Universit¨ at Regensburg Mathematik
Imaginary K¨ ahlerian killing spinors I
Nicolas Ginoux and Uwe Semmelmann
Preprint Nr. 11/2011
Imaginary K¨ ahlerian Killing spinors I
Nicolas Ginoux
∗and Uwe Semmelmann
†February 18, 2011
Abstract. We describe and to some extent characterize a new family of K¨ahler spin manifolds admitting non-trivial imaginary K¨ahlerian Killing spinors.
1 Introduction
Let (Mf2n, g, J) a K¨ahler manifold of real dimension 2n and with K¨ahler-formΩ defined bye Ω(X, Ye ) :=
g(J(X), Y) for all vectors X, Y ∈ TMf. We denote by p+ : T M −→ T1,0M, X 7→ 12(X −iJ(X)) and p−:T M −→T0,1M,X7→ 12(X+iJ(X)) the projection maps. In caseMf2nis spin, we denote its complex spinor bundle by ΣMf.
Definition 1.1 Let (Mf2n, g, J) a spin K¨ahler manifold and α∈C. A pair (ψ, φ)of sections of ΣMfis called an α-K¨ahlerian Killing spinor if and only if it satisfies, for everyX ∈Γ(TMf),
∇eXψ =−αp−(X)·φ
∇eXφ =−αp+(X)·ψ.
Anα-K¨ahlerian Killing spinor is said to be real(resp.imaginary) if and only ifα∈R(resp. α∈iR∗).
Ifα= 0, then anα-K¨ahlerian Killing spinor is nothing but a pair of parallel spinors. The classification of K¨ahler spin manifolds (resp. spin manifolds) admitting real non-parallel K¨ahlerian Killing (resp. parallel) spinors has been established by A. Moroianu in [12] (resp. by McK. Wang in [14]).
In this paper, we describe and partially classify those K¨ahler spin manifolds carrying non-trivial imaginary K¨ahlerian Killing spinors. Note first that there is no restriction in assumingα=i: obviously, changing (ψ, φ) into (ψ,−φ) changes αinto −α; moreover, (ψ, φ) is an α-K¨ahlerian Killing spinor on (Mf2n, g, J) if and only if it is an αλ-K¨ahlerian Killing spinor on (Mf2n, λ2g, J) for any constantλ >0.
K.-D. Kirchberg, who introduced this equation (see [9] for references), showed that, if a non-zero i- K¨ahlerian Killing spinor (ψ, φ) exists on (fM2n, g, J), then necessarily the complex dimension nof Mfis odd, the manifold (Mf2n, g) is Einstein with scalar curvature−4n(n+ 1), the pair (ψ, φ) vanishes nowhere and satisfiesΩe·ψ=−iψ as well asΩe·φ=iφ, see [9] and Proposition 2.1 below for further properties.
Moreover, he proved in the case n = 3 that the holomorphic sectional curvature must be constant [9, Thm. 16], in particular only the complex hyperbolic space CH3 occurs as simply-connected complete (Mf6, g, J) with non-trivial i-K¨ahlerian Killing spinors.
We extend Kirchberg’s results in several ways. First, we study in detail the critical points of the length function|ψ|ofψ. We show that, if the underlying Riemannian manifold (Mf2n, g) is connected and com- plete, then |ψ| has at most one critical value, which then has to be a (global) minimum and that the corresponding set of critical points is a K¨ahler totally geodesic submanifold (Proposition 2.3).
As a next step, we describe a whole family of examples of K¨ahler manifolds admitting non-trivial i- K¨ahlerian Killing spinors (Theorem 3.9), including the complex hyperbolic space and some K¨ahler mani- folds with non-constant holomorphic sectional curvature (Corollary 3.13). All arise as so-called doubly- warped products over Sasakian manifolds. A more detailed study of the induced spinor equation on that
∗Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, D-93040 Regensburg, E-mail:
nicolas.ginoux@mathematik.uni-regensburg.de
†Institut f¨ur Geometrie und Topologie, Universit¨at Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, E-mail:
uwe.semmelmann@mathematik.uni-stuttgart.de
Sasakian manifold allows the complex hyperbolic space to be characterized within the family (Theo- rem 3.18).
In the last section, we show that doubly-warped products are the only possible K¨ahler manifolds with non-triviali-K¨ahlerian Killing spinors as soon as both components of (ψ, φ) have the same length and are exchanged through the Clifford multiplication by a (real) vector field (Theorem 4.1). This shows an interesting analogy with H. Baum’s classification [3, 4] of complete Riemannian spin manifolds with imaginary Killing spinors.
Acknowledgment.This project benefited from the generous support of the universities of Hamburg, Potsdam, Cologne and Regensburg as well as the DFG-Sonderforschungsbereich 647. Special thanks are due to Christian B¨ar and Bernd Ammann. We also acknowledge very helpful discussions with Bogdan Alexandrov, Georges Habib and Daniel Huybrechts.
2 General integrability conditions
In this section we look for further necessary conditions for the existence of imaginary K¨ahlerian Killing spinors. Consider the vector fieldV onMfdefined by
g(V, X) :==m(hp+(X)·ψ, φi) (1)
for every vectorX onMf. We recall the following
Proposition 2.1 (see [9]) Let (ψ, φ) be an i-K¨ahlerian Killing spinor on (Mf2n, g, J) which does not vanish identically. Then the following properties hold:
i) grad(|ψ|2) = grad(|φ|2) = 2V. ii) For all vectorsX, Y ∈TMf,
g(∇eXV, Y) =<e(hp−(X)·φ, p−(Y)·φi+hp+(X)·ψ, p+(Y)·ψi). In particular,
Hess(|ψ|2)(X, Y) = Hess(|φ|2)(X, Y) = 2<e(hp−(X)·φ, p−(Y)·φi+hp+(X)·ψ, p+(Y)·ψi). iii) ∆(|ψ|2) = ∆(|φ|2) =−2(n+ 1)(|ψ|2+|φ|2), where∆ :=−trg(Hess).
iv) The vector field V is holomorphic, i.e., it satisfies: ∇eJ(X)V =J(∇eXV) for every X ∈ TMf. In particular, the vector field J(V)is Killing onMf.
v) grad(|V|2) = 2∇eVV.
Note that, from Proposition 2.1, the identity ∆(|ψ|2+|φ|2) =−4(n+ 1)(|ψ|2+|φ|2) holds onMf, therefore Mfcannot be compact.
Next we are interested in the critical points of|ψ|2 (or of |φ|2, they are the same by Proposition 2.1.i)).
We need a technical lemma:
Lemma 2.2 Under the hypotheses of Proposition 2.1, one has
∇eX∇eYV =∇e
∇eXYV +{2g(V, X)Y +g(V, Y)X−g(V, J(Y))J(X) +g(X, Y)V +g(J(X), Y)J(V)}
for all vector fieldsX, Y onMf. Therefore,
Hess(|V|2)(X, Y) = 2g(∇eXV,∇eYV) + 2 3g(X, V)g(Y, V) +|V|2g(X, Y)−g(X, J(V))g(Y, J(V)) .
Proof: Using Proposition 2.1, we compute in a local orthonormal basis{ej}1≤j≤2n ofTMf:
∇eX∇eYV =
2n
X
j=1
<e
hp−(∇eXY)·φ, p−(ej)·φi+hp+(∇eXY)·ψ, p+(ej)·ψi
+hp−(Y)·∇eXφ, p−(ej)·φi+hp−(Y)·φ, p−(ej)·∇eXφi
+hp+(Y)·∇eXψ, p+(ej)·ψi+hp+(Y)·ψ, p+(ej)·∇eXψi ej
=
2n
X
j=1
<e
hp−(∇eXY)·φ, p−(ej)·φi+hp+(∇eXY)·ψ, p+(ej)·ψi
−αhp−(Y)·p+(X)·ψ, p−(ej)·φi+αhp−(Y)·φ, p−(ej)·p+(X)·ψi
−αhp+(Y)·p−(X)·φ, p+(ej)·ψi+αhp+(Y)·ψ, p+(ej)·p−(X)·φi ej
= ∇e
∇eXYV +
2n
X
j=1
=m
hp−(Y)·p+(X)·ψ, p−(ej)·φi+hp+(Y)·p−(X)·φ, p+(ej)·ψi ej
−
2n
X
j=1
=m
hp−(Y)·φ, p−(ej)·p+(X)·ψi+hp+(Y)·ψ, p+(ej)·p−(X)·φi ej.
We compute the second line of the right-hand side of the preceding equation (the treatment of the third one is analogous). Usinghp+(X)·ψ, φi= 2ig(V, p+(X)), we obtain
hp+(Y)·p−(X)·φ, p+(ej)·ψi = hψ, p−(X)·p+(Y)·p−(ej)·φi+ 4ig(Y, p−(ej))g(V, p−(X)) +4ig(Y, p−(X))g(V, p−(ej)).
We deduce that, for everyj ∈ {1, . . . ,2n},
hp−(Y)·p+(X)·ψ, p−(ej)·φi+hp+(Y)·p−(X)·φ, p+(ej)·ψi = 2<e(hψ, p−(X)·p+(Y)·p−(ej)·φi) +4ig(Y, p−(ej))g(V, p−(X)) +4ig(Y, p−(X))g(V, p−(ej)).
The imaginary part of the right-hand side of the last equality is then given for everyj∈ {1, . . . ,2n}by 4<e(g(Y, p−(ej))g(V, p−(X)) +g(Y, p−(X))g(V, p−(ej))) = g(V, X)g(Y, ej) +g(V, J(X))g(J(Y), ej) +g(X, Y)g(V, ej) +g(J(X), Y)g(J(V), ej).
This shows that
2n
X
j=1
=m
hp−(Y)·p+(X)·ψ, p−(ej)·φi+hp+(Y)·p−(X)·φ, p+(ej)·ψi
ej = g(V, X)Y
+g(V, J(X))J(Y) +g(X, Y)V +g(J(X), Y)J(V).
Similarly, one shows that
2n
X
j=1
=m
hp−(Y)·φ, p−(ej)·p+(X)·ψi+hp+(Y)·ψ, p+(ej)·p−(X)·φi
ej = −g(V, Y)X +g(V, J(Y))J(X)
−g(V, X)Y +g(V, J(X))J(Y).
Combining the computations above, we obtain
∇eX∇eYV = ∇e
∇eXYV
+ (g(V, X)Y +g(V, J(X))J(Y) +g(X, Y)V +g(J(X), Y)J(V))
−(−g(V, Y)X+g(V, J(Y))J(X)−g(V, X)Y +g(V, J(X))J(Y))
= ∇e
∇eXYV
+ (2g(V, X)Y +g(V, Y)X−g(V, J(Y))J(X) +g(X, Y)V +g(J(X), Y)J(V)), which shows the first identity. We deduce for the Hessian of|V|2 that, for all vector fieldsX, Y onMf,
Hess(|V|2)(X, Y) = 2g(∇eX∇eVV, Y)
= 2g(∇e
∇eXVV, Y) + 2
2g(V, X)g(V, Y) +|V|2g(X, Y)−0 +g(X, V)g(V, Y) +g(J(X), V)g(J(V), Y)
= 2g(∇eXV,∇eYV) + 2 3g(X, V)g(Y, V) +|V|2g(X, Y)−g(X, J(V))g(Y, J(V)) ,
which is the second identity. This concludes the proof of Lemma 2.2.
We can now describe more precisely the set of critical values and points of|ψ|2and|V|2.
Proposition 2.3 Under the hypotheses ofProposition 2.1, assume furthermore(Mf2n, g)to be connected and complete. Then the following holds:
i) The set {V = 0} of zeros of V coincides with {∇eVV = 0}. As a consequence, the zeros of V are the only critical points of the function |V|2 on Mf2n.
ii) The subset{V = 0}is a (possibly empty) connected totally geodesic K¨ahler submanifold of complex dimension k < n in (Mf2n, g, J). Furthermore, for all x, y ∈ {V = 0}, every geodesic segment between xandy lies in{V = 0}.
iii) The function|ψ|2 has at most one critical value onMf2n, which is then a global minimum of|ψ|2. Furthermore, the set of critical points of|ψ|2 is a connected totally geodesic K¨ahler submanifold in (Mf2n, g, J).
Proof: The proof relies on simple computations and arguments.
i) Proposition 2.1.v) already implies that {∇eVV = 0} coincides with the set of critical points of |V|2. Every zero ofV is obviously a zero of∇eVV, i.e., a critical point of|V|2. Conversely, letx∈ {∇eVV = 0}.
Then 0 =gx(∇eVV, V) =|p−(Vx)·φ|2+|p+(Vx)·ψ|2, so thatp−(Vx)·φ= 0 andp+(Vx)·ψ= 0, which, in turn, implies 0 ==m(hp+(Vx)·ψ, φi) =g(Vx, Vx), that is,Vx= 0. This showsi).
ii) The subset {V = 0} - if non-empty - is the fixed-point-set in Mf2n of the flow of the holomorphic Killing fieldJ(V), therefore it is a totally geodesic K¨ahler submanifold ofMf2n (see e.g. [10, Sec. II.5]);
moreover, it cannot contain any open subset of Mf2n since otherwise V would identically vanish as a holomorphic vector field. To show the connectedness of {V = 0}, it suffices to prove the second part of the statement. Pick any two points x0, x1 in {V = 0} (or, equivalently, any critical points of |V|2) and any geodesiccin (Mf2n, g) withc(0) =x0andc(1) =x1. Consider the real-valued functionf(t) :=|V|2c(t) defined onR. Then, for any t∈Rone hasf0(t) =g(grad(|V|2), c0(t)) = 2g(∇ec0(t)V, V) and
f00(t) = Hess(|V|2)(c0(t), c0(t)).
Lemma 2.2 provides the Hessian of|V|2: for everyX ∈TMf,
Hess(|V|2)(X, X) = 2|∇eXV|2+ 2 3g(V, X)2+|V|2|X|2−g(X, J(V))2 .
By Cauchy-Schwarz inequality, |V|2|X|2−g(X, J(V))2 ≥0, so that Hess(|V|2)(X, X)≥0 for all X, in particular f is convex. This in turn implies that, if f0(0) = f0(1) = 0, then necessarily f vanishes on [0,1]. This provesii).
iii) Set, for any t∈R,h(t) :=|ψ|2c(t) where cis an arbitrary geodesic on (Mf2n, g). We show again that
h is convex. As before h00(t) = Hess(|ψ|2)(c0(t), c0(t)) ≥ 0 for every t ∈ R, where Hess(|ψ|2)(X, X) = 2(|p−(X)·φ|2+|p+(X)·ψ|2) ≥ 0 for every X ∈ TMf (Proposition 2.1). We already know that, if V = 12grad(|ψ|2) vanishes at two different points of c, then it vanishes on any geodesic segment joining the two points, therefore|ψ|2 is constant on it. This proves that|ψ|2has at most one critical value. Since his convex this critical value is necessarily a minimum. The last part of the statement is a straightforward consequence ofii) since grad(|ψ|2) = 2V by Proposition 2.1. This shows iii) and concludes the proof.
3 Doubly warped products with imaginary K¨ ahlerian Killing spinors
In this section, we describe the so-called doubly-warped products carrying non-zero imaginary K¨ahlerian Killing spinors. Doubly warped products were introduced in the spinorial context by Patrick Baier in his master thesis [1] to compute the Dirac spectrum of the complex hyperbolic space, using its representation as a doubly-warped product over an odd-dimensional sphere.
First we recall general formulas on warped products.
Lemma 3.1 Let (fM :=M ×I,eg :=gt⊕βdt2) be a warped product, where I ⊂R is an open interval, gt is a smooth 1-parameter family of Riemannian metrics on M and β ∈ C∞(M ×I,R×+). Denote by Mf−→π1 M the first projection. Then, for all X, Y ∈Γ(π∗1T M),
∇e ∂
∂t
∂
∂t = −1 2gradg
t(β(t,·)) + 1 2β
∂β
∂t
∂
∂t
∇e ∂
∂tX = ∂X
∂t +1 2g−1t ∂gt
∂t(X,·) + 1 2β
∂β
∂x(X)∂
∂t
∇eX
∂
∂t = 1 2g−1t ∂gt
∂t (X,·) + 1 2β
∂β
∂x(X)∂
∂t
∇eXY = ∇MXY − 1 2β
∂gt
∂t(X, Y)∂
∂t,
where ∂X∂t = [∂t∂, X]and∇M (resp.∇) is the Levi-Civita covariant derivative ofe (M, gt)(resp. of(fM ,eg)).
Proof: straightforward consequence of the Koszul identity.
From now on we restrict ourselves to the following particular case: the manifoldM will be equipped with aRiemannian flow.
Definition 3.2
i) A Riemannian flow is a triple (M,g,b ξ), whereb M is a smooth manifold and ξbis a smooth unit vector field whose flow is isometric on the orthogonal distribution, i.e., bg(∇bMZξ, Zb 0) =−bg(Z,∇bMZ0ξ)b for all Z, Z0∈ξb⊥, where∇bM denotes the Levi-Civita covariant derivative of(M,bg).
ii) A Riemannian flow (M,bg,ξ)b is called minimal if and only if ∇bM
ξbξb= 0, that is, if ξbis actually a Killing vector field on M.
Let (M,bg,ξ) be a minimal Riemannian flow. Letb bh denote the endomorphism-field of ξb⊥ defined by bh(Z) :=∇bMZ ξbfor everyZ ∈ξb⊥. Let∇b be the covariant derivative onξb⊥ defined for all Z ∈ Γ(bξ⊥) by
∇bXZ :=
(
[ξ, Zb ]ξb⊥ ifX =ξb
(∇bMXZ)ξb⊥ ifX ⊥ξb . Alternatively,∇b can be described by the following formulas: for all Z, Z0 ∈Γ(bξ⊥),
∇bM
bξ Z=∇b
bξZ+bh(Z) and ∇bMZ Z0=∇bZZ0−bg(bh(Z), Z0)bξ.
It is important to notice that, if (M,bg,ξ) is a (minimal) Riemannian flow andb g := r2(s2bg
ξb⊕bg
ξb⊥) for some constantsr, s >0, then (M, g, ξ:= rs1ξ) is a (minimal) Riemannian flow with corresponding objectsb given by
h=s
rbh and ∇=∇.b (2)
In this language, aSasakian manifold is a minimal Riemannian flow (M,bg,ξ) such thatb bhis a transver- sal K¨ahler structure, that is, bh2 = −Id
ξb⊥ and ∇bbh = 0. Further on in the text we shall need for normalization purposes so-called D-homothetic deformations of a Sasakian structure: a D-homothetic deformation of (M,bg,ξ) is (M, λb 2(λ2bg
ξb⊕bg
bξ⊥),λ12ξ) for someb λ ∈ R×+. The identities (2) imply that (M, λ2(λ2gb
bξ⊕bg
ξb⊥),λ12ξ) is Sasakian as soon as (M,b bg,ξ) is Sasakian.b We can now make the concept of doubly-warped product precise:
Definition 3.3 A doubly-warped productis a warped product of the form (fM ,eg) := (M×I, ρ(t)2(σ(t)2bg
ξb⊕bg
ξb⊥)⊕dt2),
whereIis an open interval,(M,bg,ξ)b is a minimal Riemannian flow,ρ, σ:I−→R×+are smooth functions andbg
ξb:=bg|
Rξ⊕Rb ξb,bg
ξb⊥ :=bg|
ξ⊥ ⊕bb ξ⊥.
As for warped products, it can be easily proved that a doubly-warped product (M ,fg) is complete as soone asI=Rand (M,bg) is complete.
It is easy to check that, setting gt :=ρ(t)2(σ(t)2bg
ξb⊕bg
ξb⊥), one has ∂g∂tt = 2ρρ0gt+2σσ0gt(π
ξb⊥,·) and the unit vector field providing the Riemannian flow on (M, gt) is ξ = ρσ1 ξ. In particular, the formulas inb Lemma 3.1 simplify:
∇e ∂
∂t
∂
∂t = 0
∇e ∂
∂tξ = 0
∇e ∂
∂tZ = ∂Z
∂t +ρ0 ρZ
∇eξ
∂
∂t = (ρσ)0 ρσ ξ
∇eξξ = −(ρσ)0 ρσ
∂
∂t
∇eξZ = ∇ξZ+h(Z)
∇eZ
∂
∂t = ρ0 ρZ
∇eZξ = h(Z)
∇eZZ0 = ∇ZZ0−gt(h(Z), Z0)ξ−ρ0
ρgt(Z, Z0)∂
∂t, where we have denoted the corresponding objects on (M, gt, ξ) without the hat “b·”.
Next we look at a possible construction of K¨ahler structures on doubly-warped products.
Lemma 3.4 Let (M ,f eg) := (M ×I, ρ(t)2(σ(t)2bg
ξb⊕bg
bξ⊥)⊕dt2) be a doubly-warped product. Assume the existence of a transversal K¨ahler structure J on (M,bg,ξ)b and define the almost complex structure Jeon MfbyJe(ξ) := ∂t∂,Je(∂t∂) :=−ξandJe(Z) :=J(Z)for allZ ∈ {ξ,∂t∂}⊥. Then(fM2n,g,e J)e is K¨ahler if and only ifbh=−ρσ0J on{ξ,∂t∂}⊥ (in particular ρσ0 must be constant).
Proof: Using the identities above we write down the condition∇eJe= 0. Denote by hand∇ the objects corresponding to gt on M. Note first that, by definition and (2), one has ∇J = 0 on {ξ,∂t∂}⊥ and Je|
{ξ, ∂∂t}⊥ =J, which does not depend ont. Hence we obtain, for allZ, Z0∈Γ(bξ⊥):
∇e ∂
∂t(Je(∂
∂t))−Je(∇e ∂
∂t
∂
∂t) = 0
∇e ∂
∂t(Je(ξ))−J(e∇e ∂
∂tξ) = 0
∇e ∂
∂t(Je(Z))−J(e∇e ∂
∂tZ) = ∂(Je(Z))
∂t −J(e ∂Z
∂t) =∂(J(Z))
∂t −J(∂Z
∂t) = 0
∇eξ(Je(∂
∂t))−Je(∇eξ
∂
∂t) = 0
∇eξ(Je(ξ))−J(e∇eξξ) = 0
∇eξ(Je(Z))−J(e∇eξZ) = h◦J(Z)−J◦h(Z)
∇eZ(Je(∂
∂t))−Je(∇eZ
∂
∂t) = −h(Z)−ρ0 ρJ(Z)
∇eZ(Je(ξ))−J(e∇eZξ) = ρ0
ρZ−J◦h(Z)
∇eZ(Je(Z0))−Je(∇eZZ0) = −gt(h(Z), J(Z0))ξ−ρ0
ρgt(Z, J(Z0))∂
∂t+gt(h(Z), Z0)∂
∂t−ρ0
ρgt(Z, Z0)ξ.
Therefore,∇eJe= 0 impliesh=−ρρ0J onξ⊥ which, in turn, impliesh◦J =J◦h. Moreover, (2) implies thath= σρbh, which yieldsbh=−ρσ0J. The reverse implication is obvious.
Remarks 3.5
1. With the assumptions of Lemma 3.4, the function ρ0 vanishes either identically or nowhere on the interval I. In the former case the vanishing ofbhis equivalent to M being locally the Riemannian product of an interval with a K¨ahler manifold; in the latter one, we may assume, up to changingσ into |ρσ0|σ(andbginto (ρσ0)2bg
ξb⊕bg
ξb⊥), thatbh=−εJ andρ0 =εσwithε∈ {±1}.
2. Given a K¨ahler doubly warped product (M ,f eg,Je) as in Lemma 3.4 and a real constantC, the map (x, t)7→(x,±t+C) provides a holomorphic isometry (fM ,eg,Je)−→(Mf0,eg0,Je0), where (Mf0,eg0) :=
(M×(C±I), g±t+C⊕dt2) andJe0is the corresponding complex structure (again as in Lemma 3.4).
If furthermoreM is spin, then this isometry preserves the corresponding spin structures. Thus, in the case whereρ06= 0, we may assume thatε= 1, i.e., thatbh=−J andρ0=σ.
Now we examine the correspondence of spinors. Let the underlying manifoldM of some minimal Rieman- nian flow (M, g, ξ) be spin and, in case M is the total space of a Riemannian submersion withS1-fibres over a spin manifoldN, letM carry the spin structure induced by that ofN. Let ΣM denote the spinor bundle of (M, g) and “·
M” its Clifford multiplication. Let the doubly warped productMfcarry the product spin structure (with Clifford multiplication denoted by “·”). Then the transversal covariant derivative∇ induces a covariant derivative - also denoted by∇ - on ΣM, which is related to the spinorial Levi-Civita covariant derivative∇M on ΣM via (see e.g. [7, eq. (2.4.7)] or [8, Sec. 4])
∇Mξ ϕ=∇ξϕ+1 4
2n−2
X
j=1
ej ·
Mh(ej) ·
M ϕ and ∇MZϕ=∇Zϕ+1 2ξ ·
M h(Z) ·
Mϕ
for everyϕ∈Γ(ΣM), where {ej}1≤j≤2n−2is a local orthonormal basis ofξ⊥⊂T M.
Lemma 3.6 Let a minimal Riemannian flow (M,bg,ξ)b carry a transversal K¨ahler structure J such that the doubly-warped product (M ,f eg,Je)is K¨ahler, where Jeis the almost-complex structure induced byJ as in Lemma 3.4. Assume furthermore M to be spin. Let Mf carry the induced spin structure. Then the following identities hold for all ϕ∈Γ(ΣfM)andZ∈ {ξ,∂t∂}⊥:
∇e ∂
∂tϕ = ∂ϕ
∂t
∇eξϕ = ∇ξϕ− ρ0
2ρΩe·ϕ− σ0 2σξ· ∂
∂t·ϕ
∇eZϕ = ∇Zϕ− ρ0
2ρ(ξ·J(Z) +Z· ∂
∂t)·ϕ, whereΩe denotes the K¨ahler form of (fM ,eg,J).e
Proof: Let (e1, . . . , e2n−2, e2n−1 :=ξ, e2n := ∂t∂) be a local positively-oriented orthonormal basis ofTMf and (ψα)αthe corresponding spinorial frame. It can be assumed thatej=ρ−1ebjwithbg(ebj,ebk) =δjkand
∂ebj
∂t = 0 (extend somebg-orthonormal basis independently of time). Splitϕ=P
αcαψα, then
∇e ∂
∂tϕ = 1 4
X
α
cα 2n
X
j,k=1
eg(∇e ∂
∂tej, ek)ej·ek·ψα+X
α
∂cα
∂t ψα
| {z }
=:∂ϕ∂t
= ∂ϕ
∂t +1 4
X
α
cα 2n−2
X
j,k=1
eg(∇e ∂
∂tej, ek)ej·ek·ψα
= ∂ϕ
∂t +1 4
X
α
cα
2n−2
X
j,k=1
{gt(∂ej
∂t , ek) +ρ0
ρδjk}ej·ek·ψα
= ∂ϕ
∂t, where we have used ∇e ∂
∂t
∂
∂t = ∇e ∂
∂tξ = 0 and ∂e∂tj = −ρρ0ej by the above choice of ej. On the other hand, the Weingarten endomorphism field of (M, gt) in Mfis given by A(ξ) := −e∇ξ∂
∂t = −(ρσ)ρσ0ξ and A(Z) :=−∇eZ ∂
∂t=−ρρ0Z for allZ ∈ {ξ,∂t∂}⊥, so that the Gauss-Weingarten formula implies
∇eξϕ = ∇Mξ ϕ+1
2A(ξ)· ∂
∂t ·ϕ
= ∇ξϕ+1 4
2n−2
X
j=1
ej ·
M h(ej) ·
M ϕ−(ρσ)0 2ρσ ξ· ∂
∂t·ϕ
= ∇ξϕ− ρ0 4ρ
2n−2
X
j=1
ej·J(ej)·ϕ−(ρσ)0 2ρσ ξ· ∂
∂t·ϕ
= ∇ξϕ− ρ0
2ρΩ·ϕ−(ρσ)0 2ρσ ξ· ∂
∂t ·ϕ,
where Ω is the 2-form associated toJ on{ξ,∂t∂}⊥, i.e., Ω(Z, Z0) =gt(J(Z), Z0) for allZ, Z0 ∈ {ξ,∂t∂}⊥. SinceΩ = Ω +e ξ∧∂t∂, we deduce that
∇eξϕ = ∇ξϕ− ρ0
2ρΩe·ϕ+ (ρ0
2ρ−(ρσ)0 2ρσ )ξ· ∂
∂t·ϕ
= ∇ξϕ− ρ0
2ρΩe·ϕ− σ0 2σξ· ∂
∂t ·ϕ.
For anyZ∈ {ξ,∂t∂}⊥, one has
∇eZϕ = ∇MZϕ+1
2A(Z)· ∂
∂t·ϕ
= ∇Zϕ+1 2ξ ·
M h(Z) ·
M ϕ− ρ0 2ρZ· ∂
∂t·ϕ
= ∇Zϕ− ρ0
2ρξ·J(Z)·ϕ− ρ0 2ρZ· ∂
∂t ·ϕ,
which shows the last identity and concludes the proof.
Later on we shall need to split spinors into different components. Recall that, on any K¨ahler spin manifold (Mf2n,eg,Je), the spinor bundle ΣMfof (Mf2n,eg) splits under the Clifford action of the K¨ahler formΩ intoe
ΣMf=
n
M
r=0
ΣrM ,f
where ΣrMf := Ker(eΩ· −i(2r−n)Id). Now if (Mf2n,eg,Je) is a doubly-warped product as above, then any ϕ ∈ ΣrMf(with r ∈ {0,1, . . . , n}) can be further split into eigenvectors for the Clifford action of
Ω =g(J·,·). Namely, since [ξ∧ ∂t∂,Ω] = 0, the automorphismξ·∂t∂ of ΣMfleaves ΣrMfinvariant; from (ξ· ∂t∂)2 =−1 one deduces the orthogonal decomposition ΣrMf= Ker(ξ· ∂t∂ +iId)⊕Ker(ξ· ∂t∂ −iId).
Since both Clifford actions of ξand ∂t∂ are ∇-parallel, so is the latter splitting. But, for anyϕ∈ΣrMf, one has
ϕ∈Ker(ξ· ∂
∂t ±iId) ⇐⇒ Ω·ϕ=i(2r−n)ϕ±iϕ
⇐⇒ Ω·ϕ=i(2r−n±1)ϕ,
that is, ΣrMf∩Ker(ξ· ∂t∂ +iId) = ΣrM and ΣrMf∩Ker(ξ· ∂t∂ −iId) = Σr−1M, where by definition ΣrM := Ker(Ω· −i(2r−(n−1)Id)) forr∈ {0,1, . . . , n−1}and{0}otherwise. Out of dimensional reasons one actually has
ΣrMf= ΣrM⊕Σr−1M (3)
for everyr∈ {0,1, . . . , n}. Beware here that, ifris even, then ΣrMfis a subspace of Σ+Mfhence ΣrMf|M is canonically identified with a subspace of Σ+Mf|M = ΣM, whereas ifr is odd then it is a subspace of Σ−Mfand is also identified as a subspace of ΣM, but this time with opposite Clifford multiplication.
Lemma 3.7 Under the hypotheses ofLemma 3.6, letϕ∈Γ(ΣrMf)for somer∈ {0,1. . . , n}and consider its decompositionϕ=ϕr+ϕr−1 w.r.t. (3). Then the identities ofLemma 3.6 read:
∇e ∂
∂tϕr = ∂ϕr
∂t
∇e ∂
∂tϕr−1 = ∂ϕr−1
∂t
∇eξϕr = ∇ξϕr+ i
2((n−2r)ρ0 ρ +σ0
σ)ϕr
∇eξϕr−1 = ∇ξϕr−1+i
2((n−2r)ρ0 ρ −σ0
σ)ϕr−1
∇eZϕ = ∇Zϕr−ρ0
ρp+(Z)· ∂
∂t·ϕr−1+∇Zϕr−1−ρ0
ρp−(Z)· ∂
∂t·ϕr
for allZ ∈ {ξ,∂t∂}⊥, where, as usual,p±(Z) =12(Z∓iJ(Z)).
Proof: The first two identities follow from ∇e ∂
∂t(ξ∧ ∂t∂) = 0 and ∂J∂t = 0. For the third and fourth ones, note that∇eξ(ξ∧∂t∂) = 0, so that
∇eξϕr+∇eξϕr−1 = ∇ξϕr+∇ξϕr−1−iρ0
2ρ(2r−n)(ϕr+ϕr−1)−iσ0
2σ(ϕr−1−ϕr)
= ∇ξϕr+ i
2((n−2r)ρ0 ρ +σ0
σ)ϕr+∇ξϕr−1+ i
2((n−2r)ρ0 ρ −σ0
σ)ϕr−1, which is the result. As for the last identity, one does not have∇eZ(ξ∧∂t∂) = 0, however
(ξ·J(Z) +Z· ∂
∂t)·ϕ = (−J(Z)· ∂
∂t ·ξ· ∂
∂t+Z· ∂
∂t)·ϕ
= −iJ(Z)· ∂
∂t·(ϕr−1−ϕr) +Z· ∂
∂t ·(ϕr+ϕr−1)
= 2p+(Z)· ∂
∂t·ϕr−1+ 2p−(Z)· ∂
∂t ·ϕr
for allZ∈ {ξ,∂t∂}⊥. This concludes the proof.
We now have all we need to rewrite the imaginary K¨ahler Killing spinor equation on doubly warped products.
Lemma 3.8 Let a spin minimal Riemannian flow (M2n−1,bg,ξ)b carry a transversal K¨ahler structure J such that the doubly-warped product(fM ,eg,Je)is K¨ahler, whereJeis the almost-complex structure induced byJ as inLemma 3.4. LetMfcarry the induced spin structure and assumen≥3to be odd. Then a pair
(ψ, φ)is an i-K¨ahlerian Killing spinor on(Mf2n,eg,Je) if and only if the following identities are satisfied by the componentsφ=φn+1
2 +φn−1
2 andψ=ψn−1
2 +ψn−3
2 w.r.t. (3):
∂φn+1 2
∂t = 0
∂φn−1 2
∂t =−i∂t∂ ·ψn−1
∂ψn−1 2 2
∂t =−i∂t∂ ·φn−1
∂ψn−3 2 2
∂t = 0
∇ξφn+1 2
= 2i(ρρ0 −σσ0)φn+1 2
∇ξφn−1
2 = 2i(ρρ0 +σσ0)φn−1
2 −∂t∂ ·ψn−1
2
∇ξψn−1
2 =−2i(ρρ0 +σσ0)ψn−1
2 +∂t∂ ·φn−1 2
∇ξψn−3
2 =−2i(ρρ0 −σσ0)ψn−3
2
∇Zφn+1
2 =p+(Z)·(ρρ0∂t∂ ·φn−1
2 −iψn−1 2 )
∇Zφn−1 2
= ρρ0p−(Z)·∂t∂ ·φn+1
2 −ip+(Z)·ψn−3 2
∇Zψn−1
2 = ρρ0p+(Z)·∂t∂ ·ψn−3
2 −ip−(Z)·φn+1 2
∇Zψn−3
2 =p−(Z)·(ρρ0∂t∂ ·ψn−1
2 −iφn−1 2 )
(4)
for everyZ ∈ {ξ,∂t∂}⊥.
Proof: Sincep+(∂t∂)·ψ= 12(∂t∂ +iξ)·ψ=12∂t∂ ·(1+iξ·∂t∂·)ψ= ∂t∂ ·ψn−1
2 and similarlyp−(∂t∂)·φ=∂t∂ ·φn−1
2 , thei-K¨ahlerian Killing spinor equation is satisfied by (ψ, φ) forX = ∂t∂ if and only if
∂φn+1 2
∂t +
∂φn−1 2
∂t = −ip+(∂
∂t)·ψ=−i∂
∂t ·ψn−1 2
∂ψn−1 2
∂t +
∂ψn−3 2
∂t = −ip−(∂
∂t)·φ=−i∂
∂t ·φn−1
2 , which gives the first four identities (use [Ω,∂t∂] = 0).
Fromp+(ξ)·ψ=−ip+(∂t∂)·ψ=−i∂t∂ ·ψn−1
2 andp−(ξ)·φ=ip−(∂t∂)·φ=i∂t∂ ·φn−1
2 we deduce that the i-K¨ahlerian Killing spinor equation is satisfied by (ψ, φ) forX=ξif and only if
∇ξφn+1 2 + i
2(−ρ0 ρ +σ0
σ)φn+1
2 = 0
∇ξφn−1 2 − i
2(ρ0 ρ +σ0
σ)φn−1 2
= −∂
∂t·ψn−1 2
∇ξψn−1 2 + i
2(ρ0 ρ +σ0
σ)ψn−1
2 = ∂
∂t ·φn−1 2
∇ξψn−3 2
+ i 2(ρ0
ρ −σ0 σ)ψn−3
2
= 0, which implies the next four equations.
Let Z ∈ {ξ,∂t∂}⊥, then thei-K¨ahlerian Killing spinor equation is satisfied by (ψ, φ) for X =Z if and only if
−ip+(Z)·ψn−1
2 = ∇Zφn+1 2 −ρ0
ρp+(Z)· ∂
∂t·φn−1 2
−ip+(Z)·ψn−3
2 = ∇Zφn−1
2 −ρ0
ρp−(Z)· ∂
∂t ·φn+1 2
−ip−(Z)·φn+1
2 = ∇Zψn−1 2 −ρ0
ρp+(Z)· ∂
∂t·ψn−3 2
−ip−(Z)·φn−1
2 = ∇Zψn−3
2 −ρ0
ρp−(Z)· ∂
∂t·φn−1
2 ,
which concludes the proof.
Next we want to describe all doubly warped products with non-zero imaginary K¨ahlerian Killing spinors.
Theorem 3.9 For n≥3 odd let (fM2n,g,e Je)be a K¨ahler spin doubly warped product as in Lemma 3.8.
If there exists a non-zeroi-K¨ahlerian Killing spinor(ψ, φ)on(Mf2n,eg,Je), then
• the minimal Riemannian flow(M2n−1,bg,ξ)b is Sasakian,
• up to changingt into−t, applying a D-homothety and translating the intervalI by a constant, one has either ρ=et orρ= sinh orρ= cosh,
• the components ψr andφr of (ψ, φ)w.r.t.(3)satisfy:
i) In case ρ=et: Then σ=et and, settingψen−3
2 :=i∂t∂ ·ψn−3
2 andϕn−1
2 :=et(φn−1
2 +i∂t∂ ·ψn−1
2 ), one has
∂
∂tφn+1
2 = 0
∂
∂tψen−3 2
= 0
∂
∂tϕn−1
2 = 0
∇b
ξbφn+1
2 = 0
∇b
ξbψen−3
2 = 0
∇ϕb n−1
2 = 0
∇bZφn+1
2 = (−1)n+12 p+(Z)b·
Mϕn−1
2
∇bZψen−3
2 = (−1)n+12 p−(Z)b·
Mϕn−1
2 . If furthermoreϕn−1
2
= 0, then forφbn−1 2
:=e−tφn−1 2
one has ∂t∂φbn−1 2
= 0and
∇φb n+1
2 = 0
∇bψen−3
2 = 0
∇b
bξφbn−1 2
= 0
∇bZφbn−1
2 = (−1)n+12 (p−(Z)b·
Mφn+1
2 +p+(Z)b·
Mψen−3
2 ).
In particular, the manifold(M2n−1,bg,ξ)b admits a non-zero transversally parallel spinor. Conversely, every non-zero transversally parallel spinor φbn−1
2
∈ Γ(Σn−1 2
M) provides a non-zero i-K¨ahlerian Killing spinor by settingφn+1
2 :=ψn−3
2 := 0andφn−1
2 :=etφbn−1
2 ,ψn−1
2 :=−eti∂t∂ ·φbn−1
2 . Moreover, for anyi-K¨ahlerian Killing spinor(ψ, φ)on that doubly warped product(fM2n,eg,Je), the component φn−1
2 is transversally parallel on(M,g,b ξ)b if and only if i∂t∂ ·ψ=−φ.
ii) In caseρ= sinh: One hasσ= coshonI=R×+and there is a one-to-one correspondence between the space of i-K¨ahlerian Killing spinors on(Mf2n,eg,Je) and that of sections (ϕn+1
2 , ϕn−1
2 ,ϕen−1
2 ,ϕen−3
2 ) of Σn+1
2 M ⊕Σn−1
2 M⊕Σn−1
2 M⊕Σn−3
2 M −→M satisfying
∇b
ξb
(∼)ϕr = (−1)2 r(n−2r)bξb·
M (∼)ϕr
∇b
ξb
(∼)ϕr−1 = −(−1)2 r(n−2r)bξb·
M (∼)ϕr−1
∇bZ
(∼)ϕr = (−1)rp+(Z)b·
M (∼)ϕr−1
∇bZ
(∼)ϕr−1 = (−1)rp−(Z)b·
M (∼)ϕr
(5)
on(M2n−1,bg,ξ), for everyb Z∈ξb⊥ (this means that(ϕn+1 2 , ϕn−1
2 )must satisfy(5)forr=n+12 and (ϕen−1
2 ,ϕen−3
2 )must satisfy(5)forr=n−12 ).
iii) In caseρ= cosh: One hasσ= sinhonI=R×+and there is a one-to-one correspondence between the space of i-K¨ahlerian Killing spinors on(Mf2n,eg,Je) and that of sections (ϕn+1
2 , ϕn−1
2 ,ϕen−1
2 ,ϕen−3
2 )
of Σn+1
2 M ⊕Σn−1
2 M⊕Σn−1
2 M⊕Σn−3
2 M −→M satisfying
∇b
bξ
(∼)ϕr = −(−1)2 r(n−2r)bξb·
M (∼)ϕr
∇b
bξ
(∼)ϕr−1 = (−1)2 r(n−2r)bξb·
M (∼)ϕr−1
∇bZ
(∼)ϕr = (−1)n+12 p+(Z)b·
M (∼)ϕr−1
∇bZ
(∼)ϕr−1 = (−1)n−12 p−(Z)b·
M (∼)ϕr
(6)
on(M2n−1,bg,ξ), for everyb Z∈ξb⊥ (this means that(ϕn+1 2
, ϕn−1 2
)must satisfy(6)forr=n+12 and (ϕen−1
2 ,ϕen−3
2 )must satisfy(6)forr=n−12 ).
Proof: We first show ρ00 = ρ on I. In order to express all equations of (4) in an intrinsic way, we have to compare all objects on (M, gt, ξ) with the corresponding ones on (M,bg,ξ). Recall thatb gt = ρ(t)2(σ(t)2bg
ξb⊕bg
ξb⊥) andξ=ρσ1 ξ. As for (2), it is elementary to check the following relations:b
∇b =∇, ξ·=ξbb·, ξ ·
M=ξbb·
M, Z·=ρZb·, Z ·
M=ρZb·
M, for allZ∈ξ⊥. Applying ∂t∂ onto
∇bZφn+1
2 =p+(Z)b·(ρ0∂t∂ ·φn−1
2 −iρψn−1
2 )
∇bZψn−3 2
=p−(Z)b·(ρ0∂t∂ ·ψn−1
2 −iρφn−1 2
) and using
∂φn+1
2
∂t =
∂ψn−3 2
∂t = 0, one obtains 0 = p+(Z)b·(ρ00∂
∂t·φn−1
2 +ρ0 ∂
∂t ·∂φn−1 2
∂t −iρ0ψn−1
2 −iρ
∂ψn−1 2
∂t )
= p+(Z)b·(ρ00∂
∂t·φn−1
2 +ρ0 ∂
∂t ·(−i∂
∂t·ψn−1
2 )−iρ0ψn−1
2 −iρ(−i∂
∂t·φn−1
2 ))
= (ρ00−ρ)p+(Z)b·∂
∂t·φn−1 2
and analogously (ρ00−ρ)p−(Z)b·∂t∂·ψn−1
2 = 0 for allZ∈ξb⊥. Fix a localg-orthonormal basis (eb j)1≤j≤2n−2of ξb⊥. PuttingZ=ej, Clifford-multiplying byejand summing overjgives (ρ00−ρ)φn−1
2 = (ρ00−ρ)ψn−1 2 = 0.
On the other hand, both equations involving
∂φn−1 2
∂t and
∂ψn−1 2
∂t provide the existence of smooth sections A±n−1
2
of Σn−1 2
M (independent oft) such that φn−1 2
=etA+n−1 2
+e−tA−n−1 2
andψn−1 2
=−eti∂t∂ ·A+n−1 2
+ e−ti∂t∂ ·A−n−1
2
. We deduce that (ρ00−ρ)A+n−1 2
= (ρ00−ρ)A−n−1 2
= 0. If both A+n−1 2
and A−n−1 2
vanished identically onM, then so wouldφn−1
2 andψn−1
2 and the identities involving∇bZφn−1
2 and∇bZψn−1 2 would provide (after contracting with the Clifford multiplication just as above) φn+1
2 = ψn−3
2 = 0, so that (ψ, φ) = 0, which is a contradiction. Thereforeρ00−ρ= 0 onI.
It follows in particular thatρ0 = 0 onIcannot hold, so we may assume thatbh=−J (hence (M2n−1,bg,ξ)b is Sasakian) andρ0 =σ(see Remarks 3.5). Furthermore, in the case where the constant (ρ0)2−ρ2does not vanish, up to replacingρby √ ρ
|(ρ02)−ρ2| (which is equivalent to performing a D-homothetic deformation of the Sasakian structure), we may assume that (ρ02)−ρ2= 1 or−1 onI. Next we rewrite the equations from Lemma 3.8 considering the new sectionsϕn+1
2 , ϕn−1
2 ,ϕen−1
2 ,ϕen−3
2 defined by
ϕn+1
2 :=φn+1 2
ϕn−1
2 :=ρ0φn−1
2 +iρ∂t∂ ·ψn−1 2
ϕen−1
2 :=iρ∂t∂ ·φn−1
2 +ρ0ψn−1 2
ϕen−3
2 :=ψn−3
2 . Note that the linear transformation (φn+1
2 , φn−1
2 , ψn−1
2 , ψn−3
2 )7→(ϕn+1 2 , ϕn−1
2 ,ϕen−1
2 ,ϕen−3
2 ) is invertible if and only if (ρ0)2−ρ26= 0. From (4) we have, for allZ ∈ξb⊥:
∂
∂tϕn+1
2 = 0
