D O I: 1 0 .1 5 0 1 / C o m m u a 1 _ 0 0 0 0 0 0 0 7 4 8 IS S N 1 3 0 3 –5 9 9 1
ON SEVEN DIMENSIONAL 3-SASAKIAN MANIFOLDS
NÜL·IFER ÖZDEM ·IR
Abstract. 3-Sasakian manifolds in dimension seven have cocalibrated and nearly parallel G2-structures. In this work, cocalibrated G2-structure is
de-formed by one of the characteristic vector …elds of the 3-Sasakian structure and a new G2structure is obtained whose metric has negative scalar curvature. In
addition, the new G2 structure has a nonzero Killing vector …eld. Then, by
using this deformation, new covariant derivative on the spinor bundle is ob-tained and the new Dirac operator is written in terms of the Dirac operator before deformation.
1. INTRODUCTION
There exist several deformations of G2 structures to obtain new G2 structures.
Some of them are conformal deformations which are extensively studied in [1, 2]. Other types of deformations use vector …elds to get new G2 structures [2].
3-Sasakian manifolds are Einstein spaces of positive scalar curvature which have three compatible orthogonal Sasakian structures [3, 4]. Relations between the spectral properties of Dirac operator and seven dimensional 3-Sasakian manifolds are inves-tigated by [5, 6, 7]. It is shown that seven dimensional 3-Sasakian manifolds have a coclosed and nearly parallel G2-structures in [8]. In this work, to obtain a new
G2-structure from a …xed G2-structure, one of the characteristic vector …elds of the
3-Sasakian structure is used for changing the fundamental 3-form. 2. PRELIMINARIES
Let us consider R7with the standard basis fe
1; :::; e7g and dual basis fe1; :::; e7g.
The fundamental 3-form on R7 is de…ned as
! = e123+ e145+ e167+ e246 e257 e347 e256; where eijk= ei^ ej^ ek. The group G
2is
G2= fg 2 GL(R7)jg ! = !g: Received by the editors: Feb.16, 2016, Accepted: March 20, 2016. 2010 Mathematics Subject Classi…cation. 53C10, 53C25, 53C27.
Key words and phrases. 3-Sasakian manifold, Riemannian manifold with structure group G2. c 2 0 1 6 A n ka ra U n ive rsity C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra . S é rie s A 1 . M a th e m a t ic s a n d S t a tis t ic s .
Also the group G2is the automorphism group of octonions. The group G2is a
com-pact, connected, simply connected and simple Lie subgroup of SO(7) of dimension 14. A G2-structure on a 7-dimensional manifold M is a reduction of the
struc-ture group of the frame bundle of M from SO(7) to G2. Let M be a Riemannian
manifold with structure group G2. The classi…cation of such manifolds are done
by Fernández and Gray by decomposing r' into G2-irreducible components. It
turned out that there are 16 such classes [1].
Sasakian and 3-Sasakian manifolds are Riemannian manifolds with some addi-tional conditions. These manifolds are studied by [3, 4]. If a (2n + 1)-dimensional Riemannian manifold is equipped with a 1-form , its dual vector …eld and an endomorphism ' : T M2n+1! T M2n+1 such that the conditions
^ (d )36= 0; ( ) = 1; '2= Id + ; g('(X); '(Y )) = g(X; Y ) (X) (Y ); rX = '(X);
(rX') (Y ) = g(X; Y ) (Y ) X
are satis…ed, this manifold called a Sasakian manifold. In addition, if a (4n + 3)-dimensional Riemannian manifold (M4n+3; g) is equipped with three Sasakian structures ( i; i; 'i), i = 1; 2; 3, such that
[ 1; 2] = 2 3; [ 1; 2] = 2 3; [ 1; 2] = 2 3 and
'3 '2= '1+ 2 3; '2 '3= '1+ 3 2; '1 '3= '2+ 3 1; '3 '1= '2+ 1 3; '2 '1= '3+ 1 2; '1 '2= '3+ 2 1;
then this manifold is called a 3-Sasakian manifold. The subbundle which is spanned by 1, 2 and 3 is called vertical subbundle and orthogonal complement of the vertical subbundle is called horizontal subbundle. Both subbundles are invariant under the endomorphisms '1, '2 and '3. Sasakian manifolds are not necessarily Einstein. But 3-Sasakian manifolds are Einstein with Einstein constant 2(2n + 1). Let M be a seven dimensional 3-Sasakian manifold. Then, this manifold is spin [3]. Hence we construct a real spinor bundle on M which is an associated vector bundle
S := PSpin(7) 7;
where 7=R8 and : Spin(7) ! End( 7) is given by restriction of the real
Clif-ford algebra Cl7-representation to Spin(7) Cl7. Then, the covariant derivative
rS on the spinor bundle is expressed locally as rSV = d (V ) + 1 4 X i;j g(rVei; ej) (eiej)
where is a local spinor section and V is a vector …eld. The …rst-order di¤erential operator D : (S) ! (S) is de…ned as D := 7 X i=1 (ei)rSei
and is called the Dirac operator of S, where (S) is the set of spinor sections [9]. 3. VECTORIAL TYPE DEFORMATION OF RIEMANNIAN
MANIFOLDS WITH STRUCTURE GROUP G2
Some of the deformations of a …xed G2 structure are conformal deformations,
deformations of a G2structure by a vector …eld and in…nitesimal deformations [1, 2].
Conformal deformations are studied by Fernández and Gray. How G2 structures
change after conformaly changing the metric is investigated in [1]. In [10] and [11] the relation between Dirac operators on associated spinor bundles is studied.
Vectorial type of deformations are studied by Karigiannis in [2]. He specially worked on deforming the fundamental 3-form by a vector …eld and obtained a new metric from the 3-form: Let (M; !; g) be a 7-dimensional Riemannian manifold with structure group G2. If ! is deformed by a vector …eld the new 3-form
e
! = ! + y !
is always positive-de…nite. Under this deformation, Karigiannis has shown that, for all vector …elds X; Y the new metric is
eg(X; Y ) = 1 (1 + g( ; ))23
(g(X; Y ) + g(X ; Y )) ;
where is the cross product associated to the …rst G2-structure !. He has also
written the new Hodge stare in terms of the old !, the old and the vector …eld corresponding to e! explicitly:
e = (1 + g( ; ))23k( + ( 1)k 1 y( ( y )))
where is a k-form [2].
In this study, we consider this type of deformations. For a …xed vector …eld , …rst we observe the map
C : (T M ) ! (T M )
u 7 ! C (X) = (1 + g( ; )) 1=3(X + X ):
The map C is one-to-one and C1-linear which has the inverse
C 1(X) = (1 + g( ; )) 2=3(X X + g(X; ) ) for a vector …eld X, see [12]. The new metriceg can also be written as
for any vector …eld X; Y . The new cross product of the new G2-structure! is founde
in terms of old cross product as
XeY = k1=3C 1(X Y ) + k 1=3(g(X; )Y g(Y; )X) ;
where k = 1 + g( ; ) and X, Y are any vector …elds. If the vector …eld is Killing, then the new covariant derivative er of the metric eg determined by! is obtained ase
e
rXY = rXY +
1
2fg( ; Y )rX + g( ; X)ry g : Note that if is a parallel vector …eld on M , then er = r is obtained.
Deformation by a vector …eld of the canonical G2-structure on seven dimensional
3-Sasakian manifolds is studied by [12]. The canonical G2-structure is deformed by
one of the charecteristic vector …elds of the 3-Sasakian structure. It is shown that the new G2-structure! is in the largest class of Ge 2 structures.
Similarly, deformations by a vector …eld of the nearly parallel G2-structure on
seven dimensional 3-Sasakian manifolds is studied by [13]. When the nearly parallel G2-structure is deformed by a charecteristic vector …eld of the 3-Sasakian structure,
it is shown that the new G2-structuree! is in the class W1 W3or W1 W2 W4.
Let (M7; g) be a seven dimensional, compact, simply-connected 3-Sasakian
man-ifold with 3-Sasakian structure ( i; i; 'i) for = 1; 2; 3. One can get a local orthonor-mal frame fe1; ; e7g such that e1= 1, e2= 2, e3= 3 and the endomorphisms
'i act on Th:= span fe 4; e5; e6; e7g by matrices: '1:= 2 6 6 4 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 3 7 7 5 ; '2:= 2 6 6 4 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 3 7 7 5 ; '3:= 2 6 6 4 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 3 7 7 5 :
Let e1; ; e7 be the co-frame corresponding to fe
1; ; e7g. The di¤erentials
of 1-forms e1, e2 and e3 is given in [8] according to this orthonormal frame. The following 3-form on the 7-dimensional 3-Sasakian manifold
! := 3e1^ e2^ e3+1 2 e
1
^ de1+ e2^ de2+ e3^ de3 ;
is a G2-structure which is coclosed and this 3-form is called the canonical G2
struc-ture of the 7-dimensional 3-Sasakian manifold [8]. This G2-structure can be written
locally as
where eijk= ei^ ej^ ek. The vector cross products of basis elements are: e1 e2= e3; e1 e3= e2; e2 e3= e1; e1 e4= e5; e1 e5= e4; e4 e5= e1; e1 e6= e7; e1 e7= e6; e6 e7= e1; e2 e4= e6; e2 e6= e4; e4 e6= e2; e2 e5= e7; e2 e7= e5; e5 e7= e2; e3 e4= e7; e3 e7= e4; e7 e4= e3; e3 e5= e6; e3 e6= e5; e6 e5= e3:
We consider the new G2-structure
e ! = ! + 1y ! = 1 2 e 1 ^ de1+ e2^ de2+ e3^ de3 + 4e123 1 2e 3 ^ de2+1 2e 2 ^ de3; in local coordinates,! can be obtained ase
e
! = e123 e145 e167 e246+ e257 e347 e356 e357+ e346 e256 e247: Note that Ce11(e1); Ce11(e2); ; C
1
e1 (e7) is a local orthonormal frame with
re-spect to the new metric ~g. From now on, the notationeei := Ce11(ei) will be used.
In this frame, one can compute [8]:
i = 1; 2; 3; j = 4; 5; 6; 7; [ei; ej] = 0;
[e1; e2] = 2e3; [e1; e3] = 2e2; [e2; e3] = 2e1;
[e4; e5] = 2e1; [e4; e6] = 2e2; [e4; e7] = 2e3;
[e5; e6] = 2e3; [e5; e7] = 2e2; [e6; e7] = 2e1:
From the Kozsul formula; one can obtain
re1e2= e3; re1e3= e2; re1e4= e5; re1e5= e4; re1e6= e7;
re1e7= e6; re2e3= e1; re2e4= e6; re2e5= e7; re2e6= e4;
re2e7= e5; re3e4= e7re3e5= e6; re3e6= e5; re3e7= e4:
Now, coclosed (canonical) G2structure on a 7-dimensional 3-Sasakian manifold will
be deformed by one of the characteristic vector …elds of the 3-Sasakian structure and then scalar curvature of the new metric is evaluated:
Theorem 3.1. There exists a Riemannian manifold with structure group G2which
in the widest class of G2structures having a nonzero Killing vector …eld whose scalar
Proof. Let M be a seven dimensional 3-Sasakian manifold. We know that this manifold is equipped with a coclosed G2structure. If we deform this G2 structure
by the characteristic vector …eld 1, one can obtain a new G2 structure which in
the widest class [12]. First we show that e1 is a Killing vector …eld with respect to the new metric eg. Note that, since the deformation is done by the Killing vector …eld 1and r 1 1= 0,
e
rXe1= 2 2=3rX 1;
is obtained for any vector …eld X. By the identity g(rX 1; 1) = 0,
eg erXe1; Y = 2 2=3eg (rX 1; Y )
= 2 2=3g C 1(rX 1) ; C 1(Y )
= 2 1=3g (rX 1; Y ) :
Since 1 is a Killing vector …eld,
eg erXe1; Y = 2 2=3g (rX 1; Y ) = 2 2=3g (rY 1; X) = eg erYe1; X :
Next to obtain the scalar curvature; we directly calculate: e R(ee1;ee1) = 3 2 1=3; R(e ee2;ee2) = 7 2 1=3; R(e ee3;ee3) = 7 2 1=3; e R(ee4;ee4) = 5 2 1=3; R(e ee5;ee5) = 5 2 1=3; R(e ee6;ee6) = 5 2 1=3; and e R(ee7;ee7) = 5 2 1=3:
Hence the scalar curvature is obtained as e S = 7 X i=1 e R(eei;eei) = 3 2 1=3:
On the other hand, e2 and e3 are not Killing vector …elds with respect to the new metriceg.
4. DIRAC OPERATOR ON SEVEN DIMENSIONAL 3-SASAKIAN MANIFOLDS
Let (M7; g) be a seven dimensional, compact, simply-connected 3-Sasakian
man-ifold with 3-Sasakian structure ( i; i; 'i) for = 1; 2; 3. The structure group of the frame bundle reduces to the subgroup SU (2) SO(7). M7is also a spin manifold.
Then we can construct spinor bundle and write the covariant derivative and the Dirac operator of this spinor bundle. In addition, one can get a local orthonormal frame fe1; ; e7g as expressed before.
In local coordinates, for a vector …eld V , the Levi-Civita covariant derivative of the metric ~g can be expressed as
e rVeei = rVeei 1 2fg(V; e1)reeie1+ g(eei; e1)rVe1g = 2 2=3(rVei rV(ei e1)) 2 5=3g(V; e1) (reie1 rei e1e1) : Since Ce1 reVeei = 1 2(rVei+ (rVei) e1) 1 2(rV(ei e1) + (rV(ei e1)) e1) 1 4g(V; e1) [reie1+ (reie1) e1] +14g(V; e1) [rei e1e1+ (rei e1e1) e1] ; e
wij(V ) =eg( erVeei;eej) can be written as:
e wij(V ) = 12g(rVei; ej) +12g((rVei) e1; ej) 1 2g(rV(ei e1); ej) 1 2g ((rV(ei e1)) e1; ej) 1 4g(V; e1)g (reie1; ej) 1 4g(V; e1)g ((reie1) e1; ej) +14g(V; e1)g (rei e1e1; ej) + 1 4g(V; e1)g ((rei e1e1) e1; ej)
Hence we get the covariant derivative on the spinor bundle:
Lemma 4.1. Let f 1; ; 8g be a local frame of spinor sections. The covariant
derivative on the spinor bundle is obtained locally as rSV~e = e1 r S V 1 2 8 X k=1 xk e1 r S V k +1 4g(V; e1) e1( (e2e3+ e4e5+ e6e7) ) 1 8 e1 0 @X i;j g (rV(ei e1) (rVei) e1; ej) (eiej) 1 A 1 8 e1 0 @X i;j g ((rV(ei e1)) e1; ej) (eiej) 1 A ;
where !( ) =e, = x1 1+ + x8 8 is a local spinor section on the manifold
Then one can obtain the Dirac operator as e De = 7 X k=1 e(eek) rSeeeke = 1 2 7 X k=1 e(eek) e1 r S ~ ek +1 4 7 X k=1 e(eek) (g(~ek; e1) e1( (e2e3+ e4e5+ e6e7) )) 1 8 7 X k=1 e(eek) ( e1fg (re~k(ei e1) (re~kei) e1; ej) (eiej) g) 1 8 7 X k=1 e(eek) ( e1(g ((r~ek(ei e1)) e1; ej) (eiej) )) :
After tedious calculations, we obtain the following theorem:
Theorem 4.2. If a spinor section is in the kernel of D and satis…es the relation 2 f (e1)d (e1) + (e2)d (e3) (e3)d (e2)g (4.1)
+2 f (e4)d (e5) + (e5)d (e4) (e6)d (e7) + (e7)d (e6)g
= e2^ d 2+ e3^ d 3+21(e3^ d 2 e2^ d 3) + 6e123 ;
then the spinor section e1( ) is in the kernel of eD.
Proof. If we use the cross product of canonical G2 structure on 7-dimensional
3-Sasakian manifold which is presented before, the new Dirac operator can be written in terms of old Dirac operator in the following form:
e De = 2 2=3 e1fD g +2 2=3 e1( (e1)d (e1) + (e2)d (e3) (e3)d (e2)) +2 2=3 e1( (e4)d (e5) + (e5)d (e4) (e6)d (e7) + (e7)d (e6)) +2 5=3 e1 e2^ d 2+ e3^ d 3+ 1 2(e3^ d 2 e2^ d 3) + 6e123 : Hence we obtain the relation given in the theorem.
The relation in the theorem is important. If we use the following real Cli¤ord representation
(e1) = I 1 2; (e2) = I I 1; (e3) = 3 1 3;
(e4) = 2 1 3; (e5) = 1 I 3; (e6) = 1 3 2;
where 1= 0 1 1 0 ; 2= 1 0 0 1 ; 3= 0 1 1 0 ; I = 1 0 0 1 ;
then one can easily check that
= (cos(3x7); sin(3x7); 0; 0; 0; 0; 0; 0)
is in kernel of Dirac operator D, locally. Unfortunately, e1( ) is not in the kernel
of eD (that is, e1( ) does not satisfy the relation (4.1)). Hence we deduce that e1(KerD) does not lie in the kernel of eD.
It is known that after the deformation of a parallel G2 structure by a parallel
vector …eld, the kernels of Dirac operators on the new and old spinor bundles are isomorphic [14]. The change in kernels of Dirac operators on the spinor bundles of a coclosed G2structure after deformation by a vector …eld has not been investigated.
This study is an example of this type of deformations. References
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Current address : Department of Mathematics, Anadolu University, 26470 Eski¸sehir, Turkey E-mail address : nozdemir@anadolu.edu.tr
