Transgression forms in dimension 4

0.35 MB·English

by  Isabel M. C. Salavessa

Transgression forms in dimension 4
6 by
0 Isabel M.C. Salavessa1 and Ana Pereira do Vale2
∗ ∗∗
0
2
Dedicated to Dmitri Alekseevsky on his 65th birthday
n
a
J
0 1 Centro de F´ısica das Interac¸c˜oes Fundamentais, InstitutoSuperior T´ecnico, Edif´ıcio Ciˆencia,
3 Piso 3, 1049-001 LISBOA,Portugal; e-mail: [email protected]
2 Centro de Matem´atica, UniversidadedoMinho, Campus deGualtar
] 4710-057 BRAGA,Portugal; e-mail: [email protected]
G
D
Abstract: WecomputeexplicittransgressionformsfortheEulerandPontrjaginclassesofaRiemannian
.
h manifold M of dimension 4 under a conformal change of the metric, or a change to a Riemannian
t
a connectionwithtorsion. Theseformulaedescribethesingularsetofsomeconnectionswithsingularitieson
m compactmanifoldsasaresidueformulaintermsofapolynomialofinvariants. Wegivesomeapplications
[ for minimal submanifolds of K¨ahler manifolds. We also express the difference of the first Chern class of

2 two almost complex structures, and in particular an obstruction to the existence of a homotopy between
v them, by a residue formula along the set of anti-complex points. Finally we take the first steps in the
9
study of obstructions for two almost quaternionic-Hermitian structures on a manifold of dimension 8 to
8
3 have homotopic fundamental forms or isomorphic twistor spaces.
2
1
4 1 Introduction
0
/
h Somek-characteristic classes Ch(E)of avector bundleE over amanifold M, can berepresented
t
a in thecohomology classes of M interms of thecurvaturetensor definedwith respecttoa certain
m
type of connections on E. If , are such connections with curvature tensors R and R,
′ ′
: ∇ ∇
v respectively, then Chern-Weil theory states that
i
X
Ch(R) = Ch(R)+dT
′
r
a
whereT is a (k 1)-form on M. Away to specify such aT is by pullingback each connection by
− ˜
the projection π : M [0,1] M, π(p,t) = p, and then take the connection = t +(1 t)
′
× → ∇ ∇ − ∇
defined on π 1E. Denoting by R˜ its curvature tensor, the Chern-Simons transgression (k 1)-
−
−
form on M, obtained by integration along [0,1] of the closed k-form Ch(R˜) on M [0,1],
×
1
T( , )(X ,...,X )= Ch(R˜)(d,X ,...,X )dt
∇ ∇′ 1 k−1 Z0 dt 1 k−1
MSC 2000: Primary: 53C42, 53C55, 53C25, 53C38; Secondary: 57R20, 57R45.
Key Words: Characteristic classes, Transgression forms, Curvature tensors, almost complex structures, almost
quaternionic K¨ahler, Singular connections, dimension 4 and 8.
∗Partially supported by Funda¸c˜ao Ciˆencia e Tecnologia through POCTI/MAT/60671/2004 and Plurianual of
CFIF.
∗∗Partially supported by Funda¸c˜aoCiˆencia e Tecnologia through POCTI/MAT/60671/2004.
1
Salavessa–Pereira do Vale 2
satisfies dT( , ) = Ch(R) Ch(R). To specify such Ts is interesting by itself, but also
′ ′
∇ ∇ −
when is a connection with singularities. This is the case when (M,g) is a Riemannian m-
′
∇
E
manifold with its Levi-Civita connection, and (E,g , ) is a Riemannian vector bundle of
E
∇
rank m such that there exists a conformal bundle map Φ : TM E which vanishes along a
→
singular set Σ. This bundle map induces on M Σ a connection ′= Φ−1∗ E, that makes Φ
∼ ∇ ∇
a parallel bundle map. This connection can be seen as a singular Riemannian connection (with
torsion)on M with respecttoadegenerated metricgˆonM, butR andCh(R)canbesmoothly
′ ′
extended to Σ by the identities R(X,Y,Z,W) = g (RE(X,Y)Φ(Z),Φ(W)), Ch(R)=Ch(RE).
′ E ′
If M is closed, and Ch gives a top rank form and an integral cohomology class, then Ch(R)
M
and Ch(R) are finite integers, representing invariants. Moreover, if the singular set Σ is
M ′ R
sufficiently small and regular, the Stokes theorem reads dT = T, where V (Σ)
R M Vǫ(Σ) − ∂Vǫ(Σ) ǫ
∼
is a tubular neighbourhood of Σ of radius ǫ, and letting ǫ 0 may describe Ch(E) Ch(R)
R → R −
as a residue of T along Σ and expressed in terms of the zeros of Φ. This type of problem is
studied in [15],[16],[17] , using currents. We provide explicit formulae of transgression forms
for the cases of the Euler and Pontrjagin classes. In section 10 we give some applications to
minimal 4-submanifolds in Ka¨hler-Einstein manifolds. In section 9 we describe an obstruction
for two almost complex structures on M to be homotopic, measured by the difference between
their Chern classes, translated to a residue formula on the set of anti-complex points. In section
11 we introduce the study of obstructions for two almost quaternionic-Hermitian structures on
a Riemannian 8-manifold to have isomorphic twistor spaces or homotopic fundamental forms.
Let (M,g) be an oriented Riemannian manifold of dimension 4 with its Levi-Civita connection,
curvature tensor RM, Ricci tensor RicciM and scalar curvature sM. Our formulae are:
Proposition 1.1 Let f :M R be a smooth map, and gˆ= efg. Set
→
P( f)= (2∆f + f 2 2sM) f +4(RicciM)♯( f) ( f 2). (1.1)
∇ k∇ k − ∇ ∇ −∇ k∇ k
Then
(RˆM) = (RM)+ 1 div (P( f))Vol , (1.2)
X X 32π2 g ∇ M
p (RˆM) = p (RM).
1 1
Theinvariance of the Pontrjagin class undera conformal change of the metric is well known (see
[6] or remark 3). We thank Sergiu Moroianu for drawing our attention to this. The authors do
not recall to have seen formula (1.2) in the literature: div (P( f)) is a 2nd-order differential
g
∇
operator on f. If f = logh where h > 0 except at a finite set of zeros p and poles p of
α i
∇
homogeneuos order 2k and 2k respectively (k ,k > 0), then:
α i α i
1
div (P( logh))Vol = 1k2(k +3)+ 1k2( k +3).
32π2 g ∇ M α− 2 α α i− 2 i − i
M
Z
P P
Theorem 1.1 Let Φ:TM E be a conformal bundle map into a Riemannian vector bundle
→
E
(E,g , ) of rank 4 over M, with coefficient of conformality given by a non-negative function
E
∇
h:M R with zero set Σ. Let gˆ= hg, and S C ( 2T M TM) defined away from Σ:
′ ∞ ∗
→ ∈ ⊗
N
Salavessa–Pereira do Vale 3
S (X,Y)= Φ 1( Φ(Y)) 1dlogh(X)Y 1dlogh(Y)X + 1g(X,Y) logh.
′ − ∇X − 2 − 2 2 ∇
Then:
(RE) = (RM) 1 d( (RˆM 1d 1( )2) )+ 1 div (P( logh))Vol
X X − 4π2 hS′∧∗ − 2 S′− 3 S′ igˆ 32π2 g ∇ M
p (RE) = p (RM) 1 d( (RˆM 1d 1( )2) )
1 1 − 2π2 hS′∧ − 2 S′− 3 S′ igˆ
where RˆM is the curvature tensor of (M,gˆ), and : TM 2TM, ( )2 : 2TM 2TM,
′ ′
S → S →
are defined by
V V V
(X),Y Z = gˆ(S (X,Y),Z)
′ gˆ ′
hS ∧ i
( )2(X Y),Z W = gˆ(S (X,Z),S (Y,W)) gˆ(S (X,W),S (Y,Z))
′ gˆ ′ ′ ′ ′
h S ∧ ∧ i −
RˆM(X Y),Z W = h RM(X Y)+φ g(X Y),Z W
gˆ g
h ∧φ = 1(∧ k∇iloghk2g+h1dlogh∧ dlogh• Hes∧s(logh))∧. i
2 − 4 2 ⊗ −
Moreover, if dΦ = 0, then (RE)= (RM)+ 1 div (P( logh))Vol andp (RE) = p (RM).
X X 32π2 g ∇ M 1 1
The angle θ [0,π] between two positive g-orthogonal almost complex structures J and
0
∈
J on M is defined by cosθ = 1 J ,J , and Σ = p M : J (p)= J (p) = cosθ 1( 1)
1 4h 0 1i { ∈ 1 − 0 } − −
is the set of anti-complex points. If J is generic, Σ is a surface of M and an orientation
1
can be given. Under the usual identification of J with its Ka¨hler form ω , the orthogonal
0 0
complement E of RJ in 2 TM, is a complex line bundle over M with complex structure
J0 0 +
”J ”. We denote by H˜(p) the orthogonal projection of J (p) into E . Let N1Σ be the total
0 V 1 J0
set of the unit normal bundle of Σ and d its Lebesgue measure. For each (p,u) N1Σ
N1Σ
∈
define κ(p,u) 1 the order of the zero of φ (r) = (1 + cosθ)(exp (ru)) at r = 0. We
(p,u) p
≥
say that (1 + cosθ) has a controlled zero set if there exist a non-negative integrable function
f :N1Σ [0,+ ] and r > 0 s.t. sup r d log(φ (r)) f(p,u) a.e. (p,u) N1Σ.
→ ∞ 0 0<r<r0| dr (p,u) | ≤ ∈
For example, this holds if for all (p,u), φ (r) is a polynomial function on r with coefficient
(p,u)
of lowest order uniformly bounded away from zero. Some weaker conditions can be given on
dk φ (0), dk+1 φ (r) and dk+2 φ (r), where k = κ(p,u), to guarantee controlled zero
drk (p,u) drk+1 (p,u) drk+2 (p,u)
set (see Prop. 9.4). For each p Σ, S(p,1) denotes the unit sphere of T Σ and σ its volume.
p ⊥ d′
∈
The function κ˜(p) = 1 κ(p,u)d u is the average order of the zero p of (1+cosθ), in
σd′ S(p,1) S(p,1)
the normal direction. Assume M is compact.
R
Theorem 1.2 If J is almost Ka¨hler then for any almost complex structure J = cosθJ +H˜
0 1 0
1
(c (M,J ) c (M,J )) ω = div((T˜J )♯)Vol (1.3)
1 1 1 0 0 0 M
M − ∧ 4π M
Z Z
where T˜ is the 1-form on M Σ, T˜(X) = 1 H˜,J H˜ . In the particular case that
∼ (1+cosθ)h∇X 0 i
EJ0H˜ is J0-anti-complex we have
∇
1
(1.3) = ∆log(1+cosθ)Vol . (1.4)
M
−4π
M
Z
In this case, assume Σ is a finite disjoint union of closed oriented submanifolds Σi of dimension
d 2. Let k be the range set of κ on N1Σi and let N1Σi = κ 1(k ). If κ is bounded a.e.
i γ iγ γ − iγ
≤
S
Salavessa–Pereira do Vale 4
and (1+cosθ) has a controlled zero set then a residue formula along Σ is obtained:
(1.3) = 1 κ˜(p)Vol = 1 k d (N1Σi). (1.5)
2 Σi 2 iγ N1Σi γ
i:Xdi=2ZΣi i:Xdi=2Xγ
1
and so c (M,J ),ω Vol c (M,J ),ω Vol = (sM + 1 ω 2)Vol ,
h 1 1 0i M ≥ h 1 0 0i M 4π 2k∇ 0k M
M M M
Z Z Z
with equality iff d 1 i. Thus, if c (M,J )= c (M,J ) (in H2(M,R)) Σ = for d = 2.
i 1 1 1 0 i i
≤ ∀ ∅
Remark 1. Pairs of almost complex structures with or without the same properties may exist on
a manifold. Alekseevsky [1] discovered examples of simply-connected non-compact Riemannian
manifoldsadmittinganon-integrable almost-K¨ahler structureandalso anintegrable non-K¨ahler
complex structure, namely some solvable groups of dimension 4(4+p+q) (q = 0).
6
We also prove in section 9:
Proposition 1.2 If M is compact, J is Ka¨hler and J is almost Ka¨hler then θ is constant.
0 1
Thus, if cosθ = 1, J and J are homotopic and define a hyper-Ka¨hler structure on M.
0 1
6 ±
If (M8,g) is an oriented 8-dimensional manifold and Q and Q are two almost quaternionic
0 1
Hermitian structures (see [2] for definitions) we define an angle θ [0,π], by cosθ = 3 Ω ,Ω ,
∈ 10h 0 1i
whereΩ arethe correspondingfundamental4-forms. Let E bethecorrespondingrank3 vector
i i
bundle generated by the twistor space of Q . In section 11 we prove:
i
Proposition 1.3 If Q is quaternionic Ka¨hler and Q is almost quaternionic Ka¨hler and M is
0 1
compact, then θ is constant. If cosθ = 1 then Ω and Ω are homotopic 4-forms in H4(M;R),
6 − 0 1 +
and if Q is also quaternionic Ka¨hler, p (E ) = p (E ). Furthermore, in the later case, (a),(b)
1 1 0 1 1
or (c) must hold: (a) E = E ; (b) E E has rank one, M is Ka¨hler and both Q ,Q are
0 1 0 1 0 1
∩
locally hyper-Ka¨hler structures; (c) E E = 0 .
0 1
∩ { }
We observe that the problem on a compact 4-manifold ”almost Ka¨hler + Einstein implies
Ka¨hler”, also called the Goldberg-conjecture, is not completely solved. Salamon in [22] gives an
example of a compact 8-dimensional almost quaternionic Ka¨hler manifold that is not quater-
nionic Ka¨hler. We thank the referee for drawing our attention to this reference.
2 Curvature tensors in dimension 4
LetV beavector spaceof dimension4andwithainnerproductg. We identify 2 := 2V with
Skew(V) (the space of skew-symmetric endomorphisms of V) and with 2V in the standard
∗V V
way, considering Skew(V) as a subset of V V with half of its usual Hilbert-Schmidt inner
∗ ⊗ V
product. Consider the vector subspaces and of the symmetric and skew-symmetric linear
⊥
R R
endomorphisms of 2, respectively. , or more generally L( 2;L(V,V)), is defined as the space
R
of curvature tensors of V (see [7], [8], or [25] for details). We recall some definitions.
V V
If R L( 2;L(V,V)) let R L(V;L( 2;V)) given by R(Z)(X Y) := R(X Y)Z.
∈ ∈ ∧ ∧
We also use the following notation: R(X,Y,Z,W) = g(R(X Y)Z,W) = R(Z,W,X,Y). If
V V ∧
R L( 2; 2), R = RT (transposed). We assume that V is with a given orientation, and
∈
so the star operator L( 2; 2) splits 2 into its eigenspaces 2, corresponding to the
V V ∗ ∈ ±
eigenvalues 1, defining respectively the space of selfdual and of anti-self-dual two forms. Then
± V V V V
Salavessa–Pereira do Vale 5
R L( 2; 2) splits as R = R+ R+ R R with R+ L( 2; 2), R L( 2; 2).
Th∈e Ricci tensor Ricci :L( 2; +2)⊕ L−(V⊕,V+−) a⊕nd t−−he scalar±cu∈rvatur±e are+given±−re∈specti±vely−by
V V → V V V V
g(Ricci(R)(X),Y) = Ricci (X,Y) = tr Z R(X,Z)(Y) and s = tr(Ricci ) = 2 R,Id =
VR V { → } R R h i
2tr(R). Thesectional curvatureof R L( 2; 2)isdenoted byσ (P)= R(X,Y,X,Y)foreach
R
∈
2-plane P spanned by an o.n.b. X,Y , and the Bianchi map b : L( 2;L(V;V)) 3V
{ } V V → ∗ ⊗
V L( 2;L(V;V)) is defined by b(R)(X ,X )(X ) = R(X ,X )(X ) + R(X ,X )(X ) +
⊂ 1 2 3 1 2 V 3 3 1V 2
R(X ,X )(X ).Thenb : , . IfR ,b(R)= 1tr( R) .Lett(R)= 1 b(R), .
2 V3 1 R→ R R⊥ → R⊥ ∈ R 2 ∗ ∗ 6h ∗i
WedenotebySym(V)thespaceofsymmetricendomorphismsofV andbySym (V)itssubspace
0
of trace-free endomorphisms.
A complex 2-plane τ of Vc is said to be totally isotropic (t.i.) if g(v,v) = 0 v τ. If (e ) is an
i
∀ ∈
o.n. basis and α= e1+ie2, β = e3+ie4, then τ = spanC α,β defines a t.i. complex plane. Set
{ }
R(ijkl) = R(e ,e ,e ,e ). The isotropic sectional curvature w.r.t R at τ is given by (see
i j k l
∈ R
[21])
R(α β,α¯ β¯) 1( )
K (R)(τ) = ∧ ∧ = R(1313)+R(1414)+R(2323)+R(2424) 2R(1234) . (2.1)
isot α β 2 4 −
k ∧ k
The Kulkarni-Nomizu product of ξ,φ 2V is a symmetric product defined by
∗
∈
ξ φ(X,Y,Z,W) = ξ(X,Z)φ(Y,W)+Nξ(Y,W)φ(X,Z) ξ(Y,Z)φ(X,W) ξ(X,W)φ(Y,Z).
• − −
We have Id = Id = 1g g. If R L( 2;L(V;V)) it is definedthe Weitzenbo¨ck operator A(R)
2
2 • ∈
V
A(R)(X,YV),Z W := Ricci g(X,Y,Z,W)+2R(Z,X,Y,W) 2R(W,X,Y,Z).
R
h ∧ i • −
If R then A(R) and A(R) = Ricci g 2R+2b(R). In this case, Ricci = s g,
R A(R) R
∈ R ∈ R • −
b(A(R)) = 4b(R). Furthermore, if b(R) = 0, A(R) : . Note that , and if R ,
± ±
→ ∗ ∈ R ∈ R
then R+R , R still lie in and R R . Straightforward computations shows:
∗ ∗ ∗ ∗ R ∗ − ∗∈ R⊥V V
Proposition 2.1 Let R and τ = spanC e1 +ie2,e3 +ie4 with e1,...,e4 an o.n. basis
∈ R { }
with orientation ǫ. K is computed at τ. Then
isot
Ricci =3g s =12 b(Id)=0 K (Id)=1
Id Id isot
Ricci =0 s =0 b( )=3 K ( )= 1ǫ
∗ ∗ ∗ ∗ isot ∗ −2
Ricci = 1s g Ricci s =s b( R )=b(R) K ( R )=K (R)
∗R∗ 2 R − R ∗R∗ R ∗ ∗ isot ∗ ∗ isot
Ricci =Ricci =t(R)g s =s =4t(R) b( R R )=0 K ( R R )=2(σ (34) σ (12)).
R∗ ∗R ∗R R∗ isot R R
∗ − ∗ ∗ − ∗ −
ForR,Q L( 2; 2),andX,Y,Z,W V weusethefollowingnotation: R(X Y) Q(Z W)=
∈ ∈ ∧ ∧ ∧
R(X Y),( Q)(Z W) Vol 4V
h ∧ ∗V V∧ i V ∈ ∗
Definition 2.1 The Euler form V(R) and the Pontrjagin form p (R) of R L( 2; 2) are the
1
X ∈
4-forms:
V V
4π2 (R) = R(e e ) R(e e ) R(e e ) R(e e )+R(e e ) R(e e )
1 2 3 4 1 3 2 4 1 4 2 3
X ∧ ∧ ∧ − ∧ ∧ ∧ ∧ ∧ ∧
= 1 R, R Vol = 1 R,R Vol (2.2)
2h ∗ ∗i V 2h∗ ∗i V
4π2p (R) = R(e e ) R(e e )+R(e e ) R(e e )+R(e e ) R(e e )
1 1 2 1 2 1 3 1 3 1 4 1 4
∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧
+R(e e ) R(e e )+R(e e ) R(e e )+R(e e ) R(e e )
2 3 2 3 2 4 2 4 3 4 3 4
∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧
= RT, RT Vol = R,R Vol (2.3)
V V
h ∗ i h ∗i
Salavessa–Pereira do Vale 6
Note that R,Q = RT,QT . Thus (RT) = (R), and p (RT) p (R) = R, R R .
1 1
h i h i X X − h ∗ − ∗i
A positive g-orthogonal complex structure J on V is a complex structure that induces the
orientation of V and it is a linear isometry. Such structures are in 1-1 correspondence with the
elements ω of 2 V of norm √2, by ω (X,Y) = g(JX,Y). The condition of orthogonality
J + J
between two of such J is equivalent to the anti-commuting condition, for the multiplication of
V
complex structures corresponds to the quaternionic multiplication of two unit pure imaginary
vectors of R4. For each J and R L( 2; 2) we define the 2-forms on V
∈
V V
Ricci (X,Y) := R(X Y),ω = R(ω )(X,Y) (2.4)
J,R J J
h ∧ i
Ψ (R)(X,Y) := 1Ricci (X,JY)+ 1Ricci (Y,JX) (2.5)
J −2 R 2 R
If R , then Ψ (R)(X,Y)=Ricci(1,1)(JX,Y). Usually Ricci is denoted by Ricci and
J J,R
∈ R ∗
named by star-Ricci but we do not use that notation to avoid confusion with the Ricci of the
curvature tensor . The star-scalar curvature is s = 2 Ricci ,ω , and b (R) 2V is
J,R J,R J J ∗
∗ h i ∈
defined by b (R)(X,Y) = 1tr(Z Jb(R)(X,Y)(Z)). If P L (V;V), that is P J = J P
J −2 → ∈ J ◦ V ◦
then both the complex trace tr (P) and the complex determinant det (P) are well defined.
J J
If X ,Y = JX ,X ,Y = JX is a real basis of V and for α = 1,2 denote ”α” := W =
1 1 1 2 2 2 α
21(Xα −iYα), α¯ := Wα¯ = Wα, then trJ(P) = αW∗αPc(Wα), detJ(P) = det[W∗αPc(Wβ)] where
(Wα,Wα = Wα) is the complex dual basis of W ,W , and Pc denotes the complex linear
ext∗ensio∗n of P ∗to Vc. If P L (V;V) 2, Pth{enα P,ωα¯} = tr (P). If R L( 2;L (V;V)),
J J J J
∈ ∩ h i ∈
then R is said to be J-invariant.
V V
Definition 2.2 If(V,J,g) isHermitian, thefirstandsecondChernformofR L( 2;L (V;V))
J
∈
w.r.t. J, are respectively
V
i 1
c (R,J) = Tr (R) c (R,J) = det (R). (2.6)
1 −2π J 2 −4π2 J
It follows that, if R L( 2;L (V;V) 2) then c (R,J) = 1 Ricci , c (R,J) = (R),
∈ J ∩ 1 2π J,R 2 X
p (R) = c (R,J) c (R,J) 2c (R,J), and b (R) = Ricci Ψ . If R and is J-
1 1 ∧ 1 V − 2 V J J,R − J ∈ R
invariant, then Ricci (JX,JY) = Ricci (X,Y), and Ψ (R)(X,Y) = Ricci (JX,Y).
R R J R
We denote by E the rank 2 subspace of 2 V defined by the orthogonal complement of R ω
J + { J}
2V = R ω E , whereVE = ω 2V : ω(JX,JY) = ω(X,Y)
+ { J}⊕ J J { ∈ + − }
andacanVoniccomplexstructurecanbegiventoE : J˜ωV(X,Y) = ω(JX,Y). Let e ,e ,e ,e
J 1 2 3 4
− { }
be a d.o.n.b. of V, giving a corresponding o.n.b. √2ω of 2 V, ω = e e +e e , ω =
σ + 1 1 ∧ 2 3 ∧ 4 2
e e e e , ω = e e +e e , and let J be defined by ω = g(J (), ). The 2-forms
1 ∧ 3− 2∧ 4 3 1 ∧ 4 2∧ 3 σ V σ σ · ·
ω ,ω span E , and J˜ω = ω corresponds to J J = J . Any such o.n. (of norm √2) basis
2 3 J 2 3 1 2 3
(ω ,ω ,ω ), where ω = ”ω ω ” defines a canonic orientation on 2 V.
1 2 3 3 1 2 +
V
3 Almost Hermitian 4-manifolds
Assume (M,J,g) is an almost Hermitian 4-manifold with its Levi-Civita connection , and we
∇
use the above notation taking V = T M. (M,J,g) is said to be almost Ka¨hler if the Ka¨hler
p
Salavessa–Pereira do Vale 7
form ω (X,Y) = g(JX,Y) is closed, and Ka¨hler if ω is parallel. The latter is equivalent to J
J J
to be almost Ka¨hler and integrable. Since J2 = Id and J is g-orthogonal, X,Y,Z T M
p
− ∀ ∈
J(JX) = J J(X), g( J(X),Y)= g(X, J(Y)). (3.1)
Z Z Z Z
∇ − ∇ ∇ − ∇
We use the following sign for curvature tensors R(X,Y)= + + , X,Y
X Y Y X [X,Y]
−∇ ∇ ∇ ∇ ∇ ∀ ∈
T M, and denote by RM the curvature tensor of M, RicciM = Ricci , sM = s , and
p J J,RM J J,RM
by , the Hilbert Schmidt inner product on kTM . The Weitzenbo¨ck formulae for ω read
∗ J
h· ·i
(see e.g. [8])
V
∆ω = tr 2ω +A(RM)(w ) (3.2)
J J J
− ∇
0= 1∆+ ω 2 = ∆ω ,ω + ω 2+ A(RM)ω ,ω . (3.3)
2 k Jk −h J Ji k∇ Jk h J Ji
Since ω is a self-dual form, dω = δω . Thus J is almost Ka¨hler iff δω = 0. In this case
J J J J
k k k k
ω is harmonic. If J is not Ka¨hler we may use the canonical Hermitian connection (see [13])
J
˜ Y = Y 1J( J(Y)).
∇X ∇X − 2 ∇X
This connection satisfies ˜g = ˜J = 0, but has torsion T˜(X,Y) = 1JdJ(X,Y). This is a
∇ ∇ −2
U(2)-connection on M, and so its curvature tensor R˜ is J-invariant. The Chern classes of M
using R˜ satisfy 2πc (M,J) = Ricci , and a direct computation shows that
1 J,R˜
Ricci (X,Y)= RicciM(X,Y)+η (X,Y), s = sM +2 η ,ω
J,R˜ J J J,R˜ J h J Ji
where η is the 2-form on M
J
η (X,Y) = 1 J J, J , (3.4)
J 4h ∇X ∇Y i
andtheinnerproductistheHilbert-SchmidtinnerproductonTM TM. If(M,J,g) isalmost
∗
⊗
Ka¨hler then (see [5] for a survey)
J = J( J). (3.5)
JX X
∇ − ∇
In this case T˜(1,1) = 0, and η ,ω = 1 J 2. Furthermore, for almost Ka¨hler J (see [9])
h J Ji −8k∇ k
sM sM = ω 2 (3.6)
J J
− k∇ k
4π c (M,J),ω = 2 R˜(ω ),ω = sM + 1 ω 2 = 1(sM +sM). (3.7)
h 1 Ji h J Ji 2k∇ Jk 2 J
We consider on E the induced connection EJ from the connection + of 2.
J ∇ ∇ +
Proposition 3.1 If (M,J,g) is almost Hermitian V
2πc (E )(X,Y) = RicciM(X,Y)+η (X,Y)= 2πc (M,J)
1 J J J 1
p ( 2T M) = c (E ) c (E ).
1 + p 1 J ∧ 1 J
Proof. Let √ω22, √ω32 be aVlocal d.o.n. frame of EJ, and (·)EJ denote the orthogonal projection
onto E . Since ω = √2 is constant, +ω is a section of E . Now,
J k Jk ∇Y J J
EJ EJs = ( + EJs)EJ = ( + +s 1 +s,ω +ω )EJ
∇Y ∇X ∇Y ∇X ∇Y ∇X − 2h∇X Ji∇Y J
= ( + +s)EJ + 1 s, +ω +ω .
∇Y ∇X 2h ∇X Ji∇Y J
Salavessa–Pereira do Vale 8
Thus
REJ(X,Y)s = (R+(X,Y)s)E + 1 s, +ω +ω 1 s, +ω +ω . (3.8)
2h ∇X Ji∇Y J − 2h ∇Y Ji∇X J
The curvature tensor of 2 satisfies R+(X,Y)ω ,ω = 2RM(X,Y)ω . E is a complex line
+ h 2 3i J J
bundle over M and so it has a (real) volume element Vol . Using (3.1) we easily see that
V EJ
another way to express η is η (X,Y) = 1Vol ( +ω , +ω ). Then
J J 2 EJ ∇X J ∇Y J
ω ω
2πc (E )(X,Y):= REJ(X,Y) 2 , 3 = RM(X,Y)ω +η (X,Y). (3.9)
1 J J J
h √2 √2i
Since c (C ω Ec) = c (C ω ) c (Ec) c (Ec) and c (Ec) = 0 we have p ( 2 TM) =
2 { J}⊕ J 1 { J} ∧ 1 J − 2 J 1 J 1 +
c (( 2 TM)c) = c (C ω Ec) = c (E ) c (E ). QED
− 2 + − 2 { J}⊕ J 1 J ∧ 1 J V
RemaVrk 2. The canonical line bundle w.r.t J is the bundle = 2(TM(1,0)) = (2,0) of the
J ∗
K
complex 2-forms of type(2,0), andonehas( 2 TM)c = C(ω ) (2,0) (0,2). ThebundleE
+ J ⊕ V ⊕ V J
is isomorphic to the realification of the anti-canonical bundle 1 = (0,2). As a complex line
V KJ−V V
bundle,ω (ω )(0,2) isthecomplexisomorphism. Therefore, c (E )= c ( 1) = c ( ). An
→ c 1 J V1 KJ− − 1 KJ
almostcomplex structuredefinesacanonicspin-cstructures= PSpincM onM withcanonicline
bundle (2,0) and complex spinor bundle S= (0, )=S+ S , S+= (0,0)+ (0,2), S = (0,1),
∗ − −
⊕
where TM acts by Clifford multiplication. The Chern class of s is also given by c ( ).
V V V V 1 KJ V
Finally we observe the following:
Lemma 3.1 If J and J are two anti-commuting g-orthogonal almost Ka¨hler complex struc-
1 2
tures on M, then J = J J is also almost Ka¨hler iff g( J (X),J X) = 0 X T M. That is
3 1 2 Z 2 1 p
∇ ∀ ∈
the case if either J or J is Ka¨hler. In that case (J ,J ,J ) is in fact an hyper-Ka¨hler structure.
1 2 1 2 3
Proof. It is sufficient to find an equivalent condition for δω =0. Using (3.1), (3.5), X T M,
3 p
∀ ∈
J (J X)+ J (X) = (J J )(J X)+ (J J )(X) =
∇J3X 3 3 ∇X 3 ∇J1J2X 1 2 3 ∇X 1 2
= J (J J X)+J ( J (J X))+ J (J X)+J ( J (X))
∇J1J2X 1 2 3 1 ∇J1J2X 2 3 ∇X 1 2 1 ∇X 2
= J ( J (J J X)) J ( J (J X))+ J (J X)+J ( J (X))
− 1 ∇J2X 1 2 3 − 1 ∇J2J1X 2 3 ∇X 1 2 1 ∇X 2
= J (J J J X)+J J ( J (J X))+ J (J X)+J ( J (X))
∇J2X 1 1 2 3 1 2 ∇J1X 2 3 ∇X 1 2 1 ∇X 2
= J (X)+ J (J X) J J (J J X)+J ( J (X))
−∇J2X 1 ∇X 1 2 − 1∇J1X 2 2 3 1 ∇X 2
( ) ( )
= J (X)+ J (J X) +J J (J X)+ J (X)
−∇J2X 1 ∇X 1 2 1 −∇J1X 2 1 ∇X 2
Using (3.5) from dω (X,J X,Z) = 0 X,Z, we have g( J (X) + J (J X),Z) =
1 2 ∀ −∇J2X 1 ∇X 1 2
g( J (X),J X). If X is a unit vector of T M then X,J X,J X,J X is an o.n.b. Hence
Z 1 2 p 1 2 3
− ∇
g(δJ ,Z) = g( J (X)+ J (J X)+ J (J X)+ J (J J X),Z) =
− 3 ∇X 3 ∇J3X 3 3 ∇J1X 3 1 ∇J3J1X 3 3 1
= g( J (X),J X) g( J (J X)+ J (X),J Z)
− ∇Z 1 2 − −∇J1X 2 1 ∇X 2 1
g( J (J X),J J X) g( J (X)+ J (J X),J Z)
− ∇Z 1 1 2 1 − −∇X 2 ∇J1X 2 1 1
= g( J (X),J X) g(J J (X),J J X) = 2g( J (X),J X).
Z 1 2 1 Z 1 1 2 Z 1 2
− ∇ − ∇ − ∇
Assume that at a given point X(p ) = 0. Thus at p , g( J (X),J X) = g( (J (X)),J X)
0 0 Z 1 2 Z 1 2
∇ ∇ ∇
= g(J X, (J (X)) = g(J X, J (X)). Hitchin ([18], lemma (6.8)) proved that hyper-
1 Z 2 1 Z 2
− ∇ − ∇
almost-K¨ahler structures are in fact hyper-Ka¨hler. QED
Salavessa–Pereira do Vale 9
4 The irreducible components of R
∈ R
has an orthogonal decomposition = 4 into O(V)-invariants subspaces ([25]), where
R R ⊕i=1Ri
= (Ker b) = R : σ = 0 = R :R = λ Ker Ricci Ker tr
1 ⊥ R
R { } { ∗} ⊂ ⊂
= (Ker tr) = R = λId
2 ⊥
R { }
= (Ker b) (Ker Ricci) = R : R = R ,tr R = tr R = 0
3
R ∩ { ∗ ∗ ∗ }
= (Ker b) (Ker Ricci) = R : R = R
4 ⊥
R ∩ { ∗ − ∗}
and they can be characterized in terms of the sectional and Ricci curvature. Set := Ker b =
B
. We also have the following characterization using K :
2 3 4 isot
R ⊕R ⊕R
Proposition 4.1 If R and b(R)= 0 then K = 0 iff σ (P) = σ (P ) P. Hence,
isot R R ⊥
∈ R − ∀
= R :b(R) = 0, σ (P) = σ (P ) P = R :K = 0 = Sym (V) g
4 R R ⊥ isot 0
R { − ∀ } { ∈ B } •
and φ φ g is an isomorphism from Sym (V) onto .
0 4
→ • R
Proof. In [25] it is proved the first equality of , and that the elements of this set satisfy
4
R
R(P,P ) = 0. Now it turns out that this condition and σ (P) = σ (P ) is equivalent
⊥ R R ⊥
−
to K = 0. We prove the less obvious implication. K = 0 means that for any o.n.b.
isot isot
X,Y,Z,W, 2R(X,Y,Z,W) = R(X,Z,X,Z)+R(X,W,X,W)+R(Y,Z,Y,Z)+R(Y,W,Y,W).
Setting P = span X,Y and replacing X by X we conclude
{ } −
0=R(X,Y,Z,W)=R(P,P⊥)
0=R(X,Z,X,Z)+R(X,W,X,W)+R(Y,Z,Y,Z)+R(Y,W,Y,W)=:Ψ(X,Y,Z,W).
From 0=Ψ(X,Y,Z,W)=Ψ(Z,Y,X,W)=Ψ(Z,X,W,Y) we have 0=Ψ(Z,Y,X,W) Ψ(X,Y,Z,W)
−
+Ψ(Z,X,W,Y) =2R(Z,W,Z,W) +2R(X,Y,X,Y) =2σ (P⊥)+2σ (P). For a proof that φ φ g
R R
→ •
defines an isomorphism see e.g. [25]. QED
Let φ Sym(V), and e a d.o.n.b. of eigenvectors of φ, with corresponding eigenvalues
i
∈
λ . Then e e with i < j are the eigenvectors of φ g (φ φ resp.) corresponding to the
i i j
∧ • •
respective eigenvalues λ +λ (2λ λ resp.). Let Λ = e e e e , Λ = e e e e ,
i j i j ±1 1∧ 2± 3∧ 4 ±2 1∧ 3∓ 2∧ 4
Λ = e e e e . We have:
±3 1∧ 4± 2∧ 3
φ g(Λ ),Λ = δ tr(φ) φ g(Λ+),Λ = δ (λ +λ λ λ )
h • ±α ±βi αβ h • 1 −βi 1β 1 2− 3− 4
φ g(Λ+),Λ = δ (λ +λ λ λ ) φ g(Λ+),Λ = δ (λ +λ λ λ )
h • 2 −βi 2β 1 3− 2− 4 h • 3 −βi 3β 1 4− 2− 3
φ φ(Λ ),Λ = 2δ (λ λ +λ λ ) φ φ(Λ ),Λ = 2δ (λ λ λ λ )
h • ±1 ±βi 1β 1 2 3 4 h • ±1 ∓βi 1β 1 2− 3 4
φ φ(Λ ),Λ = 2δ (λ λ +λ λ ) φ φ(Λ ),Λ = 2δ (λ λ λ λ )
h • ±2 ±βi 2β 1 3 2 4 h • ±2 ∓βi 2β 1 3− 2 4
φ φ(Λ ),Λ = 2δ (λ λ +λ λ ) φ φ(Λ ),Λ = 2δ (λ λ λ λ ).
h • ±3 ±βi 3β 1 4 2 3 h • ±3 ∓βi 3β 1 4− 2 3
Lemma 4.1 (φ g) = 1tr(φ)Id , and tr(φ φ)= 2λ λ = 2σ (φ).
• ±± 2 ±2 • i<j i j 2
P
Lemma 4.2 Let φ,ξ Sym(V)Vand φ,ξ = φ(e ,e )ξ(e ,e ) where e is an o.n.b. of V.
ij i j i j i
∈ h i
Then
P
ξ φ tr(φ g) = Id,φ g = 3tr(φ)
• ∈ B • h • i
φ g,ξ g = 2 φ,ξ +tr(φ)tr(ξ) (φ g) = 1tr(φ)g g φ g
h • • i h i ∗ • ∗ 2 • − •
Ricci = Tr(φ)g+2φ s = 6trφ
φ g φ g
• •
φ g, ξ g = 0 φ g = ξ g iff φ= ξ.
h • ∗ • i • •
Salavessa–Pereira do Vale 10
Proof. The equalities are proved using the eigenvalues λ and eigenvectors e of φ. Since both
i i
P = e e and P are eigenvectors of R = φ g, then R has the same eigenvalues λ +λ as
i∧ j ⊥ • ∗ ∗ ′i ′j
R. Namely, if (ijkl) is a permutation of 1234, λ +λ = λ (λ +λ )= tr(φ) (λ +λ ) =
′i ′j s s− i j − i j
λ +λ . QED
k l P
If R , then R = 1t(R) = 1tr( R) . For R we have the decomposition:
∈ R 1 3 ∗ 6 ∗ ∗ ∈ B
R = 1 tr(R)g g = 1 s g g = 1 s Id (4.1)
2 12 • 24 R • 12 R
R = 1(R+ R ) 1 tr(R)g g = = ++ (4.2)
3 2 ∗ ∗ − 12 • W W W−
R = 1(R R ) = 1(Ricci 1s g) g. (4.3)
4 2 −∗ ∗ 2 R − 4 R •
is the Weyl tensor. It applies 2 V into 2 V, with + and the self-dual and the
−
W ± ± W W
anti-selfdual part respectively, i.e, they satisfy: = = . Furthermore R =
+ sRId , R++R = 1(RicVci sRg) Vg. F∗rWom±preWvio±u∗s lem±mWas±we have: ±±
W± 12 ± − +− 2 − 4 •
PropositioVn 4.2 = Sym(V) g, and φ Sym(V) (φ g) = tr(φ)g g, (φ g) =
R4 ⊕R2 • ∀ ∈ • 2 4 • • 4
tr(φ)
(φ g) g.
− 4 •
Lemma 4.3 Let R and φ,ξ Sym(V). Then R,φ g = Ricci ,φ , R ,φ g =
R
∈ B ∈ h • i h i h∗ ∗ • i
1s tr(φ) Ricci ,φ , and φ g ,φ g = 2(trφ)2 2 φ 2 = 2tr(φ φ).
2 R −h R i h∗ • ∗ • i − k k •
Proof. Using the decomposition (4.1)-(4.3), lemma 4.2 and proposition 4.2 we have
R,φ g = 1 s g g,φ g + 1 (Ricci 1s g) g,φ g
h • i 24 Rh • • i 2h R − 4 R • • i
= 1 s (2 g,φ +tr(g)tr(φ))+ (Ricci 1s g),φ = Ricci ,φ .
24 R h i h R − 4 R i h R i
R ,φ g = Ricci ,φ = 1s g Ricci ,φ = 1s tr(φ) Ricci ,φ . QED.
h∗ ∗ • i h ∗R∗ i h2 R − R i 2 R −h R i
Proposition 4.3 If Q = R+φ g, where R and φ Sym(V), then
• ∈ B ∈
4π2 (Q) = 4π2 (R)+(sRtr(φ) Ricci ,φ +(tr(φ))2 φ 2)Vol (4.4)
R V
X X 2 −h i −k k
4π2p (Q) = 4π2p (R). (4.5)
1 1
Proof. From lemma 4.3
4π2 (Q),Vol = 1 Q ,Q = 1 R ,R + R ,φ g + 1 (φ g) ,φ g
hX Vi 2h∗ ∗ i 2h∗ ∗ i h∗ ∗ · i 2h∗ • ∗ • i
= 4π2 (R),Vol + sRtr(φ) Ricci ,φ +(tr(φ))2 φ 2.
hX Vi 2 −h R i −k k
Note that R = RT, Q = QT. Using decomposition (4.1)(4.2)(4.3) and lemmas 4.2, 4.1, we have
4π2 p (Q),Vol = Q, Q = R, R +2 R, φ g + φ g, φ g
1 V
h i h ∗ i h ∗ i h ∗ • i h • ∗ • i
= 4π2 p (R),Vol +2 , φ g = 4π2p (R)+2 ,φ g
1 V 1
h i hW ∗ • i h∗W • i
= 4π2 p (R),Vol +2 +,φ g 2 ,φ g
1 V −
h i hW • i− hW • i
= 4π2 p (R),Vol +2 +,(φ g)+ 2 ,(φ g)
h 1 Vi hW • +i− hW− • −−i
= 4π2 p (R),Vol +tr(φ)( +,Id ,Id )
h 1 Vi hW Λ+2 i−hW− Λ−2 i
= 4π2 p (R),Vol +tr(φ)(tr( +) tr( )).
1 V −
h i W − W