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arXiv:0810.0164v2 [math.DG] 14 Dec 2009
THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER
MANIFOLDS
ANDREI MOROIANU AND UWE SEMMELMANN
Abstract. The moduli space NK of infinitesimal deformations of a nearly K¨ ahler
structure on a compact 6-dimensional manifold is described by a certain eigenspace of
the Laplace operator acting on co-closed primitive (1,1) forms (c.f. [10]). Using the
Hermitian Laplace operator and some representation theory, we compute the space
NK on all 6-dimensional homogeneous nearly K¨ ahler manifolds. It turns out that the
nearly K¨ ahler structure is rigid except for the flag manifold F(1,2) = SU3/T2, which
carries an 8-dimensional moduli space of infinitesimal nearly K¨ ahler deformations,
modeled on the Lie algebra su3of the isometry group.
2000 Mathematics Subject Classification: Primary 58E30, 53C10, 53C15.
Keywords: Nearly K¨ ahler deformations, Hermitian Laplace operator.
1. Introduction
Nearly K¨ ahler manifolds were introduced in the 70’s by A. Gray [8] in the context of
weak holonomy. More recently, 6-dimensional nearly K¨ ahler manifolds turned out to be
related to a multitude of topics among which we mention: Spin manifolds with Killing
spinors (Grunewald), SU3-structures, geometries with torsion (Cleyton, Swann), stable
forms (Hitchin), or super-symmetric models in theoretical physics (Friedrich, Ivanov).
Up to now, the only sources of compact examples are the naturally reductive 3-
symmetric spaces, classified by Gray and Wolf [13], and the twistor spaces over positive
quaternion-K¨ ahler manifolds, equipped with the non-integrable almost complex struc-
ture. Based on previous work by R. Cleyton and A. Swann [6], P.-A. Nagy has shown
in 2002 that every simply connected nearly K¨ ahler manifold is a Riemannian product of
factors which are either of one of these two types, or 6-dimensional [12]. Moreover,
J.-B. Butruille has shown [5] that every homogeneous 6-dimensional nearly K¨ ahler
manifold is a 3-symmetric space G/K, more precisely isometric with S6= G2/SU3,
S3×S3= SU2×SU2×SU2/SU2, CP3= SO5/U2×S1or F(1,2) = SU3/T2, all endowed
with the metric defined by the Killing form of G.
A method of finding new examples is to take some homogeneous nearly K¨ ahler man-
ifold and try to deform its structure. In [10] we have studied the deformation problem
for 6-dimensional nearly K¨ ahler manifolds (M6,g) and proved that if M is compact,
Date: December 15, 2009.
This work was supported by the French-German cooperation project Procope no. 17825PG.
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2 ANDREI MOROIANU AND UWE SEMMELMANN
and has normalized scalar curvature scalg = 30, then the space NK of infinitesimal
deformations of the nearly K¨ ahler structure is isomorphic to the eigenspace for the
eigenvalue 12 of the restriction of the Laplace operator ∆gto the space of co-closed
primitive (1,1)-forms Λ(1,1)
0
M.
It is thus natural to investigate the Laplace operator on the known 3-symmetric exam-
ples (besides the sphere S6, whose space of nearly K¨ ahler structures is well-understood,
and isomorphic to SO7/G2∼= RP7, see [7] or [5, Prop. 7.2]). Recall that the spectrum
of the Laplace operator on symmetric spaces can be computed in terms of Casimir
eigenvalues using the Peter-Weyl formalism. It turns out that a similar method can be
applied in order to compute the spectrum of a modified Laplace operator¯∆ (called the
Hermitian Laplace operator) on 3-symmetric spaces. This operator is SU3-equivariant
and coincides with the usual Laplace operator on co-closed primitive (1,1)-forms. The
space of infinitesimal nearly K¨ ahler deformations is thus identified with the space of
co-closed forms in Ω(1,1)
0
(12) := {α ∈ C∞(Λ(1,1)
that the nearly K¨ ahler structure is rigid on S3×S3and CP3, and that the space of in-
finitesimal nearly K¨ ahler deformations of the flag manifold F(1,2) is eight-dimensional.
0
M) |¯∆α = 12α}. Our main result is
The paper is organized as follows. After some preliminaries on nearly K¨ ahler mani-
folds, we give two general procedures for constructing elements in Ω(1,1)
vector fields or eigenfunctions of the Laplace operator for the eigenvalue 12 (Corollary
4.5 and Proposition 4.11). We show that these elements can not be co-closed, thus
obtaining an upper bound for the dimension of the space of infinitesimal nearly K¨ ahler
deformations (Proposition 4.12). We then compute this upper bound explicitly on the
3-symmetric examples and find that it vanishes for S3× S3and CP3, which therefore
have no infinitesimal nearly K¨ ahler deformation. This upper bound is equal to 8 on the
flag manifold F(1,2) = SU3/T2and in the last section we construct an explicit isomor-
phism between the Lie algebra of the isometry group su3and the space of infinitesimal
nearly K¨ ahler deformations on F(1,2).
0
(12) out of Killing
In addition, our explicit computations (in Section 5) of the spectrum of the Hermitian
Laplace operator on the 3-symmetric spaces, together with the results in [11] show that
every infinitesimal Einstein deformation on a 3-symmetric space is automatically an
infinitesimal nearly K¨ ahler deformation.
Acknowledgments. We are grateful to Gregor Weingart for helpful discussions and in
particular for suggesting the statement of Lemma 5.4.
2. Preliminaries on nearly K¨ ahler manifolds
An almost Hermitian manifold (M2m,g,J) is called nearly K¨ ahler if
(∇XJ)(X) = 0,
∀ X ∈ TM,(1)
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THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS3
where ∇ denotes the Levi-Civita connection of g. The canonical Hermitian connection
¯∇, defined by
¯∇XY := ∇XY −1
is a Umconnection on M (i.e.¯∇g = 0 and¯∇J = 0) with torsion¯TXY = −J(∇XJ)Y .
A fundamental observation, which goes back to Gray, is the fact that¯∇¯T = 0 on every
nearly K¨ ahler manifold (see [2]).
We denote the K¨ ahler form of M by ω := g(J.,.). The tensor Ψ+:= ∇ω is totally
skew-symmetric and of type (3,0) + (0,3) by (1). From now on we assume that the
dimension of M is 2m = 6 and that the nearly K¨ ahler structure is strict, i.e. (M,g,J)
is not K¨ ahler. It is well-known that M is Einstein in this case. We will always normalize
the scalar curvature of M to scal = 30, in which case we also have |Ψ+|2= 4 point-wise.
The form Ψ+can be seen as the real part of a¯∇-parallel complex volume form Ψ++iΨ−
on M, where Ψ−= ∗Ψ+is the Hodge dual of Ψ+. Thus M carries a SU3structure whose
minimal connection (cf. [6]) is exactly¯∇. Notice that Hitchin has shown that a SU3
structure (ω,Ψ+,Ψ−) is nearly K¨ ahler if and only if the following exterior system holds:
2J(∇XJ)Y,
∀ X ∈ TM, ∀ Y ∈ C∞(M) (2)
?
dω = 3Ψ+
dΨ−= −2ω ∧ ω.
(3)
Let A ∈ Λ1M⊗EndM denote the tensor AX:= J(∇XJ) = −Ψ+
the endomorphism associated to Y ?Ψ+via the metric. Since for every unit vector X,
AXdefines a complex structure on the 4-dimensional space X⊥∩ (JX)⊥, we easily get
in a local orthonormal basis {ei} the formulas
|AX|2= 2|X|2,
AeiAei(X) = −4X,
where here and henceforth, we use Einstein’s summation convention on repeating sub-
scripts. The following algebraic relations are satisfied for every SU3structure (ω,Ψ+)
on TM (notice that we identify vectors and 1-forms via the metric):
JX, where Ψ+
Ydenotes
∀ X ∈ TM.
∀ X ∈ TM,
(4)
(5)
AXei∧ ei?Ψ+= −2X ∧ ω,
X ?Ψ−= −JX ?Ψ+,
(X ?Ψ+) ∧ Ψ+= X ∧ ω2,
(JX ?Ψ+) ∧ ω = X ∧ Ψ+,
∀ X ∈ TM.
∀ X ∈ TM,
∀ X ∈ TM.
∀ X ∈ TM.
(6)
(7)
(8)
(9)
The Hodge operator satisfies ∗2= (−1)pon ΛpM and moreover
∗(X ∧ Ψ+) = JX ?Ψ+,
∗(φ ∧ ω) = −φ,
∗(JX ∧ ω2) = −2X,
∀ X ∈ TM.
∀ φ ∈ Λ(1,1)
∀ X ∈ TM.
(10)
0
M.(11)
(12)
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4 ANDREI MOROIANU AND UWE SEMMELMANN
From now on we assume that (M,g) is compact 6-dimensional not isometric to the
round sphere (S6,can). It is well-known that every Killing vector field ξ on M is an
automorphism of the whole nearly K¨ ahler structure (see [10]). In particular,
Lξω = 0,LξΨ+= 0,LξΨ−= 0. (13)
Let now R and¯R denote the curvature tensors of ∇ and¯∇. Then the formula (c.f. [1])
RWXY Z
=
¯RWXY Z−1
+3
4g(Y, W)g(X, Z) +1
4g(X, Y )g(Z, W)
4g(Y, JW)g(JX, Z) −3
4g(Y, JX)g(JW, Z) −1
2g(X, JW)g(JY, Z)
may be rewritten as
RXY = −X ∧ Y + RCY
XY
and
¯RXY = −3
4(X ∧ Y + JX ∧ JY −2
3ω(X,Y )J) + RCY
XY
where RCY
XYis a curvature tensor of Calabi-Yau type.
We will recall the definition of the curvature endomorphism q(R) (c.f. [10]). Let EM
be the vector bundle associated to the bundle of orthonormal frames via a representation
π : SO(n) → Aut(E). The Levi-Civita connection of M induces a connection on EM,
whose curvature satisfies REM
the differential of π and identify the Lie algebra of SO(n), i.e. the skew-symmetric
endomorphisms, with Λ2. In order to keep notations as simple as possible, we introduce
the notation π∗(A) = A∗. The curvature endomorphism q(R) ∈ End(EM) is defined as
q(R) =1
XY= π∗(RXY) = π∗(R(X ∧ Y )), where we denote with π∗
2(ei∧ ej)∗R(ei∧ ej)∗
(14)
for any local orthonormal frame {ei}. In particular, q(R) = Ric on TM. By the same
formula we may define for any curvature tensor S, or more generally any endomorphism
S of Λ2TM, a bundle morphism q(S). In any point q : R ?→ q(R) defines an equivariant
map from the space of algebraic curvature tensors to the space of endomorphisms of E.
Since a Calabi-Yau algebraic curvature tensor has vanishing Ricci curvature, q(RCY) = 0
holds on TM. Let R0
direct calculation gives
XYbe defined by R0
XY= X ∧Y +JX ∧JY −2
3ω(X,Y )J. Then a
q(R0) =1
2
?
(ei∧ ej)∗(ei∧ ej)∗+1
2
?
(ei∧ ej)∗(Jei∧ Jej)∗−2
3ω∗ω∗.
We apply this formula on TM. The first summand is exactly the SO(n)-Casimir, which
acts as −5id. The third summand is easily seen to be2
acts as −id (c.f. [11]). Altogether we obtain q(R0) = −16
expression for q(¯R) acting on TM:
3id, whereas the second summand
3id, which gives the following
q(¯R)|TM= 4idTM. (15)
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THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS5
3. The Hermitian Laplace operator
In the next two sections (M6,g,J) will be a compact nearly K¨ ahler manifold with
scalar curvature normalized to scalg= 30. We denote as usual by ∆ the Laplace operator
∆ = d∗d+dd∗= ∇∗∇+q(R) on differential forms. We introduce the Hermitian Laplace
operator
¯∆ =¯∇∗¯∇ + q(¯R),
which can be defined on any associated bundle EM. In [11] we have computed the
difference of the operators ∆ and¯∆ on a primitive (1,1)-form φ:
(∆ −¯∆)φ = (Jd∗φ)?Ψ+.
In particular, ∆ and¯∆ coincide on co-closed primitive (1,1)-forms. We now compute
the difference ∆−¯∆ on 1-forms. Using the calculation in [11] (or directly from (15)) we
have q(R) − q(¯R) = id on TM. It remains to compute the operator P = ∇∗∇ −¯∇∗¯∇
on TM. A direct calculation using (5) gives for every 1-form θ
P(θ) = −1
= −θ − Aei∇eiθ.
In order to compute the last term, we introduce the metric adjoint α : Λ2M → TM
of the bundle homomorphism X ∈ TM ?→ X?Ψ+∈ Λ2M. It is easy to check that
α(X?Ψ+) = 2X (c.f. [10]). Keeping in mind that A is totally skew-symmetric, we
compute for an arbitrary vector X ∈ TM
?Aei(∇eiθ),X? = ?AXei,∇eiθ? = ?AX,ei∧ ∇eiθ? = ?AX,dθ?
= −?Ψ+
whence Aei(∇eiθ) = Jα(dθ). Summarizing our calculations we have proved the following
Proposition 3.1. Let (M6,g,J) be a nearly K¨ ahler manifold with scalar curvature
normalized to scalg= 30. Then for any 1-form θ it holds that
(∆ −¯∆)θ = −Jα(dθ).
The next result is a formula for the commutator of J and α ◦ d on 1-forms.
Lemma 3.2. For all 1-forms θ, the following formula holds:
(16)
(17)
4AeiAeiθ − Aei¯∇eiθ = θ − Aei¯∇eiθ = θ +1
2AeiAeiθ − Aei∇eiθ
JX,dθ? = −?JX,α(dθ)? = ?Jα(dθ),X?,
α(dθ) = 4Jθ + Jα(dJθ).
Proof. Differentiating the identity θ∧Ψ+= Jθ∧Ψ−gives dθ∧Ψ+= dJθ∧Ψ−+2Jθ∧ω2.
With respect to the SU3-invariant decomposition Λ2M = Λ(1,1)M ⊕ Λ(2,0)+(0,2)M, we
can write dθ = (dθ)(1,1)+1
wedge product of forms of type (1,1) and (3,0) vanishes we derive the equation
2α(dθ)?Ψ+and dJθ = (dJθ)(1,1)+1
2α(dJθ)?Ψ+. Since the
1
2(α(dθ)?Ψ+) ∧ Ψ+=1
2(α(dJθ)?Ψ+) ∧ Ψ−+ 2Jθ ∧ ω2.
Using (8) and (9) we obtain
1
2α(dθ) ∧ ω2=1
2Jα(dJθ) ∧ ω2+ 2Jθ ∧ ω2.
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6 ANDREI MOROIANU AND UWE SEMMELMANN
Taking the Hodge dual of this equation and using (12) gives Jα(dθ) = −α(dJθ) − 4θ,
which proves the lemma.
?
Finally we note two interesting consequences of Proposition 3.1 and Lemma 3.2.
Corollary 3.3. For any closed 1-form θ it holds that
(∆ −¯∆)θ = 0,
Proof. For a closed 1-form θ Lemma 3.1 directly implies that ∆ and¯∆ coincide on θ.
For the second equation we use Proposition 3.1 together with Lemma 3.2 to conclude
(∆ −¯∆)Jθ = −Jα(dJθ) = 4Jθ − α(dθ) = 4Jθ
since θ is closed. This completes the proof of the corollary.
(∆ −¯∆)Jθ = 4Jθ.
?
4. Special¯∆-eigenforms on nearly K¨ ahler manifolds
In this section we assume moreover that (M,g) is not isometric to the standard sphere
(S6,can). In the first part of this section we will show how to construct¯∆-eigenforms
on M starting from Killing vector fields.
Let ξ be a non-trivial Killing vector field on (M,g), which in particular implies d∗ξ = 0
and ∆ξ = 2Ric(ξ) = 10ξ. As an immediate consequence of the Cartan formula and (13)
we obtain
dJξ = Lξω − ξ?dω = −3ξ?ψ+
so by (4), the square norm of dJξ (as a 2-form) is
(18)
|dJξ|2= 18|ξ|2.(19)
In [9] we showed already that the vector field Jξ is co-closed if ξ is a Killing vector field
and has unit length. However it turns out that this also holds more generally.
Proposition 4.1. Let ξ be a Killing vector field on M. Then d∗Jξ = 0.
Proof. Let dv denote the volume form of (M,g). We start with computing the L2-norm
of d∗Jξ.
?d∗Jξ?2
L2
=
?
?
?
M?d∗Jξ,d∗Jξ?dv =?
M[|∇Jξ|2+ 5|ξ|2− |dJξ|2]dv =?
M[?∆Jξ,Jξ? − ?d∗dJξ,Jξ?]dv
=
M[?∇∗∇Jξ,Jξ? + 5|Jξ|2− |dJξ|2]dv
=
M[|∇Jξ|2− 13|ξ|2]dv
Here we used the well-known Bochner formula for 1-forms, i.e. ∆θ = ∇∗∇θ + Ric(θ),
with Ric(θ) = 5θ in our case. Next we consider the decomposition of ∇Jξ into its
symmetric and skew-symmetric parts 2∇Jξ = dJξ + LJξg, which together with (19)
leads to
|∇Jξ|2=1
4(|dJξ|2+ |LJξg|2) = 9|ξ|2+1
4|LJξg|2. (20)
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THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS7
(Recall that the endomorphism square norm of a 2-form is twice its square norm as a
form). In order to compute the last norm, we express LJξg as follows:
LJξg(X,Y ) = g(∇XJξ,Y ) + g(X,∇YJξ)
= g(J∇Xξ,Y ) + g(X,J∇Yξ) + Ψ+(X,ξ,Y ) + Ψ+(Y,ξ,X)
= −g(∇Xξ,JY ) − g(JX,∇Yξ) = −dξ(1,1)(X,JY ),
whence
?LJξg?2
On the other hand, as an application of Lemma 3.2 together with Equation (18) we get
α(dξ) = 4Jξ + Jα(dJξ) = −2Jξ, so
dξ(2,0)= −Jξ ?Ψ+.
Moreover, ∆ξ = 10ξ since ξ is a Killing vector field, which yields
L2 = 2?dξ(1,1)?2
L2. (21)
(22)
?dξ(1,1)?2
L2 = ?dξ?2
L2 − ?dξ(2,0)?2
L2 = 10?ξ?2
L2 − 2?ξ?2
L2 = 8?ξ?2
L2= 13?ξ?2
L2.
This last equation, together with (20) and (21) gives ?∇Jξ?2
this into the first equation proves that d∗Jξ has vanishing L2-norm and thus that Jξ is
co-closed.
L2. Substituting
?
Proposition 4.2. Let ξ be a Killing vector field on M. Then
∆ξ = 10ξ, and∆Jξ = 18Jξ.
In particular, Jξ can never be a Killing vector field.
Proof. The first equation holds for every Killing vector field on an Einstein manifold
with Ric = 5id. From (18) we know dJξ = −3ξ?Ψ+. Hence the second assertion follows
from:
d∗dJξ = − ∗ d ∗ dJξ
(10)
= −3 ∗ d(Jξ ∧ Ψ+) = 9 ∗ (ξ ∧ ω2)
(12)
= 18Jξ.
?
Since the differential d commutes with the Laplace operator ∆, every Killing vector
field ξ defines two ∆-eigenforms of degree 2:
∆dJξ = 18dJξand∆dξ = 10dξ
As a direct consequence of Proposition 4.2, together with formulas (18), (22), and
Proposition 3.1 we get:
Corollary 4.3. Every Killing vector field on M satisfies
¯∆ξ = 12ξ,
¯∆Jξ = 12Jξ.
Our next goal is to show that the (1,1)-part of dξ is a¯∆ -eigenform. By (22) we have
dξ = φ − Jξ?Ψ+, (23)
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8 ANDREI MOROIANU AND UWE SEMMELMANN
for some (1,1)-form φ. Using Proposition 4.1, we can write in a local orthonormal basis
{ei}:
?dξ,ω? =1
thus showing that φ is primitive. The differential of φ can be computed from the Cartan
formula:
= −d(ξ?Ψ−) = −LξΨ−+ξ?dΨ−(13)
From here we obtain
∗dφ = −4 ∗ (Jξ ∧ ω) = 4ξ ∧ ω,
whence
4dξ ∧ ω − 12ξ ∧ Ψ+(23)
(9)
=4φ ∧ ω − 16ξ ∧ Ψ+.
Using (10) and (11), we thus get
d∗dφ = − ∗ d ∗ dφ = 4φ + 16Jξ?Ψ+.
On the other hand,
2?dξ,ei∧ Jei? = ?∇eiξ,Jei? = d∗Jξ = 0,
dφ
(23)
= d(Jξ?Ψ++dξ)
(7)
= −2ξ?ω2= −4Jξ ∧ω. (24)
d ∗ dφ = = 4φ ∧ ω − 4(Jξ?Ψ+) ∧ ω − 12ξ ∧ Ψ+
d∗φ = − ∗ d ∗ φ
(11)
= ∗d(φ ∧ ω)
(24)
= X(−4Jξ ∧ ω2+ 3φ ∧ Ψ+)
(12)
= 8ξ
and finally
dd∗φ = 8dξ = 8φ − 8Jξ?Ψ+.
The calculations above thus prove the following proposition
Proposition 4.4. Let (M6,g,J) be a compact nearly K¨ ahler manifold with scalar cur-
vature scalg= 30, not isometric to the standard sphere. Let ξ be a Killing vector field on
M and let φ be the (1,1)-part of dξ. Then φ is primitive, i.e. φ = (dξ)(1,1)
d∗φ = 8ξ and ∆φ = 12φ + 8Jξ?Ψ+.
0
. Moreover
Corollary 4.5. The primitive (1,1)-form ϕ satisfies
¯∆φ = 12φ.
Proof. From (17) and the proposition above we get
¯∆φ = ∆φ − (∆ −¯∆)φ = 12φ + 8Jξ?Ψ+− (Jd∗φ)?Ψ+= 12φ.
?
In the second part of this section we will present another way of obtaining primitive
¯∆-eigenforms of type (1,1), starting from eigenfunctions of the Laplace operator. Let
f be such an eigenfunction, i.e.∆f = λf. We consider the primitive (1,1)-form
η := (dJdf)(1,1)
0
.
Lemma 4.6. The form η is explicitly given by
η = dJdf + 2df?Ψ++λ
3fω.
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THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS9
Proof. According to the decomposition of Λ2M into irreducible SU3-summands, we can
write
dJdf = η + γ?Ψ++ hω
for some vector field γ and function h. From Lemma 3.2 we get 2γ = α(dJdf) = −4df.
In order to compute h, we write
6hdv = hω ∧ ω2= dJdf ∧ ω2= d(Jdf ∧ ω2)
(12)
= 2d ∗ df = 2λf dv.
?
We will now compute the Laplacian of the three summands of η separately. First,
we have ∆df = λdf and Corollary 3.3 yields¯∆df = λdf. Since¯∆ commutes with J, we
also have¯∆Jdf = λJdf and from the second equation in Corollary 3.3 we obtain
∆Jdf =¯∆Jdf + (∆ −¯∆)Jdf = (λ + 4)Jdf.
Hence, dJdf is a ∆-eigenform for the eigenvalue λ + 4.
Lemma 4.7. The co-differential of the (1,1)-form η is given by
d∗η =?2λ
3− 4?Jdf.
Proof. Notice that d∗(fω) = −df?ω and that d∗Jdf = −∗d∗Jdf = −1
since dω2= 0. Using this we obtain
2∗d(df ∧ω2) = 0,
d∗η = ∆Jdf + 2d∗(df?Ψ+) −λ
= (λ + 4 −λ
3df?ω = (λ + 4)Jdf − 2 ∗ d(df ∧ Ψ−) −λ
3)Jdf − 4 ∗ (df ∧ ω2)
3Jdf
(12)
= (2λ
3− 4)Jdf.
?
In order to compute ∆ of the second summand of η we need three additional formulas
Lemma 4.8.
¯∆(X?Ψ+) = (¯∆X)?Ψ+.
Proof. Recall that¯∆ =¯∇∗¯∇ + q(¯R). Since Ψ+is¯∇-parallel we immediately obtain
¯∇∗¯∇(X?Ψ+) = −¯∇ei¯∇ei(X?Ψ+) = −(¯∇ei¯∇eiX)?Ψ+.
The map A ?→ A∗Ψ+is a SU3-equivariant map from Λ2to Λ3. But since Λ3does not
contain the representation Λ(1,1)
0
as an irreducible summand, it follows that A∗Ψ+= 0
for any skew-symmetric endomorphism A corresponding to some primitive (1,1)-form.
Hence we conclude
q(¯R)(X?Ψ+) = ωi∗¯R(ωi)∗(X?Ψ+) = (ωi∗¯R(ωi)∗X)?Ψ+= (q(¯R)X)?Ψ+,
where, since the holonomy of¯∇ is included in SU3, the sum goes over some orthonormal
basis {ωi} of Λ(1,1)
0
M. Combining these two formulas we obtain¯∆(X?Ψ+) = (¯∆X)?Ψ+.
?
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10ANDREI MOROIANU AND UWE SEMMELMANN
Lemma 4.9.
(∆ −¯∆)(df?Ψ+) = 6(df?Ψ+) −4λ
3fω − 2η.
Proof. From Proposition 3.4 in [11] we have
(∆ −¯∆)(df?Ψ+) = (∇∗∇ −¯∇∗¯∇)(df?Ψ+) + (q(R) − q(¯R))(df?Ψ+)
= (∇∗∇ −¯∇∗¯∇)(df?Ψ+) + 4df?Ψ+.
The first part of the right hand side reads
(∇∗∇ −¯∇∗¯∇)(df?Ψ+) = −1
From (5) we get
4Aei∗Aei∗df?Ψ+− Aei∗¯∇ei(df?Ψ+). (25)
Aei∗Aei∗df?Ψ+
= Aei∗(Aeiek∧ Ψ+(df,ek,·))
= AeiAeiek∧ Ψ+(df,ek,·) + Aeiek∧ AeiΨ+(df,ek,·)
= −4ek∧ ek?Ψ+
where we used the vanishing of the expression E = Aeiek∧ AeiejΨ+(df,ek,ej):
E = AJeiek∧ AJeiejΨ+(df,ek,ej) = AeiJek∧ AeiJejΨ+(df,ek,ej)
= Aeiek∧ AeiejΨ+(df,Jek,Jej) = −E.
It remains to compute the second term in (25). We notice that by Schur’s Lemma, every
SU3-equivariant map from the space of symmetric tensors Sym2M to TM vanishes, so
in particular (since ∇df is symmetric), one has Aei∇eidf = 0. We then compute
Aei∗¯∇eiΨ+
(6)
=(Aei∇eidf)?Ψ+−1
= 2Ψ+
2Ψ+
2Ψ+
df+ Aeiek∧ AeiejΨ+(df,ek,ej) = −8Ψ+
df,
df
=Aei∗((¯∇eidf)?Ψ+) = (Aei¯∇eidf)?Ψ++ (¯∇eidf)?Aei∗Ψ+
2(AeiAeidf)?Ψ+− 2(¯∇eidf)?(ei∧ ω)
df+ 2d∗dfω + ?Aeidf,ei?ω + 2ei∧ J¯∇eidf
df+ 2λfω + 2ei∧¯∇eiJdf = 2Ψ+
df+ 2λfω + 2dJdf + 2AJdf= 4Ψ+
=
df+ 2λfω + 2dJdf − ei∧ AeiJdf
df+ 2λfω + 2dJdf.
=
Plugging back what we obtained into (25) yields
(∇∗∇ −¯∇∗¯∇)(df?Ψ+) = −(2Ψ+
df+ 2λfω + 2dJdf),
which together with Lemma 4.6 and the first equation prove the desired formula.
?
Lemma 4.10.
∆fω = (λ + 12)fω − 2(df?Ψ+).
Proof. Since d∗(fω) = −df?ω = −Jdf we have dd∗(fω) = −dJdf. For the second
summand of ∆(fω) we first compute d(fω) = df∧ω+3fΨ+. Since d∗Ψ+=1
3d∗dω = 4ω,
Page 11
THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS11
we get d∗fΨ+= −df?Ψ++ fd∗Ψ+= −df?Ψ++ 4fω. Moreover
d∗(df ∧ ω) = − ∗ d(Jdf ∧ ω) = − ∗ (dJdf ∧ ω − 3Jdf ∧ Ψ+)
= − ∗ ([η − 2df?Ψ+−λ
= η + 2 ∗ ((df?Ψ+) ∧ ω) +2λ
= η + 2df ?Ψ++2λ
3fω] ∧ ω) + 3 ∗ (Jdf ∧ Ψ+)
3fω − 3df?Ψ+
3fω − 3df?Ψ+.
3fω, we obtain
3fω = (λ + 12)fω − 2df?Ψ+.
Recalling that η = dJdf + 2df?Ψ++λ
∆fω = −dJdf − 3df?Ψ++ 12fω + η − df?Ψ++2λ
?
Applying these three lemmas we conclude
∆(df?Ψ+) =¯∆(df?Ψ+) + (∆ −¯∆)(df?Ψ+) = (λ + 6)(df?Ψ+) −4λ
and thus
∆η = (λ + 4)dJdf + (2λ + 12)(df?Ψ+) −8λ
= λη +?4 −2λ
Finally we have once again to apply the formula for the difference of ∆ and¯∆ on
primitive (1,1)-forms. We obtain
¯∆η = ∆η − Jd∗η?Ψ+= ∆η +?2λ
Summarizing our calculations we obtain the following result.
Proposition 4.11. Let f be an ∆-eigenfunction with ∆f = λf Then the primitive
(1,1)-form η := (dJdf)(1,1)
0
satisfies
¯∆η = ληand
3fω − 2η
3fω − 4η +λ
3(λ + 12)fω −2λ
3(df?Ψ+)
3
?(df?Ψ+).
3− 4?(df?Ψ+) = λη.
d∗η =?2λ
3− 4?Jdf.
Let Ω0(12) ⊂ C∞(M) be the¯∆-eigenspace for the eigenvalue 12 (notice that¯∆ = ∆
on functions) and let Ω(1,1)
0
(12) denote the space of primitive (1,1)-eigenforms of¯∆
corresponding to the eigenvalue 12. Summarizing Corollary 4.5 and Proposition 4.11,
we have constructed a linear mapping
Φ : i(M) → Ω(1,1)
from the space of Killing vector fields into Ω(1,1)
Ψ : Ω0(12) → Ω(1,1)
Let moreover NK ⊂ Ω(1,1)
by [10] is just the space of co-closed forms in Ω(1,1)
0
(12), Φ(ξ) := dξ(1,1)
0
0
(12) and a linear mapping
Ψ(f) := (dJdf)(1,1)
0
(12),
0
.
0
(12) denote the space of nearly K¨ ahler deformations, which
(12).
0
Proposition 4.12. The linear mappings Φ and Ψ defined above are injective and the
sum Im(Φ) + Im(Ψ) + NK ⊂ Ω(1,1)
dim(NK) ≤ dim(Ω(1,1)
0
(12) is a direct sum. In particular,
0
(12)) − dim(i(M)) − dim(Ω0(12)). (26)
Page 12
12 ANDREI MOROIANU AND UWE SEMMELMANN
Proof. It is enough to show that if ξ ∈ i(M), f ∈ Ω0(12) and α ∈ NK satisfy
dξ(1,1)
0
+ (dJdf)(1,1)
then ξ = 0 and f = 0. We apply d∗to (27). Using Propositions 4.4 and 4.11 to express
the co-differentials of the first two terms we get
0
+ α = 0, (27)
8ξ + 8Jdf = 0.(28)
Since Jξ is co-closed (Proposition 4.1), formula (28) implies 0 = d∗Jξ = d∗df = 12f,
i.e. f = 0. Plugging back into (28) yields ξ = 0 too.
?
5. The homogeneous Laplace operator on reductive homogeneous
spaces
5.1. The Peter-Weyl formalism. Let M = G/K be a homogeneous space with com-
pact Lie groups K ⊂ G and let π : K → Aut(E) be a representation of K. We denote
by EM := G ×πE be the associated vector bundle over M. The Peter-Weyl theorem
and the Frobenius reciprocity yield the following isomorphism of G-representations:
L2(EM)∼=
γ∈ˆG
whereˆG is the set of (non-isomorphic) irreducible G-representations. If not otherwise
stated we will consider only complex representations. Recall that the space of smooth
sections C∞(EM) can be identified with the space C∞(G;E)Kof K-invariant E-valued
functions, i.e. functions f : G → E with f(gk) = π(k)−1f(g). This space is made
into a G-representation by the left-regular representation ℓ, defined by by (ℓ(g)f)(a) =
f(g−1a). Let v ∈ Vγ and A ∈ HomK(Vγ,E) then the invariant E-valued function
corresponding to v ⊗A is defined by g ?→ A(g−1v). In particular, each summand in the
Hilbert space direct sum (29) is a subset of C∞(EM) ⊂ L2(EM).
Let g be the Lie algebra of G. We denote by B the Killing form of g, B(X,Y ) :=
tr(adX◦adY). The Killing form is non-degenerated and negative definite if G is compact
and semi-simple, which will be the case in all examples below.
?
Vγ⊗ HomK(Vγ,E),(29)
If π : G → Aut(E) is a G-representation, the Casimir operator of (G,π) acts on E
by the formula
CasG
?
where {Xi} is a (−B)-orthonormal basis of g and π∗: g → End(E) denotes the differ-
ential of the representation π.
π=(π∗Xi)2, (30)
Remark 5.1. Notice that the Casimir operator is divided by k if one use the scalar
product −kB instead of −B.
If G is simple, the adjoint representation ad on the complexification gCis irreducible,
so, by Schur’s Lemma, its Casimir operator acts as a scalar. Taking the trace in (30)
for π = ad yields the useful formula CasG
ad= −1.
Page 13
THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS13
Let Vγ be an irreducible G-representation of highest weight γ. By Freudenthal’s
formula the Casimir operator acts on Vγby scalar multiplication with ?ρ?2− ?ρ+ γ?2,
where ρ denotes the half-sum of the positive roots and ?·? is the norm induced by −B
on the dual of the Lie algebra of the maximal torus of G. Notice that these scalars are
always non-positive. Indeed ?ρ?2− ?ρ + γ?2= −?γ,γ + 2ρ?Band ?γ,ρ? ≥ 0, since γ
is a dominant weight, i.e. it is in the the closure of the fixed Weyl chamber, whereas
ρ is the half-sum of positive weights and thus by definition has a non-negative scalar
product with γ.
5.2. The homogeneous Laplace operator. We denote by¯∇ the canonical homoge-
neous connection on M = G/K. It coincides with the Levi-Civita connection only in
the case that G/K is a symmetric space. A crucial observation is that the canonical
homogeneous connection coincides with the canonical Hermitian connection on nat-
urally reductive 3-symmetric spaces (see below). We define the curvature endomor-
phism q(¯R) ∈ End(EM) as in (14) and introduce as in (16) the second order operator
¯∆π=¯∇∗¯∇ + q(¯R) acting on sections of the associated bundle EM := G ×πE.
Lemma 5.2. Let G be a compact semi-simple Lie group, K ⊂ G a compact subgroup,
and let M = G/K the naturally reductive homogeneous space equipped with the Rie-
mannian metric induced by −B. For every K-representation π on E, let EM := G×πE
be the associated vector bundle over M. Then the endomorphism q(¯R) acts fibre-wise
on EM as q(¯R) = −CasK
sections of EM, considered as G-representation via the left-regular representation, as
¯∆ = −CasG
Proof. Consider the Ad(K)-invariant decomposition g = k⊕p. For any vector X ∈ g we
write X = Xk+Xp, with Xk∈ k and Xp∈ p. The canonical homogeneous connection is
the left-invariant connection in the principal K-fibre bundle G → G/K corresponding
to the projection X ?→ Xk. It follows that one can do for the canonical homogeneous
connection on G/K the same identifications as for the Levi Civita connection on Rie-
mannian symmetric spaces.
π. Moreover the differential operator¯∆ acts on the space of
ℓ.
In particular, the covariant derivative of a section φ ∈ Γ(EM) with respect to
some X ∈ p translates into the derivative X(ˆφ) of the the corresponding function
ˆφ ∈ C∞(G;E)K, which is minus the differential of the left-regular representation X(ˆφ) =
−ℓ∗(X)ˆφ. Hence, if {eµ} is an orthonormal basis in p, the rough Laplacian¯∇∗¯∇ trans-
lates into the sum −ℓ∗(eµ)ℓ∗(eµ) = (−CasG
to show that q(¯R) = −CasK
We claim that the differential i∗ : k → so(p)∼= Λ2p of the isotropy representation
i : K → SO(p) is given by i∗(A) = −1
(1
Next we recall that for X,Y ∈ p the curvature¯RX,Y of the canonical connection acts by
−π∗([X,Y ]k) on every associated vector bundle EM, defined by the representation π.
ℓ+CasK
ℓ). Since¯∆ =¯∇∗¯∇+q(¯R) it remains
πin order to complete the proof of the lemma.
ℓ= −CasK
2eµ∧ [A,eµ] for any A ∈ k. Indeed
2B(eµ,X)[A,eµ] +1
2eµ∧ [A,eµ])∗X = −1
2B([A,eµ],X)eµ= −[A,X].
Page 14
14ANDREI MOROIANU AND UWE SEMMELMANN
Hence the curvature operator¯R can be written for any X,Y ∈ p as
¯R(X ∧ Y ) =1
Let PSO(p)= G ×iSO(p) be the bundle of orthonormal frames of M = G/K. Then any
SO(p)-representation ˜ π defines a K-representation by π = ˜ π ◦ i. Moreover any vector
bundle EM associated to PSO(p)via ˜ π can be written as a vector bundle associated via
π to the K-principle bundle G → G/K, i.e.
EM = PSO(p)×˜ πE = G ×πE
Let {fα} be an orthonormal basis of k. Then by the definition of q(¯R) we have
q(¯R) =
2eµ∧¯RX,Yeµ= −1
2eµ∧ [[X,Y ]k,eµ] = i∗([X,Y ]k).
1
2˜ π∗(eµ∧ eν) ˜ π∗(¯R(eµ∧ eν)) =1
= −1
=
2˜ π∗(eµ∧ eν)π∗([eµ,eν]k)
2B(eν,[fα,eµ])˜ π∗(eµ∧ eν)π∗(fα)
2B([eµ,eν],fα)˜ π∗(eµ∧ eν)π∗(fα) = −1
2˜ π∗(eµ∧ [fα,eµ])π∗(fα) = −π∗(fα)π∗(fα)
= −CasK
We have shown that q(¯R) ∈ End(EM) acts fibre-wise as −CasK
f ∈ C∞(G;E)K, then the K-invariance of f implies π∗(Z)f = −Z(f) = ℓ∗(Z)f and
also CasK
1
π.
π. Let Z ∈ k and
π= CasK
ℓ, which concludes the proof of the lemma.
?
It follows from this lemma that the spectrum of¯∆ on sections of EM is the set
of numbers λγ = ?ρ + γ?2− ?ρ?2, where γ is the highest weight of an irreducible G-
representation Vγsuch that HomK(Vγ,E) ?= 0, i.e. such that the decomposition of Vγ,
considered as K-representation, contains components of the K-representation E.
5.3. Nearly K¨ ahler deformations and Laplace eigenvalues. Let (M,g,J) be a
compact simply connected 6-dimensional nearly K¨ ahler manifold not isometric to the
round sphere, with scalar curvature normalized to scalg= 30. Recall the following result
from [10]:
Theorem 5.3. The Laplace operator ∆ coincides with the Hermitian Laplace operator
¯∆ on co-closed primitive (1,1)-forms. The space NK of infinitesimal deformations of
the nearly K¨ ahler structure of M is isomorphic to the eigenspace for the eigenvalue 12
of the restriction of ∆ (or¯∆) to the space of co-closed primitive (1,1)-forms on M.
Assume from now on that M is a 6-dimensional naturally reductive 3-symmetric space
G/K in the list of Gray and Wolf, i.e. SU2× SU2× SU2/SU2, SO5/U2or SU3/T2. As
was noticed before, the canonical homogeneous and the canonical Hermitian connection
coincide, since for the later can be shown that is torsion and its curvature are parallel,
a property, which by the Ambrose-Singer-Theorem characterizes the canonical homoge-
neous connection (c.f. [5]). In order to determine the space NK on M we thus need to
apply the previous calculations to compute the¯∆-eigenspace for the eigenvalue 12 on
primitive (1,1)-forms and decide which of these eigenforms are co-closed.
Page 15
THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS 15
According to Lemma 5.2 and the decomposition (29) we have to carry out three
steps: first to determine the K-representation Λ1,1
to compute the Casimir eigenvalues with the Freudenthal formula, which gives all pos-
sible¯∆-eigenvalues and finally to check whether the G-representation Vγrealizing the
eigenvalue 12 satisfies HomK(Vγ,Λ1,1
0p defining the bundle Λ1,1
0TM, then
0p) ?= {0} and thus really appears as eigenspace.
Before going on, we make the following useful observation
Lemma 5.4. Let (G/K,g) be a 6-dimensional homogeneous strict nearly K¨ ahler man-
ifold of scalar curvature scalg= 30. Then the homogeneous metric g is induced from
−1
Proof. Let G/K be a 6-dimensional homogeneous strict nearly K¨ ahler manifold. Then
the metric is induced from a multiple of the Killing form, i.e. G/K is a normal homo-
geneous space with Ad(K)-invariant decomposition g = k ⊕ p. The scalar curvature of
the metric h induced by −B may be computed as (c.f. [3])
scalh=3
12B, where B is the Killing form of G.
2− 3CasK
λ
where λ : K → so(p) is the isotropy representation. From Lemma 5.2 we know that
CasK
2scalh
15
id. Hence we obtain the equation scalh=3
the metric g corresponding to −1
5.4. The¯∆-spectrum on S3× S3. Let K = SU2 with Lie algebra k = su2 and
G = K ×K ×K with Lie algebra g = k⊕k⊕k. We consider the 6-dimensional manifold
M = G/K, where K is diagonally embedded. The tangent space at o = eK can be
identified with
p = {(X,Y,Z) ∈ k ⊕ k ⊕ k|X + Y + Z = 0}.
Let B be the Killing form of k and define B0= −1
that the invariant scalar product
λ= −q(¯R), which on the tangent bundle was computed in Lemma 15 as q(¯R) =
2+2
5scalhand it follows scalh=5
2, i.e.
12B has scalar curvature scalg= 30.
?
12B. Then it follows from Lemma 5.4
B0((X,Y,Z),(X,Y,Z)) = B0(X,X) + B0(Y,Y ) + B0(Z,Z)
defines a normal metric, which is the homogeneous nearly K¨ ahler metric g of scalar
curvature scalg= 30.
The canonical almost complex structure on the 3-symmetric space M, corresponding
to the 3rd order G-automorphism σ, with σ(k1,k2,k3) = (k2,k3,k1), is defined as
J(X,Y,Z) =
2
√3(Z,X,Y ) +
1
√3(X,Y,Z).
The (1,0)-subspace p1,0of pCdefined by J is isomorphic to the complexified adjoint
representation of SU2on suC
(notice that E∼=¯E because every SU2∼= Sp1representation is quaternionic).
Lemma 5.5. The SU2-representation defining the bundle Λ(1,1)
ducible summands Sym4E and Sym2E.
2. Let E = C2denote the standard representation of SU2
0
TM splits into the irre-
Page 16
16ANDREI MOROIANU AND UWE SEMMELMANN
Proof. The defining SU2-representation of Λ(1,1)TM is p1,0⊗p0,1 ∼= Sym2E ⊗Sym2E∼=
Sym4E ⊕ Sym2E ⊕ Sym0E from the Clebsch-Gordan formula. Since we are interested
in primitive (1,1)-forms, we still have to delete the trivial summand Sym0E∼= C.
?
Since G = SU2× SU2× SU2, every irreducible G-representation is isomorphic to one
of the representations Va,b,c= SymaE ⊗ SymbE ⊗ SymcE. The Casimir operator of the
SU2-representation SymkE (with respect to B) is −1
of G is the sum of the three SU2-Casimir operators. Hence all possible¯∆-eigenvalues
with respect to the metric B0are of the form
8k(k +2) and the Casimir operator
3
2(a(a + 2) + b(b + 2) + c(c + 2)). (31)
for non-negative integers a,b,c. It is easy to check that the eigenvalue 12 is obtained
only for (a,b,c) equal to (2,0,0), (0,2,0) or (0,0,2). The restrictions to SU2(diagonally
embedded in G) of the three corresponding G-representations are all equal to the SU2-
representation Sym2E, thus dimHomSU2(V2,0,0,Λ(1,1)
other summands. Hence the eigenspace of¯∆ on primitive (1,1)-forms for the eigenvalue
12 is isomorphic to V2,0,0⊕ V0,2,0⊕ V0,0,2and its dimension, i.e. the multiplicity of the
eigenvalue 12, is equal to 9.
0
p) = 1, and similarly for the two
Since the isometry group of the nearly K¨ ahler manifold M = SU2× SU2× SU2/SU2
has dimension 9, the inequality (26) yields
dim(NK) ≤ dim(Ω(1,1)
We thus have obtained the following
0
(12)) − dim(i(M)) − dim(Ω0(12)) = −dim(Ω0(12)) ≤ 0.
Theorem 5.6. The homogeneous nearly K¨ ahler structure on S3× S3does not admit
any infinitesimal nearly K¨ ahler deformations.
Finally we remark that there are also no infinitesimal Einstein deformations neither.
In [11] we showed that the space of infinitesimal Einstein deformations of a nearly K¨ ahler
metric g, with normalized scalar curvature scalg= 30, is isomorphic to the direct sum
of¯∆-eigenspaces of primitive co-closed (1,1)-forms for the eigenvalues 2,6 and 12. It is
clear from (31) that neither 2 nor 6 can be realized as¯∆-eigenvalues.
Corollary 5.7. The homogeneous nearly K¨ ahler metric on S3×S3does not admit any
infinitesimal Einstein deformations.
5.5. The¯∆-spectrum on CP3. In this section we consider the complex projective
space CP3= SO5/U2, where U2is embedded by U2⊂ SO4⊂ SO5. Let G = SO5with
Lie algebra g and K = U2with Lie algebra k. We denote the Killing form of G with
B. Then we have the B-orthogonal decomposition g = k ⊕ p, where p can be identified
with the tangent space in o = eK. The space p splits as p = m ⊕ n, where m resp.
n can be identified with the horizontal resp. vertical tangent space at o of the twistor
space fibration SO5/U2→ SO5/SO4= S4. We know from Lemma 5.4 that B0= −1
defines the homogeneous nearly K¨ ahler metric g of scalar curvature scalg= 30.
12B
Page 17
THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS 17
Let {ε1,ε2} denote the canonical basis of R2. Then the positive roots of SO5 are
α1= ε1, α2= ε2, α3= ε1+ ε2, α4= ε1− ε2, with ρ =3
root space corresponding to the root α. Then
mC= gα1⊕ g−α1⊕ gα2⊕ g−α2,
The invariant almost complex structure J may be defined by specifying the (1,0)-
subspace p1,0of pC:
p1,0= {X − iJX | X ∈ p} = gα1⊕ gα2⊕ g−α3,
It follows that J is not integrable, since the restricted root system {α1,α2,−α3} is
not closed under addition (cf. [4]). We note that replacing −α3by α3yields an inte-
grable almost complex structure. This corresponds to the well-known fact that on the
twistor space the non integrable almost complex structure J is transformed into the the
integrable one by replacing J with −J on the vertical tangent space.
Let Ck denote the U1-representation on C defined by (z,v) ?→ zkv, for v ∈ C and
z ∈ U1∼= C∗. Then, since U2= (SU2× U1)/Z2, any irreducible U2-representation is
of the form Ea,b= SymaE ⊗ Cb, with a ∈ N, b ∈ Z and a ≡ b mod 2. As usual let
E = C2denote the standard representation of SU2.
With this notation we obtain the following decomposition of p1,0considered as U2-
representation
p1,0∼= E0,−2⊕ E1,1
Since p0,1is obtained from p1,0by conjugation we have p0,1 ∼= E0,2⊕ E1,−1. The defin-
ing U2-representation of Λ(1,1)TM is p1,0⊗ p0,1, which obviously decomposes into 5
irreducible summands, among which, two are isomorphic to the trivial representation
E0,0. Considering only primitive (1,1)-forms we still have to delete one of the trivial
summands and obtain
Lemma 5.8. The U2-representation defining the bundle Λ(1,1)
decomposition into irreducible summands
Λ(1,1)
0
p = E0,0⊕ E1,3⊕ E1,−3⊕ E2,0.
Let Va,bbe an irreducible SO5-representation of highest weight γ = (a,b) with a,b ∈ N
and a ≥ b ≥ 0, e.g. V1,0= Λ1and V1,1= Λ2. The scalar product induced by the Killing
form B on the dual t∗ ∼= R2of the maximal torus of SO5is −1
scalar product. By the Freudenthal formula we thus get
CasVa,b= ?γ,γ + 2ρ?B= −1
Notice that we have V1,1= soC
It follows (c.f. Remark 5.1) that all possible¯∆-eigenvalues with respect to the metric
induced by B0are of the form 2(a(a + 3) + b(b + 1)). The eigenvalue 12 is realized if
and only if (a,b) = (1,1). We still have to decide whether the SO5-representation V1,1
actually appears in the decomposition (29) of L2(Λ1,1
2ε1+1
2ε2. Let gα⊂ gCbe the
nC= gα3⊕ g−α3.
withE0,−2∼= g−α3
andE1,1∼= gα1⊕ gα2. (32)
0
TM has the following
6times the Euclidean
6(a(a + 3) + b(b + 1)). (33)
5and CasV1,1= −1, which is consistent with CasG
ad= −1.
0TM). However this follows from
Page 18
18ANDREI MOROIANU AND UWE SEMMELMANN
Lemma 5.9. The SO5-representation V1,1 restricted to U2 ⊂ SO5 has the following
decomposition as U2-representation:
V1,1∼= (E0,0⊕ E2,0) ⊕ (E0,−2⊕ E1,1⊕ E0,2⊕ E1,−1)
and in particular
dimHomU2(V1,1,Λ1,1
and
0pC) = 2 dimHomU2(V1,1,C) = 1.
Proof. We know already that V1,1= soC
that soC
in (32). Hence it remains to explicit the adjoint representation of U2on uC
that its restriction to U1acts trivially, whereas its restriction to SU2decomposes into
C ⊕ suC
The eigenspace of¯∆ on primitive (1,1)-forms for the eigenvalue 12 is thus isomorphic
to the sum of two copies of V1,1, i.e. the eigenvalue 12 has multiplicity 2 · 10 = 20.
It is now easy to calculate the smallest eigenvalue and the corresponding eigenspace
of the Laplace operator ∆ on non-constant functions. We do this for¯∆, which coincides
with ∆ on functions. Then we have to replace Λ(1,1)
the trivial representation C and to look for SO5-representations Va,bcontaining the zero
weight. It follows from Lemma 5.9 and (33) that the ∆-eigenspace on functions Ω0(12)
is isomorphic to V1,1and is thus 10-dimensional. Since the dimension of the isometry
group of the nearly K¨ ahler manifold SO(5)/U2is 10, the inequality (26) shows that
dim(NK) ≤ dim(Λ(1,1)
so there are no infinitesimal nearly K¨ ahler deformations in this case neither.
5is the complexified adjoint representation and
5= uC
2⊕(p1,0⊕p0,1). The decomposition of the last two summands is contained
2. It is clear
2, i.e. uC
2∼= E0,0⊕ E2,0.
?
0
p in the calculations above with
0
(12)) − dim(i(M)) − dim(Λ0(12)) = 20 − 10 − 10 = 0,
Finally, we remark like before that there are also no other infinitesimal Einstein
deformations, since by (33), the eigenvalues 2 and 6 do not occur in the spectrum of¯∆
on Λ(1,1)
0
M. Summarizing, we have obtained the following:
Theorem 5.10. The homogeneous nearly K¨ ahler structure on CP3= SO5/U2does not
admit any infinitesimal nearly K¨ ahler or Einstein deformations.
5.6. The¯∆-spectrum on the flag manifold F(1,2). In this section we consider the
flag manifold M = SU3/T2, where T2⊂ SU3is the maximal torus. Let g = su3and let
k = t, the Lie algebra of T2. We have the decomposition
g = k ⊕ p
and
p = m ⊕ n.
Denoting by Eij, Sij are ”real and imaginary” part of the projection of the vector
Xij∈ gl3(equal to 1 on ith row and jth column and 0 elsewhere) onto su3:
Eij= Xij− Xji
the subspaces m and n are explicitly given by
Sij= i(Xij+ Xji),
m = span{E12,S12,E13,S13} = span{e1,e2,e3,e4},
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THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS 19
n = span{E23,S23} = span{e5,e6}.
The dual of the Lie algebra t of the maximal torus T2can be identified with
t∗∼= {(λ1,λ2,λ3) ∈ R3| λ1+ λ2+ λ3= 0}.
If {εi} denotes the canonical basis in R3then the set of positive roots is given as
φ+= {αij= εi−εj| 1 ≤ i < j ≤ 3} and the half-sum of the positive roots is ρ = ε1−ε3
Let B denote the Killing form of SU3. By Lemma 5.4, B0 = −1
homogeneous nearly K¨ ahler metric g of scalar curvature scalg= 30.
12B defines the
The almost complex structure J is explicitly defined on p by
J(e1) = e2,J(e3) = −e4,J(e5) = e6.
Alternatively we may define the (1,0)-subspace of pC:
p1,0= gα12⊕ gα31⊕ gα23= span{X12,X31,X23},
where gαis the root space for α. It follows that J is not integrable, since the restricted
root system {α12,α31,α23} is not closed under addition (c.f. [4]).
Let E = C3be the standard representation of SU3with conjugate representation¯E.
Any irreducible representations of SU3is isomorphic to one of the representations
Vk,l:= (SymkE ⊗ Syml¯E)0,
where the right hand side denotes the kernel of the contraction map
SymkE ⊗ Syml¯E → Symk−1E ⊗ Syml−1¯E,
i.e. Vk,lis the Cartan summand in SymkE ⊗ Syml¯E. The weights of SymkE are
aε1+ bε2+ cε3, with a,b,c ≥ 0, a + b + c = k.
If v1,v2,v3are the weight vectors of E, then these weights correspond to the weight
vectors va
SymlE, we see that the weights of Vk,lare
1· vb
2· vc
3in SymkE. Since the weights of Syml¯E are just minus the weights of
(a−a′)ε1+(b−b′)ε2+(c−c′)ε3,
From the given definition of the almost complex structure J it is clear that the T2-
representation on p1,0splits in three one-dimensional T2-representations with the weights
α12,α31,α23.
a,b,c,a′,b′,c′≥ 0, a+b+c = k, a′+b′+c′= l. (34)
Since the weights of a tensor product representation are the sums of weights of each
factor and since ε1+ ε2+ ε3= 0 on the Lie algebra of the maximal torus of SU3, we
immediately obtain
Corollary 5.11. The weights of the T2-representation on Λ1,1p∼= p1,0⊗ p0,1are
±3ε1, ±3ε2, ±3ε3,and 0.
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20ANDREI MOROIANU AND UWE SEMMELMANN
It remains to compute the Casimir operator of the irreducible SU3-representations
Vk,l. The highest weight of Vk,lis γ = kε1− lε3and ρ = ε1− ε3, thus
CasVk,l= ?γ,γ + 2ρ?B= −1
Here we use again the Freudenthal formula and the fact that the Killing form B induces
−1
have V1,1 = suC
previous cases.
It follows that all possible¯∆-eigenvalues (with respect to the metric B0) are of the
form 2(k(k + 2) + l(l + 2)). Obviously the eigenvalue 12 can only be obtained for
k = l = 1. Moreover, the restriction of the SU3-representation V1,1contains the zero
weight space. In fact, from (34), the zero weight appears in Vk,lif and only if there exist
a,b,c,a′,b′,c′≥ 0, a+b+c = k, a′+b′+c′= l such that (a−a′)ε1+(b−b′)ε2+(c−c′)ε3=
0, which is equivalent to k = l. We see that dimHomT2(V1,1,Λ(1,1)
Hence the eigenspace of¯∆ on primitive (1,1)-forms for the eigenvalue 12 is isomorphic
to the sum of four copies of V1,1, i.e. the eigenvalue 12 has multiplicity 4 · 8 = 32.
Computing the the smallest eigenvalue and the corresponding eigenspace of the
Laplace operator ∆ on non-constant functions we find V0,0 for the eigenvalue 0 and
V1,1for the eigenvalue 12. All other possible representations give a larger eigenvalue.
Hence, the ∆-eigenspace on functions Ω0(12) is isomorphic to two copies of V1,1, i.e. the
eigenvalue 12 has multiplicity 8 · 2 = 16.
Since the dimension of the isometry group of the nearly K¨ ahler manifold SU3/T2is
8, we obtain from (26)
6(k(k + 2) + l(l + 2)). (35)
6times the Euclidean scalar product on t∗⊂ R3(easy calculation). Notice that we
3and CasV1,1= −1, which is consistent with CasG
ad= −1 as in the
0
p) = 2 · 2 = 4.
dim(NK) ≤ dim(Ω(1,1)
0
(12)) − dim(i(M)) − dim(Ω0(12)) = 8. (36)
In the next section we will show by an explicit construction that actually the equality
holds, so the flag manifold has an 8-dimensional space of infinitesimal nearly K¨ ahler
deformations.
Before describing this construction we note that there are no infinitesimal Einstein
deformations other than the nearly K¨ ahler deformations. It follows from (35) that the
eigenvalue 2 does not occur in the spectrum of¯∆ on Λ(1,1)
realized on the SU3-representations V = V1,0or V = V0,1. However it is easy to check
that HomT2(V,Λ(1,1)
0
p) = {0}.
Corollary 5.12. Every infinitesimal Einstein deformation of the homogeneous nearly
K¨ ahler metric on F(1,2) = SU3/T2is an infinitesimal nearly K¨ ahler deformation.
0
M. The eigenvalue 6 could be
6. The infinitesimal nearly K¨ ahler deformations on SU3/T2
In this section we describe by explicit computation the space of infinitesimal nearly
K¨ ahler deformations of the flag manifold F(1,2) = SU3/T2. The Lie algebra u3 is
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THE HERMITIAN LAPLACE OPERATOR ON NEARLY K¨AHLER MANIFOLDS 21
spanned by {h1,h2,h3,e1,...,e6}, where
h1= iE11,h2= iE22,h3= iE33,
e1= E12− E21,
e2= i(E12+ E21),
e3= E13− E31,
e4= i(E13+ E31),
e5= E23− E32,
e6= i(E23+ E32).
We consider the bi-invariant metric g on SU3induced by −B/12, where B denotes the
Killing form of su3. It is easy to check that |ei|2=1 and |hi− hj|2= 1 with respect to
g. We extend this metric to U3in the obvious way which makes the frame {ei,√2hj}
orthonormal. This defines a metric, also denoted by g, on the manifold M = F(1,2).
From now on we identify vectors and 1-forms using this metric and use the notation
eij= ei∧ ej, etc.
An easy explicit commutator calculation yields the exterior derivative of the left-
invariant 1-forms eion U3:
de1 = −2e2∧ (h1− h2) + e35+ e46,
de2 = 2e1∧ (h1− h2) + e45− e36,
de3 = 2e4∧ (h3− h1) − e15+ e26,
de4 = −2e3∧ (h3− h1) − e25− e16,
de5 = −2e6∧ (h2− h3) + e13+ e24,
de6 = 2e5∧ (h2− h3) + e14− e23.
(37)
Let J denote the almost complex structure on M = F(1,2) whose K¨ ahler form is
ω = e12− e34+ e56(It is easy to check that ω, which a priori is a left-invariant 2-form
on U3, projects to M because Lhiω = 0). J induces an orientation on M with volume
form −e123456. Let Ψ++iΨ−denote the associated complex volume form on M defined
by the adT3-invariant form (e2+ iJe2) ∧ (e4+ iJe4) ∧ (e6+ iJe6). Explicitly,
Ψ+= e136+ e246+ e235− e145,
Using (37) we readily obtain
Ψ−= e236− e146− e135− e245.
d(e12) = −d(e34) = d(e56) = Ψ+, (38)
so
dω = 3Ψ+,anddΨ−= −2ω2.
The pair (g,J) thus defines a nearly K¨ ahler structure on M (a fact which we already
knew).
We fix now an element ξ ∈ su3⊂ u3, and denote by X the right-invariant vector field
on U3defined by ξ. Consider the functions
xi= g(X,ei),vi= g(X,hi). (39)
The functions viare projectable to M and clearly v1+ v2+ v3= 0. Let us introduce
the vector fields on U3
a1= x6e5− x5e6,a2= x3e4− x4e3,a3= x2e1− x1e2.
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22 ANDREI MOROIANU AND UWE SEMMELMANN
One can check that they project to M. Of course, one has
Ja1= x5e5+ x6e6, Ja2= x3e3+ x4e4, Ja3= x1e1+ x2e2.
The commutator relations in SU3yield
dv1= a2− a3, dv2= a3− a1, dv3= a1− a2. (40)
Using (37) and some straightforward computations we obtain
d(Ja1) = (−a1+ a2+ a3)?Ψ++ 4(v2− v3)e56,
d(Ja2) = (a1− a2+ a3)?Ψ++ 4(v1− v3)e34,
d(Ja3) = (a1+ a2− a3)?Ψ++ 4(v1− v2)e12.
We claim that the 2-form
ϕ = v1e56− v2e34+ v3e12
on M is of type (1,1), primitive, co-closed, and satisfies ∆ϕ = 12ϕ. The first two
assertions are obvious (recall that v1+ v2+ v3= 0). In order to prove that ϕ is co-
closed, it is enough to prove that dϕ ∧ ω = 0. Using (38) and (40) we compute:
dϕ ∧ ω = [(a2− a3) ∧ e56− (a3− a1) ∧ e34+ (a1− a2) ∧ e12] ∧ (e12− e34+ e56)
= (a1− a2) ∧ e1256− (a3− a2) ∧ e1234+ (a1− a2) ∧ e3456= 0.
Finally, using (41), we get
(41)
(42)
∆ϕ = d∗dϕ = − ∗ d ∗ [(a2− a3) ∧ e56− (a3− a1) ∧ e34+ (a1− a2) ∧ e12]
= − ∗ d[Ja2∧ e12+ Ja3∧ e34+ Ja3∧ e56− Ja1∧ e12− Ja1∧ e34− Ja2∧ e56]
= − ∗ [d(Ja2) ∧ (e12− e56) + d(Ja3) ∧ (e34+ e56) − d(Ja1) ∧ (e12+ e34)]
= − ∗ [(a1+ a2+ a3)?Ψ+∧ (e12− e56+ e34+ e56− e12− e34)
−2(a2?Ψ+) ∧ (e12− e56) − 2(a3?Ψ+) ∧ (e34+ e56) + 2(a1?Ψ+) ∧ (e12+ e34)
+4(v1− v3)e34∧ (e12− e56) + 4(v1− v2)e12∧ (e34+ e56)
−4(v2− v3)e56∧ (e12+ e34)]
= − ∗ [4(2v1− v2− v3)e1234+ 4(v1+ v3− 2v2)e1256+ 4(2v3− v1− v2)e3456]
= − ∗ [12v1e1234− 12v2e1256+ 12v3e3456] = 12ϕ.
Taking into account the inequality (36), we deduce at once the following
Corollary 6.1. The space of infinitesimal nearly K¨ ahler deformations of the nearly
K¨ ahler structure on F(1,2) is isomorphic to the Lie algebra of SU3. More precisely,
every right-invariant vector field X on SU3defines an element ϕ ∈ NK via the formulas
(39) and (42).
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Deformations of Nearly K¨ ahler Structures,
Andrei Moroianu, CMLS,´Ecole Polytechnique, UMR 7640 du CNRS, 91128 Palaiseau,
France
E-mail address: am@math.polytechnique.fr
Uwe Semmelmann, Mathematisches Institut, Universit¨ at zu K¨ oln, Weyertal 86-90
D-50931 K¨ oln, Germany
E-mail address: uwe.semmelmann@math.uni-koeln.de
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