We shall now work out in detail the abstract spinor correspondence stated in Lemma 2.4 for the case that M carries a metric almost contact structure. The following result serves as a preparation.
Lemma 3.14. Given a metric almost contact structure with characteristic connection on M, the lift of η∧F to its cone M¯ is given by
1
a3r3(∂ryω)∧ω.
Proof. Since∂ry[a31r3(∂ryω)∧ω] = 0 we just need to show the equality onT M. ForX, Y ∈T M we have
F(X, Y) =g(X, ψY) =− 1
a2r2g(X, J Y¯ +η(Y)ar∂r) =− 1
a2r2ω(X, Y) and
η(X) =g(X, arJ ∂r) = 1
arω(X, ∂r) which provesF =−a21r2ω andη=−ar1∂ryω onT M.
We recall the definition of the connections
∇sXY = ∇gXY + 2sTc(X, Y) and ¯∇sXY = ¯∇¯gXY + 2sT¯(X, Y)
for s∈Rfrom the beginning of this chapter. Theorem 3.2 yieldsTc =T + 2aη∧F and since T¯=a2r2T andTc =a2r2Tc, we getTc−T¯ as the lift of 2a3r2η∧F to ¯M. With Lemma 3.14 we obtainTc−T¯=2r(∂ryω)∧ω.
Theorem 3.15. Assume that the almost contact metric manifold(M, g, ψ, η) admits a charac-teristic connection and is spin. Then there is for α= 12aorα=−12a:
1. A one to one correspondence between Killing spinors with torsion
∇sXϕ=αXϕ
onM and parallel spinors of the connection∇¯s+4sr(∂ryω)∧ω onM¯ with cone constanta
∇¯sXϕ+2s
r(Xy(∂ryω)∧ω)ϕ= 0,
2. A one to one correspondence between ∇¯s-parallel spinors on M¯ with cone constant aand spinors on M satisfying
∇sXϕ−2asXy(η∧F)ϕ=αXϕ.
In particular, fors=14 we get the correspondence
spinors onM spinors onM¯
∇cXϕ=αXϕ ∇¯Xϕ=−2r1Xy((∂ryω)∧ω)ϕ
∇cXϕ=αXϕ+a2Xy(η∧F)ϕ ∇¯Xϕ= 0
Remark 3.16. Assuming a Killing spinor with torsion on M, we get the correspondence to a spinor on ¯M as in case (1) of the Theorem. In particular this spinor is parallel in the cone direction.
Remark 3.17. Since ¯∇ = ¯∇¯g+ 12T¯ is the characteristic connection of the almost hermitian structure on ¯M, we can write
T¯ = N¯+dωJ,
wheredωJ=dω◦J. Thus one can rewrite all equations above. For example the correspondence (1) of Theorem 3.15 is given with spinors on ¯M satisfying
∇¯¯gXϕ+sXy[ ¯N+dωJ+2
r(∂ryω)∧ω]ϕ = 0.
Equivalently, one can use the description ofTconM given byTc =η∧dη+dFψ+N−η∧(ξyN) ([FI02]) to rewrite the second correspondence. Note that this also implies that ¯T = ¯N+dωJ is the lift of
a2r2T =a2r2(Tc−2aη∧F) =a2r2(η∧(dη−2aF) +dFψ+N−η∧(ξyN)) to ¯M, in particular we have∂ry( ¯N+dωJ) = 0.
3.3 Examples
In this Section, we shall discuss several examples of metric almost contact structures and the special spinor fields that exist on them and on their cones. In particular, we shall describe several situations where the cone carries a parallel spinor field for the characteristic connection ¯∇of its almost hermitian structure.
Example 3.18. For a metric almost contact manifold (M, g, ψ, η), the deformation gt:=tg+ (t2−t)η⊗η, ξt:= 1
tξ, ηt:=tη, t >0
is often used for different purposes and constructions (compare Example 1.10). Since Tanno used it in [Ta68] it is the so calledTanno deformation. It has the property that if the original manifold is K-contact or Sasaki, then the deformed manifold (M, gt, ξt, ηt, ψ) has again this property.
In [Be12, Cor.2.18] it was proved that almost any Sasakianη-Einstein manifold satisfying Ricg=λg−νη⊗η for someλ, ν∈R
carries Killing spinors with torsion. On an Einstein-Sasaki manifold (M, g, ψ, η) of dimension n= 2k+ 1>5 this spinors are constructed as follows. Consider the one dimensional subbundles of the spinor bundle Σtof (M, gt) defined by
L1(Σt) :={ϕ∈Σt|ψ(X)ϕ=−iXϕ∀X⊥ξ}, L2(Σt) :={ϕ∈Σt|ψ(X)ϕ=iXϕ∀X ⊥ξ}. Defineϵ=±1 to be the number satisfyinge1ψ(e1)...ekψ(ek)ξϕ=ϵik+1ϕfor a local orthonormal frame e1, ψ(e1), .., ek, ψ(ek), ξ on M. Theorem 2.22 from [Be12] then states that the spinors
ϕ1∈L1(Σt) andϕ2∈L2(Σt) are Killing spinors with torsion forst=4(kk+1−1)(1t−1) with Killing numbers
β1,t= ϵ 2
2kt−(k+ 1) t(k−1) = ϵ
2(1−4st) and β2,t= (−1)k+1β1,t (∗) respectively. Fort= 1, there is no deformation, and indeed the parameterstis then zero and the two spinors are just classical Riemannian Killing spinors. Since (M, gt, ξt, ηt, ψ) with fundamental 2-formFtis Sasakian, the characteristic torsion of∇cis given byTc=ηt∧dηt= 2ηt∧Ft. Thus, the Killing equation
∇gXtϕi+st(XyTc)ϕi = βi,tXϕi, i= 1,2 can equivalently be reformulated as
∇gXtϕi+1
4(XyTc)ϕi−(1−4st)1
4(XyTc)ϕi = βi,tXϕi.
If 1−4st = 0, both Killing numbers βi,t vanish by equation (∗) and the Killing equation is reduced to ∇cϕi = 0 – the spinor fields ϕi are ∇c-parallel and, as observed before, the cone construction is not possible. The condition 1−4st>0 is equivalent to t > k+12k and we observe that in this case, the last equation is exactly of the form treated in Theorem 3.15, case (2) for s= 1/4 and a= 2|βi,t|= 1−4st>0. Recall that we know from Theorem 3.12 that the cone ( ¯M ,¯gt) of the Tanno deformation is a locally conformally K¨ahler manifold (classχ4). Hence, we can conclude from Theorem 3.15, case (2):
Theorem 3.19. Let (M, g, ψ, η) be an Einstein Sasaki manifold of dimension 2k+ 1 > 5.
Consider its Tanno deformation(M, gt, ξt, ηt, ψ)fort > k+12k and the cone( ¯M ,¯gt, Jt), constructed with cone constanta= 1−4st>0, and endowed with the conformally K¨ahler structure described before. Then the two Killing spinors with torsion on(M, gt, ξt, ηt, ψ)induce each a spinor on the cone( ¯M ,g¯t, Jt)that is parallel with respect to its characteristic connection∇¯.
Although Killing spinors with torsion do exist on (M, gt, ξt, ηt, ψ) for 0< t < k+12k , Theorem 3.15, case (2) cannot be applied because the signs do not match. Of course, case (1) does still hold and therefore we obtain a spinor field satisfying a more complicated equation on ¯M. Fort= 1 (meaningst= 0), Theorem 3.19 is the classical cone correspondence between Riemannian Killing spinors on Einstein-Sasaki manifolds and Riemannian parallel spinors on their cone [B¨a93].
Example 3.20. The Heisenberg groupH is defined to be the following Lie subgroup of Gl(5,R):
H :=
1 u v z
0 1 0 x
0 0 1 y
0 0 0 1
|u, v, x, y, z∈R
.
The vector fieldsu1=∂u,u2=∂x+u∂z,u3=∂v,u4 =∂y+v∂z and u5 =∂z form a basis of the left invariant vector fields. Forρ >0 we consider the metric ([KV85])
g= 1
ρ(du2+dx2+dv2+dy2) + (dz−udx−vdy)2
and get an orthonormal framee1=√ρu1, e2=√ρu2,e3=√ρu3,e4=√ρu4 ande5=u5. We consider the almost contact structures given by
F1:=e1∧e2−e3∧e4 andF2:=e1∧e2+e3∧e4,
both with the sameξ:=e5. Becker-Bender calculates in [Be12] that the characteristic connection for both structures is given by its torsionTc =−ρe1∧e2∧e5−ρe3∧e4∧e5. She also proves thatϕ1and ϕ2, defined via the equations
ψ2(X)ϕ1=−iXϕ1 ∀X ⊥ξandψ2(X)ϕ2=iXϕ2∀X⊥ξ,
where ψj is the (1,1) tensor to the 2-form Fj forj = 1,2, are Killing spinors with torsion for s=−34 with Killing numberρand−ρrespectively:
∇−X34ϕ1=ρXϕ1 and∇−X34ϕ2=−ρXϕ2
for∇−34 :=∇g−32Tc.
On ¯M we get two almost hermitian structures, constructed from F1 and F2: We consider the orthonormal basis on ¯M given by Xi := ar1ei for i = 1..5 and X6 := ∂r. Then the almost hermitian structures on ¯M are given by
Ω1=−X1∧X2+X3∧X4+X5∧X6 and Ω2=−X1∧X2−X3∧X4+X5∧X6. We look at the corresponding characteristic connections ¯∇1and ¯∇2, coming from the connections
∇1and∇2with torsionsT1=Tc−2aη∧F1andT2=Tc−2aη∧F2onM, and thes-dependent connections ¯∇s,1 := ¯∇g+ 2sT¯1 and ¯∇s,2 := ¯∇g+ 2sT¯2. The equivalence of the characteristic connections forF1 andF2onM implies, that the connections ¯∇s,i+4sr(∂ryΩ)∧Ω are the same fori = 1,2. With Theorem 3.15 we get for both orientations on ¯M, constructed with a= 2ρ, the existence of a spinorϕsatisfying
∇¯−X34,iϕ− 3
2rXy((∂ryΩi)∧Ωi)ϕ= 0 fori= 1,2.
Thus we have two linear independent spinors on ¯Msatisfying this equation, which, due to Remark 3.17 is equivalent to
∇¯¯gXϕ−3
4Xy[ ¯Ni+dΩi+2
r(∂ryΩi)∧Ωi]ϕ= 0
where ¯Ni denotes the Nijenhuis tensor of the almost hermitian structure Ωi. We calculate the types of the structuresF1 andF2and with Theorem 3.12 we get immediately
Lemma 3.21. The structure F1 is of typeC7 and the structureF2 is of typeC6.
Thus the almost hermitian structure onM¯ induced byF1 is of mixed typeχ3⊕χ4 and the almost hermitian structure onM¯ induced byF2 is of typeχ4.
Proof. With the givenFiwe get in the basis defined aboveψie1=−e2fori= 1,2 andψ1e3=e4, ψ2e3=−e4; the other values ofψkej we get from the skew symmetry ofψk. With the equation (∇gXF2)(Y, Z) = 12Tc(X, ψ2Y, Z) +12Tc(X, Y, ψ2Z) and the fact thatTc=−ρe1∧e2∧e5−ρe3∧ e4∧e5=−ρF2∧η we get
δFk(ξ) =−∑
i
(∇geiFk)(ei, ξ) =−∑
i
1
2Tc(ei, ψkei, ξ)
=1 2ρ∑
i
(e1∧e2∧e5+e3∧e4∧e5)(ei, ψkei, ξ) =1 2ρ∑
i
(e1∧e2+e3∧e4)(ei, ψkei) and thus we haveδF1(ξ) = 0 and δF2(ξ) =−2ρ. We calculate
(∇gXFk)(Y, Z) = 1
2Tc(X, ψkY, Z) +1
2Tc(X, Y, ψkZ)
=−1
2ρ[F2∧η(X, ψkY, Z) +F2∧η(X, Y, ψkZ)]
=−1
2ρ[F2(X, ψkY)η(Z) +F2(ψkY, Z)η(X) +F2(Y, ψkZ)η(X) +F2(ψkZ, X)η(Y)].
Sinceψ1andψ2commute we haveF2(X, ψkY) =g(X, ψ2ψkY) =g(X, ψkψ2Y) =−g(ψkX, ψ2Y) =
−F2(ψkX, Y) and get
(∇gXFk)(Y, Z) =−1
2ρ[F2(X, ψkY)η(Z)−F2(X, ψkZ)η(Y)]. (II.6) The structureF1is of typeC7ifδF1(ξ) = 0 and
(∇gXF1)(Y, Z) =η(Z)(∇gYF1)(X, ξ)−η(Y)(∇gψ1XF1)(ψ1Z, ξ).
The right side equals 1
2[η(Z)Tc(Y, ψ1X, ξ)−η(Y)Tc(ψ1X, ψ21Z, ξ)] =−1
2ρ[η(Z)F2(Y, ψ1X) +η(Y)F2(ψ1X, Z)]
and due to the calculation above, this proves the first statement.
With the equation (II.6) we have (∇gXF2)(Y, Z) =−1
2ρ(g(X, ψ22Y)η(Z)−g(X, ψ22Z)η(Y)) =−δF2(ξ)
4 (g(X, Y)η(Z)−g(X, Z)η(Y)) which proves thatF2 is of typeC6.
Example 3.22. Another example (see [Be12]) is given by the homogeneous spaceM := SO(3)× SL(2,R)/SO(2) with the embedding
SO(2)∋A(t) :=
cost −sint sint cost
7→
[ A(t), A
(t 2
)−1] .
As an orthonormal basis of a reductive complement ofso(2) inso(3)×sl(2,R) we choose
e1 := D1
0 0 0
0 0 −1
0 1 0
,0
, e2:=D1
0 0 1
0 0 0
−1 0 0
,0
, e3:= 1 2D2
0,
1 0 0 −1
,
e4:=1 2D2
0,
0 1 1 0
, e5:=
c1
0 −1 0
1 0 0
0 0 0
, c2
1 2
0 −1
1 0
,
such thatc1+c2̸= 0,D21=c1(c1+c2),D22=−c2(c1+c2) and the numbersc1,−c2and (c1+c2) have the same signature. We consider the almost contact structure (M, ξ, F) defined via
ξ:=e5 andF =e1∧e2+e3∧e4.
Then the characteristic connection∇c has torsionTc =−c1e1∧e2∧e5−c2e3∧e4∧e5. Lemma 3.23. The almost contact structure(M, ξ, F)is normal. Furthermore, the almost her-mitian structure on M¯, constructed with a = −c14−c2, induced by the almost contact structure (M, ξ, F) is of classχ3 and thus the structure(M, ξ, F)is of mixed classC3⊕..⊕ C8.
Proof. We use Theorem 3.11 and prove that the almost contact structure (M, ξ, F) is normal, satisfies δF = (−c1−c2)η and never satisfies (∇gXF)(Y, Z) = aη(Y)g(X, Z)−aη(Z)g(X, Y) (Thus the structure on ¯M is never K¨ahler and for a=−c14−c2 it really is of typeχ3).
First, the definition ofF implies
ψ(e1) =−e2, ψ(e2) =e1, ψ(e3) =−e4, ψ(e4) =e3, ψ(e5) = 0
and thuse1(ψ(X)) =e2(X),e2(ψ(X)) =−e1(X),e3(ψ(X)) =e4(X),e4(ψ(X)) =−e3(X) and we get
Tc(X, ψY, ψZ) =η(X)(−c1e1∧e2(ψY, ψZ)−c2e3∧e4(ψY, ψZ))
=η(X)(−c1e1∧e2(Y, Z)−c2e3∧e4(Y, Z)).
This implies
Tc(X, Y, Z) = (−c1e1∧e2∧η−c2e3∧e4∧η)(X, Y, Z) =SX,Y,ZTc(X, ψY, ψZ), whereSdenotes the cyclic sum. We have
N(X, Y, Z) =(∇gXF)(Z, ψY)−(∇gYF)(Z, ψX) + (∇gψXF)(Z, Y)−(∇gψYF)(Z, X) +η(X)(∇gYF)(ξ, ψZ)−η(Y)(∇gXF)(ξ, ψZ).
Using the equality (∇gXF)(Y, Z) = 12(Tc(X, ψY, Z) +Tc(X, Y, ψZ)) we get N(X, Y, Z) =Tc(X, Y, Z)−SX,Y,ZTc(X, ψY, ψZ) = 0.
Secondly we calculate δF(X) =−∑
i
(∇geiF)(ei, X) =−1 2
∑
i
Tc(ei, ψei, X)
=−1
2[−c1e1∧e2∧e5(e1,−e2, X)−c1e1∧e2∧e5(e2, e1, X)
−c2e3∧e4∧e5(e3,−e4, X)−c2e3∧e4∧e5(e4, e3, X)
=η(X)(−c1−c2)].
Finally, to show that (M, ξ, F) never satisfies (∇gXF)(Y, Z) =aη(Y)g(X, Z)−aη(Z)g(X, Y) we calculate
(∇ge1F)(e1, ξ) =c1
2 andaη(e1)g(e1, ξ)−aη(ξ)g(e1, e1) =−a as well as
(∇ge3F)(e3, ξ) =c2
2 andaη(e3)g(e3, ξ)−aη(ξ)g(e3, e3) =−a.
Thus we can chooseasuch that the structure on ¯M is integrable ifc1=c2. But in the construc-tion ofM we needed the condition thatc1and−c2have the same signature and thatc1+c2̸= 0, which contradictsc1=c2.
In this example we only have Killing spinors with torsion satisfying∇sXϕ=αXϕforα= 0. But since the construction of ¯M explicitly depends on 2α=a̸= 0, we cannot lift these spinors to ¯M.