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**LOWER BOUNDS FOR THE EIGENVALUES OF THE** **Spin c DIRAC OPERATOR ON SUBMANIFOLDS**

### Roger Nakad, Julien Roth

**To cite this version:**

### Roger Nakad, Julien Roth. LOWER BOUNDS FOR THE EIGENVALUES OF THE Spin c DIRAC

### OPERATOR ON SUBMANIFOLDS. Archiv der Mathematik, Springer Verlag, 2015, 104 (5), pp.451-

### 461. �hal-01112320�

OPERATOR ON SUBMANIFOLDS

ROGER NAKAD AND JULIEN ROTH

ABSTRACT. We prove lower bounds for the eigenvalues of theSpin^{c}Dirac operator on
submanifolds. These estimates are expressed in terms of extrinsic and intrinsic quanti-
ties. We also give estimates involving the Energy-Momentum tensor as well as conformal
bounds. The limiting cases of these estimates give rise to particular spinor fields, called
generalized twisted Killing spinors, which are also studied.

1. INTRODUCTION ANDPREMILINARIES

The limiting cases of estimates for eigenvalues of the Dirac operator on compact (with or without boundary) manifolds give rise to examples of special geometries. For instance, equality in the classical inequalitiy of Friedrich [3]

λ16 n 4(n−1)inf

M ScalM,

wherenis the dimanesion of the manifoldM andScalM its scalar curvature, forces the manifold to be Einstein with positive scalar curvature, due to the fact that the eigenspinor associated with the first eigenvalue is then a real Killing spinor.

This is also the case for the conformal inequality of Hijazi [9]

λ16 n 4(n−1)µ1,

whereµ1is the first eigenvalue of the Yamabe operator as well for the inequality involving the Energy-momentum tensor [10]

λ_{1}61
4inf

M(Scal_{M}+|Q_{ϕ}|^{2}),

whereQ_{ϕ}is the Energy-Momentum tensor associated with a first eigenspinorϕ.

On the other hand, in the recent years, many estimates have been proved for the
eigenvalues of theSpin^{c}Dirac operator. A great difference between both cases if that the
equality cases forSpin^{c}lower bounds are less and therefore give larger classes of limiting
manifolds (see [8, 13, 14] for instance).

In this note, we prove a new lower bound for the eigenvalues of theSpin^{c}Dirac operator
on submanifolds of Spin^{c} manifolds (see Theorem 2.1). This generalizes for theSpin^{c}
case, previous estimates by Hijazi-Zhang [11, 12] and Ginoux-Morel [5]. The limiting
case of this estimate is characterized by the existence of particular spinor fields called
generalized twisted Killing spinors. We will study these particular spinor fields and show
that under a natural assumption on the dimension and the codimension of the submanifold,
they are in facttwisted Killing spinorswhich generalize naturally the usual Killing spinors
for twisted spinor bundles.

Finally, we also prove conformal estimates (Theorem 2.3) as well as estimates involving the Energy-Momentum tensor associated with an eigenspinor (Theorem 2.2).

2010Mathematics Subject Classification. 53C27, 53C42.

Key words and phrases. Submanifolds, Dirac Operator, Eigenvalues,Spin^{c}spinors, Twisted Killing spinors.

1

2 R. NAKAD AND J. ROTH

We recall briefly some basic facts about Spin^{c} manifolds and their hypersurfaces
(the reader can refer to [1, 2, 4, 5]). Let(fM^{m+n},eg)be a RiemannianSpin^{c}manifold and
M^{m} a submanifold isometrically immersed into Mf. Assume thatM is also Spin^{c} and
denote byN M the normal bundle of the immersion ofM intoMf. We denote byieΩ(resp.

iΩ) the curvature2-form of the corresponding auxiliary line bundle. Since the manifolds
M andMfareSpin^{c}, there exists aSpin^{c}structure on the bundleN M. We denote byΣN
theSpin^{c}bundle ofN M and let

Σ :=

ΣM⊗ΣN ifmornis even,

ΣM⊗ΣN⊕ΣM⊗ΣN otherwise

It is well known that there is a natural isomorphism betweenΣandΣMf_{|M}. Moreover, we
denote by∇the covariant derivative onΣdefined by

∇:=

∇^{ΣM} ⊗Id + Id⊗∇^{ΣN} ifmornis even,

∇^{ΣM} ⊗Id + Id⊗∇^{ΣN} ⊕ ∇^{ΣM}⊗Id + Id⊗∇^{ΣN} otherwise.

We have the following identification between the Clifford multiplication
X·_{M} ϕ= (X·ω_{⊥}·ψ)|_{M},

(1)

whereϕ=ψ|M,ψ∈Γ(ΣMf),ω_{⊥} :=ω_{n} ifnis even andω_{⊥} =−iωn ifnis odd, with
ω_{n} =i[^{n+1}_{2} ]ν_{1}· · ·ν_{n}the complex volume element of the normal bundle. We also recall the
Spin^{c}Gauss formula

∇eXϕ=∇Xϕ+1 2

m

X

j=1

ej·B(X, ej)·ϕ, (2)

where∇e is the spinorial connection onMfandBis the second fundamental form ofM in Mf. We will denote byHthe mean curvature.

Now, let us define the following Dirac operators D=

m

X

j=1

ej· ∇e_{j}, De:=

m

X

j=1

ej·∇ee_{j},

and

DH= (−1)^{n}ω⊥·De = (−1)^{n}ω⊥·D+1

2H·ω⊥·.

Clearly, from the spinorial Gauss formula (2), we haveDe =D−^{1}_{2}H·. Moreover (see [1,
Lemma 2.1]),DandDHare formally self-adjoint andD_{H}^{2} =De^{∗}DewhereDe^{∗}is the formal
adjoint ofDewith respect to theL^{2}-scalar productR

M(·,·)dvg.

We finish this section of preliminaries by the two following lemmas. The first one gen-
eralizes the classical Lichnerowicz formula in the context of twistedSpin^{c} spinor bun-
dles. Before stating the lemma, we need to introduce the following function associ-
ated to a spinor field ϕ ∈ Γ(Σ), R^{N}_{ϕ} := 2Pn

i,j=1

D

ei·ej·Id⊗ R^{N}_{e}_{i}_{,e}_{j}ϕ,_{|ϕ|}^{ϕ}2

E
, on
Mϕ ={x∈M |ϕ(x)6= 0}. Here,R^{N}_{e}_{i}_{,e}_{j} stands for the spinorial curvature of the normal
bundleN M.

Proposition 1.1(Twisted Lichnerowicz formula). For any spinor fieldϕ ∈Γ(Σ), point- wise onMϕ, we have

D^{2}ϕ, ϕ

=h∇^{∗}∇ϕ, ϕi+1

4(ScalM +R^{N}_{ϕ})|ϕ|^{2}+ i

2hΩ·ϕ, ϕi. (3)

Proof:We give the proof in the case wheremornis even. The other case is similar. We compute the square of the Dirac operatorDacting onϕ=α⊗σ. We have

(4) D^{2}ϕ=∇^{∗}∇ϕ+1

2

m

X

i,j=1

e_{i}·e_{j}· R_{e}_{i}_{,e}_{j}ϕ,

where∇^{∗}is the formal adjoint of∇andRis the spinorial curvature associated with the
connection∇. From the definition of∇, we see easily that

R_{e}_{i}_{,e}_{j}ϕ= (R^{M}_{e}

i,e_{j}σ)⊗α+σ⊗(R^{N}_{e}

i,e_{j}α).

Then, a classical computation on each factor gives the desired formula.

We have this second elementary lemma

Proposition 1.2. For any spinor fieldϕ∈Γ(Σ), we have
hiΩ·ϕ, ϕi ≥ −c_{m}

2 |Ω||ϕ|^{2}.
(5)

Proof:Ifmornis even, thenΣM⊗ΣN and a spinorϕ∈Σcan be writtenϕ=σ⊗α.

Hence, we have

hiΩ·ϕ, ϕi = hiΩ·(σ⊗α), σ⊗αi=hi(Ω ·

Mσ)⊗α, σ⊗αi

= hi(Ω ·

Mσ), σi|α|^{2}≥ −c_{m}

2 |Ω||σ|^{2}|α|^{2}≥ −c_{m}
2 |Ω||ϕ|^{2}.

Note that we use the fact that the scalar product onΣis the product of the scalar products
onΣM andΣN. We also use the classical estimate onM, that ishiΩ·σ, σi ≥ −^{c}_{2}^{m}|Ω||σ|^{2}
(see [8]). Ifmandnare odd, thenΣ = ΣM⊗ΣN⊕ΣM⊗ΣN and a spinorϕ∈Σis of
the formϕ= (α⊗σ, σ^{0}⊗α^{0}). Thus, we have

hiΩ·ϕ, ϕi = hiΩ·(σ⊗α), σ⊗αi+hiΩ·(σ^{0}⊗α^{0}), σ^{0}⊗α^{0}i

≥ −cm

2 |Ω| |σ|^{2}|α|^{2}+|σ^{0}|^{2}|α^{0}|^{2}

≥ −cm

2 |Ω||ϕ|^{2}.

This concludes the proof.

2. EIGENVALUE ESTIMATES FOR SUBMANIFOLDS

Now, we have all the ingredients to state the eigenvalue estimates. We begin by the follow-
ing basic estimates involving intrinsic terms (scalar curvature, curvature of the line bundle
overM) and extrinsic terms (mean curvature and spinorial normal curvature). This result
generalizes in theSpin^{c} setting the estimate of Hijazi-Zhang [11] (extending to any codi-
mension by Ginoux-Morel [5]).

Theorem 2.1. Let(M^{m}, g)be a compact RiemannianSpin^{c} manifold isometrically im-
mersed into a RiemannianSpin^{c}manifold(Mf^{m+n},eg). Consider a non-trivial eigenspinor
fieldϕ ∈ Γ(Σ)for the submanifold Dirac operatorDH, i.e.DHϕ = λϕ. Assume that
m≥2and

Scal_{M}+R^{E}_{ϕ}−c_{m}|Ω^{M}|> m−1

m ||H||^{2}>0
onMϕ, then, we have

λ^{2}≥ 1
4inf

M_{ϕ}

r m

m−1(Scal_{M} +R^{N}_{ϕ} −c_{m}|Ω|)− ||H||

^{2}
.
Moreover, if equality holds, thenϕis a twisted generalized Killing spinor.

4 R. NAKAD AND J. ROTH

Proof: Let λ be an eigenvalue of the submanifold Dirac operator D_{H} andq a smooth
function, nowhere equal to_{m}^{1}. We consider the following modified connection∇^{λ,q}defined
by

∇^{λ,q}_{X} ψ=∇Xψ+ 1−q

2(1−mq)X·H·ψ+qλX·ω_{⊥}·ψ,

for any spinor fieldψ∈Γ(Σ). Letϕbe an eigenspinor forDHassociated with the eigen- valueλ. Using the Twisted Lichnerowicz formula (3), we can easily compute

Z

M

|∇^{λ,q}ϕ|^{2}vg =
Z

M

(1 +mq^{2}−2q)h
λ^{2}−1

4

Scal_{M} +R^{N}_{ϕ}

1 +mq^{2}−2q−(m−1)kHk^{2}
(1−mq)^{2}

|ϕ|^{2}

− i

2(1 +mq^{2}−2q) <Ω·ϕ, ϕ >i
vg

(6)

Then, using Inequality (5) and by assumingm(ScalM+R^{N}_{ϕ}−cm|Ω|)>(m−1)kHk^{2}>0,
we can chooseqso that

(1−mq)^{2}= (m−1)kHk
q m

m−1(ScalM +R^{N}_{ϕ} −cm|Ω|)− kHk
,
(7)

onMϕ. Inserting (7) in (6), and since the complement ofMψ inM is of measure0, we conclude because the left member of (6) is nonnegative.

If equality occurs,∇^{λ,q}_{X} ϕ= 0, which implies that|ϕ|is constant onM and
Dϕ= m(1−q)

2(1−mq)H·ϕ+mqλω⊥·ϕ.

On the other hand, sinceϕis an eigenspinor forDHand the link betweenDHandD, we get

0 = λω_{⊥}·ϕ+H

2 ·ϕ− m(1−q)

2(1−mq)H·ϕ−mqλω_{⊥}·ϕ

= (1−mq)^{2}λω_{⊥}·ϕ−(m−1)H
2 ·ϕ.

Moreover, equality also implies2|λ|=q

m

m−1(ScalM+R^{N}_{ϕ} −cm|Ω|)− ||H||. From the
expression ofqand the above relation, we haveω_{⊥}·ϕ= sgn(λ)_{||H||}^{H} ·ϕand thusϕsatisfies

∇Xϕ = −_{m}^{f}X ·ω_{⊥}·ϕ, withf = ^{sgn(λ)}_{2} q _{m}

m−1(ScalM +R^{N}_{ϕ} −cm|Ω|). That is,ψis
a generalized twisted Killing spinor. Note that heref isa prioria function. We will see
in Section 3 that under some assumptions on the dimensionsmandn, the functionf is

constant.

Now, we define the Energy-Momentum tensor associated with a spinor fieldψ∈Γ(Σ)on Mψby

Q^{ψ}_{ij} =1

2(ei·ω_{⊥}· ∇e_{j} +ej·ω_{⊥}· ∇e_{i}, ψ

|ψ|^{2})
Note that

Q^{ψ}_{ij}= 1

2(ei·M∇e_{j} +ej·M∇e_{i}, ψ

|ψ|^{2})

so it is the intrinsic energy momentum tensor and it is the one appearing in the Einstein Dirac equation. We have the following estimate involving the Energy-Momentum tensor.

Theorem 2.2. Let(M^{m}, g)be a compact RiemannianSpin^{c} manifold isometrically im-
mersed into a RiemannianSpin^{c}manifold(Mf^{m+n},eg). Consider a non-trivial eigenspinor
fieldϕ ∈ Γ(Σ)for the submanifold Dirac operator DH, i.e. DHϕ = λϕ. Assume that
m≥2and

ScalM +R^{N}_{ϕ} + 4|Q^{ϕ}|^{2}−cm|Ω^{M}|>||H||^{2}>0

onM_{ϕ}, then, we have
λ^{2}≥ 1

4inf

M_{ϕ} Scal_{M}+R^{N}_{ϕ} + 4|Q^{ϕ}|^{2}−c_{m}|Ω|)− ||H||^{2}
.

Moreover, if equality holds, thenϕis a twisted (symmetric) EM-spinor.

Proof:For any real functionqthat never vanishes, consider the modified covariant deriva- tive defined onΓ(Σ)by

∇^{Q}_{e}_{i}ψ=∇e_{i}ψ− 1

2mqei·H·ψ+ (−1)^{n+1}qλei·ω⊥·ψ+X

j

Q^{ψ}_{ij}ej·ω⊥·ψ

Again, we can compute, for an eigenspinorϕ Z

M

|∇^{Q}ϕ|^{2}vg =
Z

M

(1 +mq)^{2}h
λ^{2}−1

4

Scal_{M} +R^{N}_{ϕ} + 4|Q^{ϕ}|^{2}

(1 +mq^{2}) −kHk^{2}
mq^{2}

i|ϕ|^{2}vg

−1 4 Z

M

(1 +mq^{2})h 2
mq(1 +mq^{2})

kHk^{2}−< H·ϕ, ω_{⊥}·ϕ >^{2}

|ϕ|^{4}

i|ψ|^{2}v_{g}

−i 2

Z

M

<Ω·ϕ, ϕ >vg

(8)

Now, we use again (5) and if moreover,ScalM+R^{N}_{ϕ} −cM|Ω|+ 4|Q^{ϕ}|^{2}>kHk^{2}>0, we
take

q= v u u t

kHk m(q

ScalM +R^{N}_{ϕ} −cM|Ω|+ 4|Q^{ϕ}|^{2}− kHk)
,

and then by the Cauchy Schwarz inequality, we have
kHk^{2}−< H·ϕ, ω⊥·ϕ >^{2}

|ϕ|^{4} ≥0.

If equality holds, then ∇^{Q}ϕ = 0and equality occurs in the Cauchy-Schwarz inequality,
that is,kHk^{2}−< H·ϕ, ω_{⊥}·ϕ >^{2}

|ϕ|^{4} = 0. Thus, proceeding as in the proof of Theorem
2.1, we deduce that ∇Xϕ = −Q(X) · ω_{⊥} · ϕ, that is ψ is a twisted (symmetric)

Energy-Momentum spinor (EM-spinor).

Note that, by a completely similar computation, we can obtain a lower bound in- volving both symmetric and skew-symmetric Energy-Momentum tensors as in [7]. We do not write it in this note.

Finally, following the idea of Hijazi [9], we consider a conformal change of the metrc

¯

g =e^{2u}eg. LetΣ−→Σ,¯ ψ−→ψ¯be the corresponding isometry between the two spinor
bundles. Recall that for 2 spinorsψandϕonΣand for any vector fieldXonMf, we have

(ϕ, ψ) = ( ¯ϕ,ψ)¯ ¯g and X¯¯·ψ¯=X·ψ Note that we have

D(e^{−(m−1)}^{2} ^{u}ψ) =e^{−(m+1)}^{2} ^{u}Dψ

whereDdenotes the Dirac operator w.r.t the metricg. Moreover, the corresponding mean curvature is given by

H =e^{−2u}(H−mgrad^{N}u)

Now, assume thatgrad^{N}u= 0, thenD_{H}is also conformally covariant and we have
D_{H}(e^{−(m−1)}^{2} ^{u}ψ) =e^{−(m+1)}^{2} ^{u}D_{H}ψ

From now on, we will consider regular conformal change of the metricg, i.e.,grad^{N}u= 0.

6 R. NAKAD AND J. ROTH

Theorem 2.3. Let(M^{m}, g)be a compact RiemannianSpin^{c} manifold isometrically im-
mersed into a RiemannianSpin^{c}manifold(Mf^{m+n},eg). Consider a non-trivial eigenspinor
fieldϕ∈Γ(Σ)for the submanifold Dirac operatorDH, i.e.DHψ=λψ. For any regular
conformal change of metric¯g=e^{2u}eg, assume thatm≥3and

ScalM e^{2u}+R^{N}_{ϕ} + 4|Q^{ψ}|^{2}−cm|Ω^{M}|>||H||^{2}>0
onMϕ, then, we have

λ^{2}≥1
4 inf

Mψ

q

ScalM e^{2u}+R^{N}_{ψ} + 4|Q^{ψ}|^{2}−cm|Ω|)− ||H||

^{2}
.

Proof:Forψ∈Γ(Σ)an eigenspinor ofDHwith eigenvalueλ, letϕ:=e^{−(n−1)u}^{2} ψ. then,
we haveDHϕ=λe^{−u}ϕ. Recall that

∇eiψ=∇eiψ−1

2e_{i}·du·ψ−1
2e_{i}(u)ψ,

and ei = e^{−u}ei. Now, it is straightforward to get Q^{ϕ}_{i,j} = e^{−u}Q^{ψ}_{i,j}, hence |Q^{ϕ}|^{2} =
e^{−2u}|Q^{ψ}|^{2}. Equation (8) is also true on(M , g). If, we apply it tof ϕ, we get

Z

M

|∇^{Q}ϕ|^{2}vg =
Z

M

(1 +mq)^{2}h

(λe^{−u})^{2}−1
4

Scal_{M}+R^{N}_{ϕ} + 4|Q^{ϕ}|^{2}

(1 +mq^{2}) −kH˜k^{2}_{g}
mq^{2}

i|ϕ|^{2}_{g}vg

−1 4

Z

M

(1 +mq^{2})h 2
mq(1 +mq^{2})

kHk˜ ^{2}_{g}−<H·ϕ, ω˜ _{⊥}·ϕ >^{2}_{g}

|ϕ|^{4}_{g}

i|ϕ|^{2}_{g}vg

−i 2

Z

M

<Ω·ϕ, ϕ >gvg

(9)

sinceH˜ =e^{−u}HandR^{N}_{ϕ} =e^{−2u}R^{N}_{ψ}, we have

Z

M

|∇^{Q}ϕ|^{2}v_{g} =
Z

M

(1 +mq)^{2}e^{−2u}h

(λ)^{2}−1
4

ScalMe^{2u}+R^{N}_{ψ} + 4|Q^{ψ}|^{2}

(1 +mq^{2}) −kHk^{2}
mq^{2}

i|ϕ|^{2}v_{g}

−1 4

Z

M

(1 +mq^{2})e^{−2u}h 2
mq(1 +mq^{2})

kHk^{2}−< H·ψ, ω_{⊥}·ψ >^{2}

|ψ|^{4}

i|ϕ|^{2}v_{g}

−i 2

Z

M

<Ω·ϕ, ϕ >gvg

(10)

Now, we use again (5) and if moreover,Scal_{M} e^{2u}+R^{N}_{ψ} −c_{M}|Ω|+ 4|Q^{ψ}|^{2}>kHk^{2}>0,
we take

q= v u u t

kHk m(q

Scal_{M}e^{2u}+R^{N}_{ψ} −c_{M}|Ω|+ 4|Q^{ψ}|^{2}− kHk)
,

and then we use the Cauchy Schwarz inequality

kHk^{2}−< H·ψ, ω_{⊥}·ψ >^{2}

|ψ|^{4} ≥0,

to get the desired result.

By taking u as first eigenfunction of the Yamabe operator on M, we get the follow- ing corollary, whereµ1is the first eigenvalue of the Yamabe operator.

Corollary 2.4. Under the assunptions of Theorem 2.3 and if
µ_{1}+R^{N}_{ψ} + 4|Q^{ψ}|^{2}−c_{m}|Ω^{M}|>||H||^{2}>0
onM_{ϕ}, then, we have

λ^{2}≥1
4 inf

M_{ψ}

q

µ1+R^{N}_{ψ} + 4|Q^{ψ}|^{2}−cm|Ω|)− ||H||^{2}
.

3. GENERALIZED TWISTEDKILLING SPINORS

We have seen in Theorem 2.1 that if equality occurs, then, the eigenspinorϕis in fact a generalized twisted Killing spinor, that is satisfies the equation

(11) ∇Xϕ=f X·ω_{⊥}·ϕ=f X ·

Mϕ, wherefis a real function. We prove the following

Proposition 3.1. Let (M^{m}, g) be a compact Riemannian Spin^{c} manifold isometrically
immersed into a RiemannianSpin^{c}manifold(fM^{m+n},eg). Letϕ∈Γ(Σ)be a generalized
twisted Killing spinor with real-valued functionf. Ifm > n+ 4, thenf is constant.

Proof:We define the following forms forp∈ {1,· · ·, m}, ωp(X1,· · ·, Xp) =D

(X1∧X2∧ · · · ∧Xp) ·

Mϕ, ϕE ,

We have the following easy facts (see [6, 8] for instance). For anyk>0, the formsω4k+1

andω4k+2are imaginary-valued whereas the formsω4k+3andω4kare real-valued. More- over, we have for anyp>0

dωp= ((−1)^{p}f−f)ωp+1.
In particular, we have for anyk>1

(12) df∧ω2k = 0.

Assume thatf is not constant and letx∈Msuch thatdf 6= 0, on a neighborhoodV ofx.

Hencedf^{⊥}is of dimensionm−1and we consider{e1,· · ·, em−1}an orthonormal frame
ofdf^{⊥}. From this, we have

ω_{2k}(e_{i}_{1},· · ·, e_{i}_{2k}) = 0,

for any subset{i_{1},· · · , i_{2k}} of{1,· · ·, m−1}. Thus, forl = _{m−1}

2

, we deduce that
the spinor fieldsϕ,e_{i}_{1} ·

M e_{i}_{2} ·

Mϕ,· · · ande_{i}_{1} ·

Me_{i}_{2}· · ·e_{i}_{2l} ·

M ϕare orthonormal onV. Consequently, the space spanned by these spinors is a vector subspace ofΣxof complex dimension

1 +

m−1 2

+

m−1 4

+· · ·+

m−1 2l

= 2^{m−2}.

Since the complex dimension ofΣis d(m, n) =

(

2^{m+n}^{2} ifm+nis even
2^{m+n−1}^{2} ifm+nis odd

we conclude that2^{m−2} 6 d(m, n). Therefore,f is constant if 2^{m−2} > d(m, n)which
corresponds to the condition expressed in the statement of the Proposition. Indeed, on one
hand, if m+nis even, then2^{m−2} > d(m, n)is equivalent tom−2 > ^{n+m}_{2} , that is
m > n+ 4. On the other hand, if fm+nis odd, then2^{2m−2}> d(m, n)is equivalent to
m−2 > ^{n+m−1}_{2} , that ism > n+ 3. But sincem+nis odd, thenmcannot be equal to

n+ 4. hence we also havem > n+ 4.

Hence, we deduce the following

8 R. NAKAD AND J. ROTH

Corollary 3.2. Let(M^{m}, g)be a compact RiemannianSpin^{c} manifold isometrically im-
mersed into a RiemannianSpin^{c}manifold(Mf^{m+n},eg). ifm > n+ 4and equality occurs
in Theorem 2.1, thenM admits a twisted Killing spinor.

Acknowledgment.The first author is indebted to the Center for Advanced Mathemati- cal Sciences (CAMS, Lebanon) and the University of Paris-Est, Marne-la-Vall´ee for their hospitality and support.

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[14] R. Nakad & J. Roth,TheSpin^{c}Dirac operator on hypersurfaces and applications, Diff. Geom. Appl.,31
(1), (2013), pp 93-103

(J. Roth) LAMA, UNIVERSITE´PARIS-ESTMARNE-LA-VALLEE´ , CITE´DESCARTES, CHAMPS SURMARNE, 77454 MARNE-LA-VALLEE CEDEX´ 2, FRANCE

E-mail address:julien.roth@u-pem.fr

(R. Nakad) NOTREDAMEUNIVERSITY-LOUAIZE, FACULTY OFNATURAL ANDAPPLIED SCIENCES, DE- PARTMENT OFMATHEMATICS ANDSTATISTICS, P. O. BOX72, ZOUKMIKAEL, LEBANON

E-mail address:rnakad@ndu.edu.lb