Now, we assume that Nn is a differentiable manifold carrying an almost complex struc-ture. An almost complex structure on a differentiable manifold Nn is given by a (1, 1)-tensor J satisfying J2 = −IdT N. The pair (N, J) is then referred to as an almost complex manifold. An almost complex manifold should have an even real dimension, i.e. n = 2m. The integer m is called the complex dimension of the manifold N. The endomorphism J can be extended by C-linearity to the complexified tangent bundle TCN =T N ⊗RC, then
TCN =T1,0N ⊕T0,1N,
where T1,0N (resp. T0,1N) denotes the eigenbundle ofTCN corresponding to the eigen-value i(resp. −i) of J. The bundle T1,0N is given by
T1,0N =T0,1N ={X−iJ X |X ∈Γ(T N)}.
Fix a Hermitian metricg compatible with the almost complex structure, i.e. a Rieman-nian metric g with the proprerty
g(J X, J Y) = g(X, Y),
then n(X, Y) = g(X, J Y) is a real 2-form on N. We will call (N, J, g) a Hermitian manifold. On a Hermitian manifold (N, J, g), the almost complex structure is called a complex structure if and only ifT1,0N is formally integrable, i.e. [T1,0N, T1,0N]⊂T1,0N. This integrability condition is equivalent to say that the Nijenhuis tensor NJ vanishes.
The Nijenhuis tensor NJ is the (2,1)-tensor defined by
NJ(X, Y) = [X, Y] +J[J X, Y] +J[X, J Y]−[J X, J Y],
1.3. SPINC STRUCTURES ON COMPLEX MANIFOLDS 59 for any X, Y ∈Γ(T N). A K¨ahler manifold is a Hermitian manifold (N, J, g) such that J is a complex structure and ∇J = 0, where ∇ is the Levi-Civita connection on N. Consider (N, J, g) a Hermitian manifold and{e1, J e1,· · · , em, J em}an orthonormal ba-sis of T N. We point out that, as a complex vector space, T N is C-isomorphic (resp.
C-anti-isomorphic) to T1,0N (resp. T0,1N). We define a complex vectorZj by Zj = 1
2(ej −iJ ej).
It is easy to check that {Z1,· · · , Zm} forms a basis of T1,0N and {Z1,· · · , Zm}forms a basis of T0,1N. Moreover, we have for any k, l= 1,· · · , m
g(Zk, Zl) = g(Zk, Zl) = 0, g(Zk, Zl) = g(Zk, Zl) = 1
2δkl.
Hence, the dual space T1,0∗ N of T1,0N is C-isomorphic to T0,1N by T0,1N −→ T1,0∗ N
Zj 7−→ Zj∗
=: 2g(Zj, .)
Similary,T0,1N isC-isomorphic toT0,1∗ N. The set{Z1∗,· · · , Zm∗}forms a basis ofT1,0∗ N and {Z1∗,· · · , Zm∗} forms a basis of T0,1∗ N.
Proposition 1.3.1. Every Hermitian manifold (Nn, J, g) has a canonical Spinc struc-ture whoseSpincbundle is given byΣN = Λ0,∗N =⊕mr=0Λ0,rN,whereΛ0,rN = Λr(T0,1∗ N) is the bundle of complex r-forms of type (0, r). The auxiliary line bundle of this canon-ical Spinc structure is given by KN−1, where KN = Λm(T1,0∗ N) is the canonical bundle of N [Fri00].
Proof. A Hermitian structure on N2m produces a reduction of the SO2m principal bundle PSO2mN to the subgroup Um, where Um is viewed canonically as a subgroup of SO2m. We denote the Um-reduction by U(N). Besides, there exists a homomorphism F : Um −→Spinc2m such that the following diagram commutes
Spinc2m
Um SO2m×S1
?
ξc
F
-f
Here, f is defined by f(A) = (A,detA), A ∈ Um and the homomorphism F can be explicitly described. In fact,
F(A) = hYm
j=1
cos(θj
2) + sin(θj
2)fjJ(fj)
, ei2Pmj=1θji ,
60 CHAPTER 1. INTRODUCTION TO COMPLEX SPIN GEOMETRY where {f1,· · · , fm}is a unitary basis in Cm with respect to which A has diagonal form A = (eiθ1,· · · , eiθk) and J is the complex structure of Cm. The lift F induces a Spinc structure on N via the Spinc2m-principal fiber bundle
PSpinc2mN = U(N)×UmSpinc2m. The corresponding S1-principal bundle is given by
PS1N = U(N)×detS1.
By definition, the auxiliary line bundle of this canonical Spinc structure is given by L=PS1N ×ρC= U(N)×ρ◦detC= Λm(T N) = Λm(T0,1∗ N).
Now, using thatClm 'Σ2m we can easily check that ρc2m◦F =ρfm, where ρfm : Um −→
End(Clm) is the extension of the standard representation of Um to Clm. Hence, the Spinc bundle is given by
ΣN = U(N)×ρc
2m◦F Σ2m 'U(N)×
ρem Clm =Cl(T N) = Λ∗(T N) = Λ∗(T0,1∗ N) = Λ0,∗N.
For any other Spinc structure the associated Spinc bundle can be written as [Fri00]
ΣN = Λ0,∗N ⊗ L,
where L2 =KN ⊗L and L is the auxiliary bundle associated with this Spinc structure.
In this case, the 2-form n can be considered as an endomorphism of ΣN via Clifford multiplication and it acts on a spinor ψ locally by [Fri00]
n·ψ =−
m
X
j=1
ej·J ej·ψ,
where {e1, J e1,· · · , em, J em} is any local frame of tangent vector fields. Moreover, we have the well-known orthogonal splitting
ΣN =⊕ml=0ΣlN,
where ΣlN denotes the eigensubbundle corresponding to the eigenvaluei(m−2l) ofn, with complex rank m
l
. Moreover, for any Z ∈Γ(T1,0N) and for any ψ ∈Γ(ΣrN), we have Z·ψ ∈Γ(Σr+1N) and Z ·ψ ∈Γ(Σr−1N).
Remark 1.3.1. If we choose the function f to bef(A) = (A,det1A), we get onN another Spinc structure called the anti-canonical Spinc structure for which ΣN = Λ∗,0N and the auxiliary line bundle is given by KN.
Proposition 1.3.2. Consider (N2m, J, g) a Hermitian manifold. For any Spinc struc-ture, we have
(Σ0N)2 =KN ⊗L,
where L is the auxiliary line bundle associated with the Spinc structure.
1.3. SPINC STRUCTURES ON COMPLEX MANIFOLDS 61 Proof. First, we can prove that there is an isomorphism between Λ0,rN ⊗Σ0N and ΣrN for any r = 0,· · · , m [Fri00, Kir86]. Hence, ΣN = Λ0,∗N ⊗ Σ0N. But, ΣN = Λ0,∗N ⊗ L,where L2 =KN ⊗L. Finally, (Σ0N)2 =KN ⊗L.
For the canonical Spinc structure and since L= (KN)−1, we deduce that Σ0N is trivial.
Hence, ifM is K¨ahler, the canonical Spincstructure carries parallel spinors lying in Σ0N (the complex constant functions).
Remark 1.3.2. We can view the canonical Spinc structure on K¨ahler manifolds dif-ferently using the definition of Spinc structure given in the Introduction. In fact, the complex fiber bundle Λ0,∗N = ⊕mr=0Λ0,rN, of rank 2m, has a scalar product (the exten-sion of the metric) and a connection (the extenexten-sion of the Levi-civita connection on N) which are compatible. Moreover, the map γ :T N −→End(Λ0,∗N) defined by
γ(X)ψ = 1
√2(X+iJ X)[∧ψ −√
2Xyψ,
satisfies the conditions (30), (31) et (32) for every X ∈ Γ(T N), ψ ∈ Γ(Λ0,rN) and r= 0,· · · , m. Hence, every K¨ahler manifold has a Spinc structure.
In 1997, A. Moroianu [Moro97] classified all simply connected Spinc manifolds car-rying parallel spinors:
Theorem 1.3.1. A simply connected Spinc manifold N carries a parallel spinor if and only if it is isometric to the Riemannian product N1×N2 of a simply connected K¨ahler manifold N1 and a simply connected Spin manifold N2 carrying a parallel spinor.
In this case, the Spinc structure ofN is the product of the canonical Spinc structure of N1 and the Spin structure of N2 and the parallel spinor is the product ψ1⊗ψ2 of a parallel spinorψ1of the canonical Spincstructure ofN1(this spinor lies in Σ0N1) and the parallel spinor of the Spin manifold N2. Moreover, the connection on the auxiliary line bundle associated with this product Spinc structure is the extension of the Levi-Civita connection to (KN1)−1. Hence, the curvature 2-formiΩ is given byiΩ =iρN1,whereρN1 is the Ricci 2-form onN1defined byρN1(X, Y) = ricN1(X, J Y) for everyX, Y ∈Γ(T N1).
Here, ricN1 is the Ricci curvature ofN1 considered as a bilinear symmetric form.
Chapter 2
Lower Bounds for the Eigenvalues of the Spin c Dirac Operator 1
2.1 Introduction
On a compact Riemannian Spin manifold (Nn, g) of dimension n > 2, T. Friedrich [Fri80] showed that any eigenvalue λ of the Dirac operator satisfies
λ2 >λ21 := n
4(n−1)inf
N S, (2.1)
The limiting case of (2.1) is characterized by the existence of a special spinor called a real Killing spinor. This is a sectionψ of the spinor bundle satisfying, for everyX ∈Γ(T N),
∇Xψ =−λ1 n X·ψ.
In [Hij95], O. Hijazi defined, on the complement set of zeroes of any spinor fieldφ, a field of symmetric endomorphisms `φ associated with the field of quadratic forms, denoted also by `φ, called the energy-momentum tensor which is given, for any vector field X, by
`φ(X) = Re
X· ∇Xφ, φ
|φ|2
.
The associated symmetric bilinear form is then given for every X, Y ∈Γ(T N) by
`φ(X, Y) = 1 2Re
X· ∇Yφ+Y · ∇Xφ, φ
|φ|2
.
Note that, if the spinor field φ is an eigenspinor, C. B¨ar showed that the zero set is contained in a countable union of (n −2)-dimensional submanifolds and has locally finite (n−2)-dimensional Hausdroff density [B¨ar99]. In 1995, O. Hijazi [Hij95] modified the connection ∇ in the direction of the endomorphism `ψ where ψ is an eigenspinor associated with an eigenvalue λ of the Dirac operator and established that
λ2 >inf
N (1
4S+|`ψ|2). (2.2)
1This chapter is the subject of a published paper [Nak10]
63
64 CHAPTER 2. LOWER BOUNDS The limiting case of (2.2) is characterized by the existence of a spinor field ψ satisfying for all X ∈Γ(T N),
∇Xψ =−`ψ(X)·ψ. (2.3)
The trace of `ψ being equal to λ, Inequality (2.2) improves Inequality (2.1) since, by the Cauchy-Schwarz inequality, |`ψ|2 > (tr(`nψ))2, where tr(`ψ) denotes the trace of `ψ. N. Ginoux and G. Habib showed in [GH09] that the Heisenberg manifold is a limiting manifold for (2.2) but equality in (2.1) cannot occur.
Using the conformal covariance of the Dirac operator, O. Hijazi [Hij86] showed that, on a compact Riemannian Spin manifold (Nn, g) of dimensionn>3, any eigenvalue of the Dirac operator satisfies
λ2 >λ21 := n
4(n−1)µ1, (2.4)
where µ1 is the first eigenvalue of the Yamabe operator given by L:= 4n−1
n−24+S,
where 4 is the Laplacian acting on functions. In dimension 2, C. B¨ar [B¨ar92] proved that any eigenvalue of the Dirac operator on N satisfies
λ2 >λ21 := 2πχ(N)
Area(N, g), (2.5)
where χ(N) is the Euler-Poincar´e number of N. The limiting case of (2.4) and (2.5) is also characterized by the existence of a real Killing spinor. In terms of the energy-momentum tensor, O. Hijazi [Hij95] proved that, on such manifolds any eigenvalue of the Dirac operator satisfies the following
λ2 >λ21 :=
πχ(N)
Area(N,g) + inf
N |`ψ|2 if n = 2,
1
4µ1+ inf
N |`ψ|2 if n>3.
(2.6)
Again, the trace of`ψ being equal toλ, Inequality (2.6) improves Inequalities (2.4) and (2.5). The limiting case of (2.6) is characterized by the existence of a spinor field ϕ satisfying for all X ∈Γ(T N),
∇Xϕ=−`ϕ(X)· ϕ, (2.7)
where ϕ=e−n−12 uψ, the spinor field ψ is an eigenspinor associated with the first eigen-value of the Dirac operator and ψ is the image of ψ under the isometry between the spinor bundles of (Nn, g) and (Nn, g =e2ug). Suppose that on a Spin manifoldN, there exists a spinor field φ such that, for all X ∈Γ(T N),
∇Xφ=−E(X)·φ, (2.8)
2.1. INTRODUCTION 65 whereE is a symmetric 2-tensor defined onT N. It is easy to see that E must be equal to`φ. If the dimension of N is equal to 2, T. Friedrich [Fri98] proved that the existence of a pair (φ, E) satisfying (2.8) is equivalent to the existence of a local immersion of N into the Euclidean space R3 with Weingarten tensor equal to 2E. In [Mor05], B.
Morel showed that ifNn is a hypersurface of a manifold carrying a parallel spinor, then the energy-momentum tensor (associated with the restriction of the parallel spinor) is, up to a constant, the second fundamental form of the hypersurface. G. Habib [Hab07]
studied Equation (2.8) for an endomorphism E, not necessarily symmetric. He showed that the symmetric part of E is `φ and the skew-symmetric part of E is qφ defined on the complement set of zeroes of φ by
qφ(X, Y) = g(qφ(X), Y) = 1 where ψ is an eigenspinor associated with an eigenvalueλ and gets that
λ2 >inf
N (1
4S+|`ψ|2+|qψ|2). (2.9) The Heisenberg group and the solvable group are examples of limiting manifolds [Hab07].
For a better understanding of the tensor qφ, he studied Riemannian flows and proved that, if the normal bundle carries a parallel spinor, the tensor qφ plays the role of the O’Neill tensor of the flow. In this chapter, we prove the corresponding inequalities for Spinc manifolds:
Theorem 2.1.1. Let (Nn, g) be a compact Riemannian Spinc manifold of dimension n >2. Then, any eigenvalue of the Dirac operator to which is attached an eigenspinor ψ satisfies
We will only consider deformations of the connection in the direction of the sym-metric endomorphism `ψ and hence under the same conditions as Theorem 2.1.1, one gets
In 1999, A. Moroianu and M. Herzlich [HM99] proved that on Spinc manifolds of di-mensionn >3, any eigenvalue of the Dirac operator satisfies
λ2 >λ21 := n
4(n−1)µ1, (2.12)
where µ1 is the first eigenvalue of the perturbed Yamabe operator defined by LΩ =L−cn|Ω|g.
66 CHAPTER 2. LOWER BOUNDS The limiting case of (2.12) is characterized by the existence of a real SpincKilling spinor ψ satisfying Ω·ψ =ic2n|Ω|gψ, i.e. a section ψ satisfying
∇Xψ =−λ1
n X·ψ and Ω·ψ =icn 2|Ω|gψ.
In terms of the energy-momentum tensor we prove:
Theorem 2.1.2. Under the same conditions as Theorem 2.1.1, any eigenvalue λ of the Dirac operator to which is attached an eigenspinor ψ satisfies
λ2 >
1
4µ1 + infN |`ψ|2 if n >3,
πχ(N) Area(N,g)− 12
R
N|Ω|gvg
Area(N,g) + infN |`ψ|2 if n= 2,
(2.13) where µ1 is the first eigenvalue of the perturbed Yamabe operator.
Using the Cauchy-Schwarz inequality in dimension n > 3, we have that Inequality (2.13) implies Inequality (2.12). As a corollary of Theorem 2.1.2, we compare the lower bound to a conformal invariant (the Yamabe number) and to a topological invariant, in case of 4-dimensional manifolds whose auxiliary line bundle has self dual curvature (see Corollary 2.3.1 and Corollary 2.3.2). Finally, we study the limiting case of (2.11) and (2.13), and we give an example.