2

∈ Proposition 5.1. Let (M,θ,ξ,J,gθ ) be a spin contact metric manifold. We have for any ψ Γ(Σgθ M)     d
gt ,g  1 1 1
 D θ θ ψ =− D ψ − δτ.ψ = λ ◦{D ,λ ◦ L } ψ. dt H /t=0 2 H,τ 2 2 ξ H ξ ξ

Remark 5.1. This allows to consider formula (10) of Proposition 4.2 from a variational point of view.

tτ − 1 tτ Proof of Proposition 5.1. We have G t = e . Hence, b t = e 2 , and we have gθ ,gθ gθ ,gθ

t 

   g ,gθ 1 1 − 1 1 − 1 − 1 D θ = 2 tτ ◦∇ 2 tτ + 2 tτ ◦ 2 tτ ◦ 2 tτ + ∇ H εi. e − 1 tτ e e At e (εi) e εi. − 1 tτ 2 e 2 (εi ) e 2 (εi ) i i 


   1 1 − 1 1 − 1 − 1 = 2 tτ ◦∇ 2 tτ + 2 tτ ◦ 2 tτ ◦ 2 tτ gθ e − 1 tτ e e At e (εi) e (εj ), εk εi.εj .εk 4 e 2 (εi ) i,j,k + ∇ εi. − 1 tτ e 2 (εi ) i R. Petit / Differential Geometry and its Applications 22 (2005) 229–252 251 


 

  1 − 1 1 − 1 − 1 − 1 = ∇ 2 tτ 2 tτ + t 2 tτ 2 tτ 2 tτ gθ − 1 tτ e (εj ), e (εk) gθ At e (εi) e (εj ), e (εk) εi.εj .εk 4 e 2 (εi ) i,j,k + ∇ εi. − 1 tτ . e 2 (εi ) i Using (20), we obtain       
t  1 d gθ ,gθ 1 d − tτ 1 d tτ D = g ∇ (e 2 ) = (ε ), ε + g ∇ (e ) = (ε ), ε dt H /t=0 4 θ εi dt /t 0 j k 2 θ εi dt /t 0 j k i,j,k       1 d tτ 1 d tτ + g ∇ (e ) = (ε ), ε − g ∇ (e ) = (ε ), ε ε .ε .ε 2 θ εj dt /t 0 i k 2 θ εk dt /t 0 i j i j k  + ∇ 1 εi. d − tτ (e 2 (εi ))/t=0 i dt 1 



 = −g (∇ τ)(ε ), ε + g (∇ τ)(ε ), ε + g (∇ τ)(ε ), ε 8 θ εi j k θ εi j k θ εj i k i,j,k
 1  − g (∇ τ)(ε ), ε ε .ε .ε − ε .∇ θ εk i j i j k 2 i τ(εi ) i 1 
 1  = g (∇ τ)(ε ), ε ε .(ε .ε − ε .ε ) − ε .∇ 8 θ εk i j i k j j k 2 i τ(εi ) i,j,k i 1 
 1 
 1  =− g (∇ τ)(ε ), ε ε .ε .ε − g (∇ τ)(ε ), ε ε − ε .∇ 4 θ εk i j i j k 4 θ εj i j i 2 i τ(εi ) i,j,k i,j i 1 
 1 
 1  = g (∇ τ)(ε ), ε ε − g (∇ τ)(ε ), ε ε − ε .∇ 4 θ εk i i k 4 θ εj j i i 2 i τ(εi ) i,k i,j i 1
 1 = d (trace τ)+ δτ − D . 4 H gθ 2 H,τ = Since tracegθ τ 0, we deduce the first equality. The second equality follows directly from formula (10) and Proposition 3.4. 2

References

[1] H. Baum, Lorentzian twistor spinors and CR-geometry, Differential Geom. Appl. 11 (1999) 69–96. [2] D.E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Math., vol. 509, Springer-Verlag, New York, 1976. [3] J.P. Bourguignon, P. Gauduchon, Spineurs, opérateurs de Dirac et variations de métriques, Comm. Math. Phys. 144 (1992) 581–599. [4] S. Dragomir, On pseudo-Hermitian immersions between strictly pseudoconvex CR manifolds, Amer. J. Math. 117 (1995) 169–202. [5] E. Falbel, C. Gorodski, On contact sub-Riemannian symmetric spaces, Ann. Scient. Ec. Norm. Sup. 28 (1995) 571–589. [6] T. Friedrich, K.D. Kirchberg, Eigenvalue estimates for the Dirac operator depending on the Weyl tensor, J. Geom. Phys. 41 (2002) 196–207. 252 R. Petit / Differential Geometry and its Applications 22 (2005) 229–252

[7] H.B. Lawson, M.L. Michelsohn, Spin Geometry, Princeton Math. Ser., vol. 38, Princeton University Press, Princeton, 1989. [8] S. Maier, Generic metrics and connections on spin- and spinc-manifolds, Comm. Math. Phys. 188 (1997) 407–437. [9] J. Morgan, The Seiberg–Witten Equations and Applications to the Topology of Smooth Four-Manifolds, Mathematical Notes, vol. 44, Princeton University Press, Princeton, 1996. [10] T. Mrowka, P. Ozsvath, B. Yu, Seiberg–Witten monopoles on Seifert fibered spaces, Comm. Anal. Geom. 5 (1997) 685– 793. [11] N. Mok, Y.T. Siu, S.K. Yeung, Geometric superrigidity, Invent. Math. 113 (1993) 57–83. [12] L.I. Nicolaescu, Adiabatic limits of the Seiberg–Witten equations on Seifert manifolds, Comm. Anal. Geom. 6 (1998) 331–392. [13] L.I. Nicolaescu, Geometric connections and geometric Dirac operators on contact metric manifolds, dg-ga/0101155. [14] R. Petit, Harmonic maps and strictly pseudoconvex CR manifolds, Comm. Anal. Geom. 10 (2002) 575–610. [15] M. Rumin, Formes différentielles sur les variétés de contact, J. Differential Geom. 39 (1994) 281–330. [16] M. Rumin, Sub-riemannian limit of the differential form spectrum of contact metric manifolds, Geom. Anal. Funct. Appl. 10 (2000) 407–452. [17] N. Tanaka, A Differential Geometric Study on Strongly Pseudoconvex CR Manifolds, Lecture Notes in Math., vol. 9, Kyoto University, Kyoto, 1975. [18] S. Tanno, Variational problems on contact riemannian manifolds, Trans. Amer. Math. Soc. 314 (1989) 349–379. [19] H. Urakawa, Yang–Mills connections over compact strongly pseudoconvex CR manifolds, Math. Z. 216 (1994) 541–573. [20] S. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978) 25–41. [21] S.K. Yeung, On vanishing theorems and rigidity of locally symmetric manifolds, Geom. Anal. Func. Appl. 11 (2001) 175–198.