andrei moroianu, uwe semmelmann - uni-stuttgart.de...andrei moroianu, uwe semmelmann abstract. we...

UniversitatStuttgart

FachbereichMathematik

Generalized Killing Spinors on SpheresAndrei Moroianu, Uwe Semmelmann

Preprint 2013/012

Fachbereich MathematikFakultat Mathematik und PhysikUniversitat StuttgartPfaffenwaldring 57D-70 569 Stuttgart

E-Mail: [email protected]

WWW: http://www.mathematik.uni-stuttgart.de/preprints

ISSN 1613-8309

c© Alle Rechte vorbehalten. Nachdruck nur mit Genehmigung des Autors.LATEX-Style: Winfried Geis, Thomas Merkle

mailto:[email protected]

http://www.mathematik.uni-stuttgart.de/preprints

GENERALIZED KILLING SPINORS ON SPHERES

ANDREI MOROIANU, UWE SEMMELMANN

Abstract. We study generalized Killing spinors on round spheres Sn. We show that onthe standard sphere S8 any generalized Killing spinor has to be an ordinary Killing spinor.Moreover we classify generalized Killing spinors on Sn whose associated symmetric endomor-phism has at most two eigenvalues and recover in particular Agricola–Friedrich’s canonicalspinor on 3-Sasakian manifolds of dimension 7. Finally we show that it is not possible todeform Killing spinors on standard spheres into genuine generalized Killing spinors.

2010 Mathematics Subject Classification: Primary: 53C25, 53C27, 53C40

Keywords: generalized Killing spinors, parallel spinors.

1. Introduction

A generalized Killing spinor on a spin manifold (M, g) is a non-zero spinor Ψ ∈ Γ(ΣM)satisfying for all vector fields X the equation ∇XΨ = A(X) ·Ψ, where A is some symmetricendomorphism field. If A is a non-zero multiple of the identity, Ψ is called a Killing spinor[3, 5]. We will call generalized Killing spinors with A 6= λid genuine generalized Killingspinors.

Generalized Killing spinors arise naturally as the restrictions of parallel spinors on spinmanifolds M to hypersurfaces M ⊂ M (see [4, 11, 12, 16, 17]). In this case the endomorphismA is half of the second fundamental form of M . The converse is true under certain conditions,e.g. when both the manifold (M, g) and the spinor Ψ are real analytic [2].

In low dimensions any generalized Killing spinor Ψ defines a G-structure on M , where Gis the stabilizer of Ψ at some point. The intrinsic torsion of this G-structure is determinedby the endomorphism A, and since A is assumed to be symmetric, some part of the intrinsictorsion has to vanish. This leads to interesting reformulations of the existence of generalizedKilling spinors, e.g. they correspond to half-flat SU(3)-structures [8, 13] in dimension 6 andto co-calibrated G2-structures [9, 10] in dimension 7.

Date: September 25, 2013.This work was done during a “Research in Pairs” stay at CIRM, Luminy. We warmly thank the CIRM for

hospitality. The first author was partially supported by the contract ANR-10-BLAN 0105 “Aspects Conformesde la Geometrie”.

1

2 ANDREI MOROIANU, UWE SEMMELMANN

In [17] we started an investigation of generalized Killing spinors on Einstein manifolds,motivated by an analogue of the Goldberg conjecture. We showed that any generalized Killingspinor on the standard spheres S2 and S5, as well as on any 4-dimensional Einstein manifoldsof positive scalar curvature has to be an ordinary Killing spinor and we have constructedexamples of genuine generalized Killing spinors on S3. Moreover, we gave an account of theother examples of genuine generalized Killing spinors on Einstein manifolds which can befound in the recent literature on S3 × S3 and CP3 (cf. [9, 15, 18]), and on 7-dimensional3-Sasakian manifolds (cf. [1]).

In the present article we concentrate on the existence question for generalized Killingspinors on standard spheres. It is a classical theorem that any Einstein hypersurface ofpositive scalar curvature in the Euclidean space Rn+1 is locally isometric to Sn. Thus spheresare the only hypersurfaces in Rn+1 admitting generalized Killing spinors. Our problem canbe rephrased into the question: Is it possible to realize standard spheres as hypersurfaces ofnon-flat manifolds with reduced holonomy, e.g. Calabi-Yau or hyperkahler manifolds?

Even on such simple manifolds as the standard spheres, the problem of proving existenceor non existence of genuine generalized Killing spinors turns out to be extremely difficult.In this article we obtain the following partial results: in Section 3 we show that on S8

any generalized Killing spinor has to be an ordinary Killing spinor. The same statementis true for any 8k-dimensional standard sphere if a natural vector field associated to thespinor does not vanish identically. In Section 4 we consider generalized Killing spinors forwhich the symmetric endomorphism A has exactly two eigenvalues. We show that this ispossible only in dimension 3 and 7, where the generalized Killing spinors coincides with theexamples mentioned above. In the last section we investigate deformations of generalizedKilling spinors. Using the Weitzenbock formula for trace-free symmetric tensors we prove arigidity result for Killing spinors on spheres, similar in some sense with the rigidity of Einsteinmetrics [6, Sect. 4.63].

2. Preliminaries

We refer to [5, 14] for basic definitions in spin geometry and list below some of the mostimportant facts which will be needed in the sequel. Let (Mn, g) be an n-dimensional Rie-mannian spin manifold with real spinor bundle ΣM . The Levi-Civita connection ∇ inducesa connection on ΣM , also denoted by ∇. In addition the real spinor bundle ΣM is endowedwith a ∇-parallel Euclidean scalar product 〈., .〉.

Throughout this article we will identify 1-forms and bilinear forms with vectors and endo-morphisms respectively, by the help of the Riemannian metric.

The Clifford multiplication with tangent vectors is parallel with respect to ∇ and skew-symmetric with respect to 〈., .〉:

(1) 〈X ·Ψ,Φ〉 = −〈Ψ, X · Φ〉, ∀ X, Y ∈ TM, ∀ Ψ,Φ ∈ ΣM.

GENERALIZED KILLING SPINORS 3

In particular 〈X ·Ψ,Ψ〉 = 0 for any vector field X and spinor Ψ. The Clifford multiplicationwith 2-forms is defined via the equation

(2) (X ∧ Y ) ·Ψ = X · Y ·Ψ + g(X, Y ) Ψ.

Using (1) and the basic Clifford formula X · Y ·+Y ·X ·+2g(X, Y )id = 0, we easily get

(3) 〈X · Y ·Ψ,Ψ〉 = −g(X, Y )〈Ψ,Ψ〉, ∀ X, Y ∈ TM, Ψ ∈ ΣM,

which together with (2) shows that Clifford product with 2-forms is also skew-symmetric.

The curvature RΣM of the spinor bundle and the Riemannian curvature are related by

(4) RΣMX,Y Ψ = 1

2R(X ∧ Y ) ·Ψ ∀ X, Y ∈ TM, Ψ ∈ ΣM,

where R : Λ2M → Λ2M denotes the curvature operator defined by

g(R(X ∧ Y ), U ∧ V ) := g(RX,YU, V ), RX,Y := [∇X ,∇Y ]−∇[X,Y ].

Note that with our convention the curvature operator on the standard sphere acts on 2-formsas minus the identity.

A generalized Killing spinor [2, 4, 12, 17] on (M, g) is a spinor Ψ satisfying the equation

(5) ∇XΨ = A(X) ·Ψ, ∀ X ∈ TM,

where A ∈ Γ(End(TM)) is some symmetric endomorphism field, sometimes called the endo-morphism associated to Ψ. Clearly a generalized Killing spinor Ψ has constant length and byrescaling we may always assume that |Ψ|2 = 1.

After taking a further covariant derivative in Eq. (5) and skew-symmetrizing one obtainsthe curvature equation (see [17, Eq. (9)]):

(6) (d∇A)(X, Y ) = [(∇XA)Y − (∇YA)X] ·Ψ = 2A(X) ∧ A(Y ) ·Ψ + 12R(X ∧ Y ) ·Ψ.

Moreover, one has the following constraint equations ([17, Eqs. (11) and (12)]):

(7) 0 = δ∇A + dtrA,

(8) scal = 4(trA)2 − 4trA2,

where δ∇A := −∑n

i=1(∇eiA)ei denotes the divergence of A.

It is well known that the standard sphere Sn admits the maximal possible number of realKilling spinors trivializing the spinor bundle ΣM , cf. [3]. About the existence of generalizedKilling spinors much less is known. We quote the following previous results:

• There are no genuine generalized Killing spinors on S2,S4 and S5, cf. [17].• There are examples of genuine generalized Killing spinors on S3 of the form Ψ = ξ ·Φ,

where ξ is a unit length left-invariant Killing vector field and Φ is a Killing spinor withKilling constant 1

2. In this example the symmetric endomorphism A has eigenvalue 1

2

of multiplicity 1, and eigenvalue −32

of multiplicity 2, cf. [17].


• There is a genuine generalized Killing spinor on S7, which again is of the form Ψ = ξ·Φ,where ξ is a unit length Killing vector field on S7 and Φ is a certain Killing spinor.Like in dimension 3, the eigenvalues of A are 1

2and −3

2, this time with multiplicities

3 and 4, respectively, cf. [1].

3. Generalized Killing spinors on S8k

The aim of this section is to show that every generalized Killing spinor on S8 is a Killingspinor, as well as a partial result in the same direction for all spheres S8k.

Recall that in dimension 8k the real spin representation splits as Σ8k = Σ+8k⊕Σ−8k, where Σ±8k

are the ±1-eigenspaces of the multiplication with the volume element and are interchangedby Clifford multiplication with vectors. Correspondingly, Ψ splits as Ψ = Ψ+ + Ψ−. Let ηbe the vector field on S8k given by

(9) g(η,X) = 〈X ·Ψ+,Ψ−〉, ∀ X ∈ TS8k.

If the form η does not vanish identically, we have the following:

Theorem 3.1. Let Ψ be a generalized Killing spinor on S8k. If the one-form defined in (9)is non-vanishing on a dense subset, then Ψ is a Killing spinor.

Proof. We assume that Ψ is scaled to have unit length. Denoting a := tr(A) and using thefact that the scalar curvature of S8k equals 8k(8k− 1), Eq. (8) reads a2− trA2 = 2k(8k− 1).From (5) we get:

(10) ∇XΨ± = A(X) ·Ψ∓.

Let S− denote the open set of points p ∈ S8k with Ψ−p 6= 0. It is easy to see that S− is

dense. Indeed, if U were a non-empty open subset of S8k \S−, then (10) yields A(X) ·Ψ+ = 0for all X ∈ TU , so A|U = 0. By (10) again, Ψ+ is parallel (and non-zero) on U , so the Riccitensor of S8k vanishes on U , which is absurd. A similar argument shows that the set S+

where Ψ+ is non-vanishing is also dense, so the set S := S− ∩ S+ is dense in S8k.

We denote by h := |Ψ−|2 the length function of Ψ−. Since Ψ has unit length, |Ψ+|2 = 1−h.From (10), the derivative of h in the direction of any tangent vector X reads

dh(X) = 2〈∇XΨ−,Ψ−〉 = 2〈A(X) ·Ψ+,Ψ−〉 = 2η(A(X)) = 2g(A(η), X),

whence

(11) dh = 2A(η).


Taking the covariant derivative in the direction of Y in (9), assuming that X is parallel atsome point and using (10) yields

g(∇Y η,X) = 〈X · A(Y ) ·Ψ−,Ψ−〉 + 〈X ·Ψ+, A(Y ) ·Ψ+〉= −g(X,A(Y )) |Ψ−|2 + g(X,A(Y )) |Ψ+|2

= (1− 2h)g(A(Y ), X),

so

(12) ∇Y η = (1− 2h)A(Y ), ∀ Y ∈ TS8k.

Taking the covariant derivative with respect to some vector field X in this equation, using(11) and skew-symmetrizing, yields:

RY,Xη = (1− 2h)((∇YA)X − (∇XA)Y )− 4g(A(η), Y )A(X) + 4g(A(η), X)A(Y ),

and since the curvature of the round sphere satisfies RY,XZ = g(X,Z)Y − g(Y, Z)X for allvectors X, Y, Z, we get

(1−2h)((∇YA)X−(∇XA)Y ) = 4g(A(η), Y )A(X)−4g(A(η), X)A(Y )+g(X, η)Y −g(Y, η)X.

Using this last equation in the curvature equation (6) we obtain that for every vectors X, Ythe following relation holds:

(2h− 1)(2A(X) · A(Y ) + 2g(A(X), A(Y ))− 1

2X · Y − 1

2g(X, Y )

)·Ψ

= (4g(A(η), Y )A(X)− 4g(A(η), X)A(Y ) + g(X, η)Y − g(Y, η)X) ·Ψ(13)

(we have used the well known formula X∧Y = X ·Y +g(X, Y ) and the fact that the curvatureendomorphism of the round sphere is minus the identity).

In (13) we take the Clifford product with X and sum over an orthonormal basis X = ei.Using the standard formulas in Clifford calculus this yields

(2h− 1)(−2aA(Y ) + 2A2(Y ) + 8k−1

2Y)·Ψ

= (−4ag(A(η), Y )− 4A(η) · A(Y ) + η · Y + 8kg(η, Y )) ·Ψ.

Taking the scalar product with Ψ in this formula gives

0 = −4ag(A(η), Y ) + 4g(A(η), A(Y )) + (8k − 1)g(η, Y ), ∀ Y ∈ TS8k,

whence

(14) A2(η) = aA(η)− 8k−14η.

We now take the Clifford product with A(X) in (13) and sum over an orthonormal basisX = ei to obtain

(2h− 1)(−2trA2A(Y ) + 2A3(Y ) + 1

2aY − 1

2A(Y )

)·Ψ

=(−4trA2g(A(η), Y )− 4A2(η) · A(Y ) + A(η) · Y + ag(η, Y )

)·Ψ.


Taking again the scalar product with Ψ and using (8) yields

0 = (8k(8k−1)−4a2)g(A(η), Y )+4g(A2(η), A(Y ))−g(A(η), Y )+ag(η, Y ), ∀ Y ∈ TS8k,

whence

(15) A3(η) = (a2 − 2k(8k − 1) + 14)A(η)− a

4η.

Plugging (14) into this equation shows that A(η) = 18kaη, so from (14) again we get

a2

64k2η = a2

8kη − 8k−1

4η.

As η is non-vanishing on a dense subset, we obtain a2 = 16k2 on S8k. This, together with(8), shows that the square norm of the trace-free symmetric tensor A− a

8kid vanishes:

|A− a8k

id|2 = tr(A− a8k

id)2 = trA2 − a4k

trA+ a2

8k= trA2 − a2

8k= 16k2 − 2k(8k − 1)− 2k = 0.

This implies that A = a8k

id = ±12id and thus finishes the proof.

Corollary 3.2. Every generalized Killing spinor Ψ on S8 is a Killing spinor.

Proof. For every p ∈ S+ the injective map X ∈ TpS8 7→ X · Ψ+ ∈ (Σ−8 )p is bijective sincedim TpS8 = dim(Σ−8 )p = 8. Consequently, the vector field η is non-vanishing on S.

4. Generalized Killing spinors with two eigenvalues

In this section we consider generalized Killing spinors Ψ on the sphere (M, g) := Sn (n ≥ 3)and assume that the associated symmetric endomorphism A has at each point at most twoeigenvalues λ and µ. If these eigenvalues coincide at each point, then it is well known thattheir common value is constant on M , so Ψ is a Killing spinor. We assume from now on thatλ 6= µ at least at some point of M , and thus on some non-empty open set S (it turns outthat they are actually constant on M , cf. Lemma 4.1). We will denote by Tλ ⊂ TM andTµ ⊂ TM the eigenspaces corresponding to λ and µ respectively. These two subspaces aremutually orthogonal at each point and are well-defined distributions on S.

We start with calculating the derivative d∇A at points of S in three different cases. First,let X, Y ∈ Tµ:

(∇XA)Y − (∇YA)X = X(µ)Y − (∇XA)Y + µ∇XY − Y (µ)X + A(∇YX)− µ∇Y X= (µ− λ)(∇XY )λ − (µ− λ)(∇YX)λ +X(µ)Y − Y (µ)X

= [X, Y ]λ +X(µ)Y − Y (µ)X.

A similar calculation for a pair of vectors U, V ∈ Tλ leads to

(∇VA)U − (∇UA)V = (λ− µ)[U, V ]µ + V (λ)U − U(λ)V.

Finally, on a mixed pair of vectors X ∈ Tµ, V ∈ Tλ, we find

(∇XA)V − (∇VA)X = (λ− µ)(∇XV )µ − (µ− λ)(∇VX)λ − V (µ)X +X(λ)V.


Substituting the equations above into the curvature equation (6), with R = −id for thesphere, we obtain for every X, Y ∈ Tµ and U, V ∈ Tλ:

(2µ2 − 12)X ∧ Y ·Ψ = (µ− λ)[X, Y ]λ ·Ψ + (X(µ)Y − Y (µ)X) ·Ψ,(16)

(2λ2 − 12)V ∧ U ·Ψ = (λ− µ)[V, U ]µ ·Ψ + (V (λ)U − U(λ)V ) ·Ψ,(17)

(2λµ− 12)X ∧ V ·Ψ = (λ− µ)((∇XV )µ + (∇VX)λ) + (X(λ)V − V (µ)X) ·Ψ.(18)

Lemma 4.1. The eigenvalues λ and µ are constant on Sn.

Proof. Since the sphere is connected, it is enough to show that λ and µ are constant on theopen set S. Taking the scalar product with X · Ψ in equation (16) for X ∈ Tµ implies thatµ is constant in Tµ-directions. Similarly the second equation gives that λ is constant in allTλ-directions.

Let p and q denote the dimensions of Tλ and Tµ respectively (which are constant on S).Then (8) yields

(19) (pλ+ qµ)2 − (pλ2 + qµ2) = 14n(n− 1);

Differentiating this relation with respect to some vector V ∈ Tλ gives V (µ)(µ(q−1)+qλ) = 0.Assuming that V (µ) is different from zero on some open set S ′ ⊂ S, then

(20) µ(q − 1) + qλ = 0

on S ′. Differentiating again with respect to V and using the fact that λ is constant in Tλ-directions, we get (q − 1)V (µ) = 0. The assumption that V (µ) is different from zero on S ′

implies that q = 1, so (20) implies λ = 0, which contradicts (19). Hence µ is constant on Sand a similar argument shows that λ is constant too.

Lemma 4.2. One of the eigenvalues λ and µ has to be equal to ±12.

Proof. Assume first that λµ = 14. Then for any vector fields X, Y ∈ Tµ and U, V ∈ Tλ, taking

the scalar product in (18) with Y ·Ψ and U ·Ψ yields

g(∇XV, Y ) = 0 = g(∇VX,U),

i.e. (∇XV )µ = 0 = (∇VX)λ. Thus Tλ and Tµ are two non-trivial parallel distributions onSn, which is clearly a contradiction. This shows that λµ 6= 1

4.

Since even-dimensional spheres do not have any non-trivial distributions, it follows thatn = 2k + 1 is odd. By changing the notations if necessary, we can assume that dim(Tµ) >dim(Tλ). If µ2 = 1

4we are done, so for the remaining part of the proof we assume that µ2 6= 1

4.

From (16) it follows that for every x ∈ Sn and X, Y ∈ Tµx with X ⊥ Y , the vector [X, Y ]λ

is non-zero (note that this expression is tensorial in X and Y , so it only depends on theirvalues at x). Consequently, the map Y 7→ [X, Y ]λ from the orthogonal complement of X in


Tµx to Tλ

x is injective. From the dimensional assumption it follows that dim(Tµx) = k + 1 and

dim(Tλx) = k, so in particular the above map is bijective. It follows that for every X ∈ Tµ

x

and V ∈ Tλx there exists a unique Y ∈ Tµ

x, Y ⊥ X, such that [X, Y ]λ = V . Applying (16)and (18) to these vectors yields

(λ− µ)((∇XV )µ + (∇VX)λ) ·Ψ = (2λµ− 12)X · V ·Ψ = (2λµ− 1

2)X · [X, Y ]λ ·Ψ

= 1µ−λ(2λµ− 1

2)(2µ2 − 1

2)X ·X · Y ·Ψ

= − |X|2

µ−λ(2λµ− 12)(2µ2 − 1

2)Y ·Ψ.

This shows that for every X ∈ Tµx and V ∈ Tλ

x, the vector (∇VX)λ vanishes, thus Tλ is atotally geodesic distribution. From (17) we deduce that λ2 = 1

4unless k = 1. It remains to

rule out the case where n = 3.

In this case Tλ is one-dimensional, so we can consider a unit vector V which spans it ateach point. Then V is geodesic and taking the scalar product with X ·Ψ in (18) shows thatg(∇XV,X) = 0 for every X ∈ Tµ. Thus V is a unit Killing vector field on S3. It is wellknown that every such vector satisfies |∇XV | = |X| for every X orthogonal to V . Comparingthe norms of the two spinors in (18) yields 2λµ − 1

2= ±(λ − µ), which can be rewritten as

(2λ± 1)(2µ∓ 1) = 0. This proves the lemma.

Up to a change of orientation we thus may from now on assume that λ = 12.

Lemma 4.3. The distribution Tλ is totally geodesic. Moreover, the following equations holdfor any vectors X, Y ∈ Tµ and V ∈ Tλ:

(2µ+ 1)X ∧ Y ·Ψ = [X, Y ]λ ·Ψ,(21)

X · V ·Ψ = −(∇XV )µ ·Ψ.(22)

Proof. We have λ = 12

and µ 6= λ constant. Equation (21) thus follows directly from (16).

Next, taking in (18) the scalar product with V ·Ψ, gives 0 = g((∇VX)λ, V ) = −g(X,∇V V ),and by polarization (∇VU +∇UV )µ vanishes for every vector fields U, V in Tλ. On the otherhand, (17) implies [V, U ]µ = 0, so adding these two relations we obtain that (∇UV )µ = 0, i.e.Tλ is totally geodesic.

In particular this can also be expressed by the fact that (∇VX)λ vanishes for every X ∈ Tµ

and V ∈ Tλ, so (22) follows directly from (18).

Remark 4.4. With a similar argument we get (∇XY +∇YX)λ = 0 for all vectors X, Y ∈ Tµ.Thus the distribution Tµ would also be totally geodesic if integrable.

Corollary 4.5. For every x ∈ Sn there is a representation of the real Clifford algebra Cl(Tλx)

on Tµx.


Proof. For V ∈ Tλx and X ∈ Tµ

x we define

ρV (X) := (∇XV )µ.

Then (22) can be re-written as ρV (X) ·Ψ = V ·X ·Ψ, whence

(ρV ρV (X)) ·Ψ = V · ρV (X) ·Ψ = V · V ·X ·Ψ = −|V |2X ·Ψ,

showing that ρV ρV = −|V |2id. This proves the lemma.

Lemma 4.6. The second eigenvalue of A is µ = −32.

Proof. Taking in (21) the scalar product with V ·Ψ and applying (22), gives

g([X, Y ], V ) = −(2µ+ 1)〈V ·X · Y ·Ψ,Ψ〉 = −(2µ+ 1)〈X · V ·Ψ, Y ·Ψ〉= −(2µ+ 1)g(∇XY, V )

This equation can be rewritten as g((2µ + 2)∇XY −∇YX, V ) = 0. Interchanging X and Yand subtracting the resulting equations we obtain (2µ+ 3)[X, Y ]λ = 0.

If µ 6= −32, the distribution Tµ is totally geodesic (see Remark 4.4), and since Tµ is also

totally geodesic, both distributions would be parallel, which is of course impossible on Sn.

Lemma 4.7. The multiplicities p and q of λ and µ are related by q = p+ 1.

Proof. Introducing the values λ = 12

and µ = −32

in (8) we obtain the equation

14n(n− 1) = a2 − trA2 = (p

2− 3q

2)2 − p

4− 9q

4.

Substituting n = p+ q immediately leads to p = q − 1.

Corollary 4.8. The pair (p, q) of multiplicities of λ and µ is one of (1, 2), (3, 4) or (7, 8).

Proof. By Corollary 4.5 and Lemma 4.7, there exists a Clp representation on Rp+1. From theclassification of real Clifford algebras (cf. [14]), this can only happen when p is 1, 3 or 7.

We thus see that a generalized Killing spinor whose associated endomorphism has twoeigenvalues can only exist on Sn for n = 3, n = 7 or n = 15. We will now further investigatethe geometry determined by Ψ and at the end we will consider these three cases separately.

For every V ∈ Tλ consider the skew-symmetric endomorphism ρV of Tµ defined above byρV (X) := −(∇XV )µ. Equation (22) then reads

(23) X · V ·Ψ = ρV (X) ·Ψ, ∀ X ∈ Tµ, ∀ V ∈ Tλ.

For every U, V ∈ Tλ with g(U, V ) = 0 we pick some arbitrary vector X ∈ Tµ with |X| = 1and write using (21) and (23):

U · V ·Ψ = (X · U) · (X · V ) ·Ψ = (X · U) · ρV (X) ·Ψ = ρV (X) · (X · U) ·Ψ= ρV (X) · ρU(X) ·Ψ ∈ Tλ ·Ψ.


This shows that Λ2Tλ · Ψ ⊂ Tλ · Ψ. Moreover, this also shows that for every X ∈ Tµ andU, V ∈ Tλ

(24) 〈U · V ·Ψ, X ·Ψ〉 = 0.

Lemma 4.9. The sub-bundle Tλ ·Ψ of ΣSn is parallel with respect to the modified connection∇X := ∇X − 1

2X·.

Proof. For X ∈ Tµ and V ∈ Tλ we have

(∇X − 12X·)(V ·Ψ) = (∇XV ) ·Ψ + V · A(X) ·Ψ− 1

2X · V ·Ψ

= (∇XV ) ·Ψ− 32V ·X ·Ψ− 1

2X · V ·Ψ

= (∇XV ) ·Ψ− V ·X ·Ψ = (∇XV ) ·Ψ + ρV (X) ·Ψ= (∇XV )λ ·Ψ ∈ Tλ ·Ψ,

and for U, V ∈ Tλ, keeping in mind that Tλ is totally geodesic and that Λ2Tλ ·Ψ ⊂ Tλ ·Ψ:

(∇U − 12U ·)(V ·Ψ) = (∇UV ) ·Ψ + V · A(U) ·Ψ− 1

2U · V ·Ψ

= (∇UV ) ·Ψ + 12V · U ·Ψ− 1

2U · V ·Ψ

= (∇UV ) ·Ψ + V ∧ U ·Ψ ∈ Tλ ·Ψ.

Since ∇ is flat on ΣSn, it follows that Tλ ·Ψ can be trivialized with ∇-parallel (i.e. Killing)spinors. We denote by K the p-dimensional vector space of Killing spinors on Sn obtainedin this way. By definition, for every Φ ∈ K, there exists a vector field ξΦ ∈ Tλ satisfyingξΦ · Ψ = Φ. Clearly 〈Ψ,Φ〉 = 0, and as Ψ has unit norm, |ξΦ|2 = |Φ|2. For every tangentvector X we have g(ξΦ, X) = 〈X · Ψ,Φ〉. Using the obvious fact that A(X)λ = 1

2Xλ and

A(X)µ = −32Xµ, we compute using (24):

g(∇XξΦ, X) = 〈X · ∇XΨ,Φ〉+ 〈X ·Ψ,∇XΦ〉 = 〈X · A(X) ·Ψ,Φ〉+ 12〈X ·Ψ, X · Φ〉

= 〈X · A(X) ·Ψ,Φ〉 = 〈(Xµ +Xλ) · (12Xµ − 3

2Xλ) ·Ψ, ξΦ ·Ψ〉

= −32〈Xµ ·Xλ ·Ψ, ξΦ ·Ψ〉+ 1

2〈Xλ ·Xµ ·Ψ, ξΦ ·Ψ〉 = 0.

This shows that ξΦ is a Killing vector field on Sn for every Killing spinor Φ ∈ K. There existsthus a linear map F from K to Λ2Rn+1 which associates to each Φ ∈ K a skew-symmetricmatrix FΦ ∈ Λ2Rn+1 such that (ξΦ)x = FΦ(x) for every x ∈ Sn ⊂ Rn+1. In fact FΦ is relatedto the covariant derivative of ξΦ by

(25) ∇XξΦ = FΦ(X), ∀ X ∈ TSn.As |ξΦ|2 = |Φ|2, we obtain (FΦ)2 = −|Φ|2idRn+1 . If we choose now an orthonormal basisΦ1, . . . ,Φp of K, and denote by Fi := FΦi

for simplicity, the previous relation becomes

(26) (Fi)2 = −id, Fi Fj + Fj Fi = 0 for i 6= j.

We now consider the three cases above separately.


The case n = 3. In this case the distribution Tλ is 1-dimensional, and the unit vectorfield generating it (unique up to a sign) is Killing. The symmetric tensor A thus coincideswith the one defined in [17, Sect. 4.2]. Of course, the space of generalized Killing spinorswith respect to this tensor A is 4-dimensional, since the spin representation in dimension 3has a quaternionic structure.

The case n = 7. We have seen that ξ1, ξ2, ξ3 is an orthonormal basis of Tλ at each pointconsisting of unit Killing vector fields. It is well known that every unit Killing vector field onthe round sphere is Sasakian. The relation (26) just tells that the triple ξ1, ξ2, ξ3 defines a3-Sasakian structure.

We remark that the spinor Ψ is exactly the canonical spinor constructed by Agricola andFriedrich [1] on any 3-Sasakian manifold of dimension 7.

The case n = 15. It would have been interesting to obtain examples of generalized Killingspinors with two eigenvalues on S15 similar to those constructed above in dimension 3 and 7.Unfortunately this turns out to be impossible.

Assuming the existence of such a spinor Ψ, we would obtain from the construction abovean orthonormal set of Killing vector fields ξ1, . . . , ξ7 on S15 whose defining endomorphismsFi ∈ Λ2R16 satisfy (26). This shows that there exists a representation of the real Cliffordalgebra Cl7 on R16 such that Fi(x) = ei · x for every x ∈ R16 and 1 ≤ i ≤ 7. By definition ofFi we thus have (ξi)x = ei · x for every x ∈ S15 and 1 ≤ i ≤ 7. As Cl7 = R(8) ⊕ R(8), thisrepresentation decomposes in a direct sum R16 = Σ1⊕Σ2 of two 8-dimensional representationsof Cl7. Each xi ∈ Σi (i ∈ 1, 2) defines a vector cross product Pxi on R7 by the formula(u ∧ v) · xi = Pxi(u, v) · xi.

Using (25) we can write for every x = (x1, x2) ∈ S15 and i 6= j ∈ 1, . . . , 7:

(∇ξiξj)x = Fj(ξi)x = Fj(Fi(x)) = ej · ei · x = (ej ∧ ei · x1, ej ∧ ei · x2)

= (Px1(ej, ei) · x1, Px2(ej, ei) · x2).

Recall now that ξ1, . . . , ξ7 span a totally geodesic distribution on S15. This implies that thereexist functions f1, . . . , f7 on S15 such that

(∇ξiξj)x =7∑

k=1

fk(x)(ξk)x =7∑

k=1

fk(x)Fk(x) =7∑

k=1

fk(x)ek · x =7∑

k=1

fk(x)(ek · x1, ek · x2).

Comparing these last two equations yields Px1(ej, ei) = Px2(ej, ei) for every (x1, x2) ∈ S15 ⊂R16 and for every i 6= j ∈ 1, . . . , 7 . This implies that the vector cross product Px isindependent of x, which is of course a contradiction. There are thus no solutions on thesphere S15.

We have proved the following

Theorem 4.10. Let Ψ be a generalized Killing spinor on the sphere Sn whose associatedsymmetric endomorphism A has at most two eigenvalues λ and µ at each point. Then λ and


µ are both constant. If λ = µ, then A = ±12id and Ψ is a Killing spinor. If λ 6= µ, then up

to a permutation of λ and µ and a change of orientation one has λ = 12, µ = −3

2and n = 3

or n = 7.

• If n = 3, the 12-eigenspace of A is spanned by a unit left-invariant Killing vector field

ξ on S3 and Ψ = ξ · Φ for some Killing spinor Φ with constant 12.

• If n = 7, the 12-eigenspace of A is spanned by three Killing vector fields ξ1, ξ2, ξ3

defining a 3-Sasakian structure on S7 and Ψ is the canonical spinor of the 3-Sasakianstructure introduced in [1].

5. Deformations of generalized Killing spinors

In this section we study the deformation problem for generalized Killing spinors on spheres,and show in particular that Killing spinors are rigid, in the sense that they cannot be deformedinto generalized Killing spinors.

For every spin manifold (M, g), the set GK(M, g) of generalized Killing spinors is a Frechetmanifold. On the round sphere Sn, the (finite dimensional) vector spaces K 1

2(Sn) and K− 1

2(Sn)

consisting of Killing spinors with Killing constants ±12

respectively, are Frechet submanifoldsof GK(Sn).

Theorem 5.1. The submanifolds K± 12(Sn) are connected components of GK(Sn).

Proof. Let M be the connected component of GK(Sn) containing K 12(Sn) and let Ψt be a

curve inM starting at some point of K 12(Sn), i.e. a smooth 1-parameter family of spinors on

Sn satisfying

(27) ∇XΨt = At(X) ·Ψt,

where At ∈ Γ(End+(TSn)) is symmetric for all t and A0 = 12id. Without any loss in generality

we can assume that Ψt has unit norm for every t. We will denote the derivative with respect tot by a dot and drop the subscript whenever the objects are evaluated at t = 0. Differentiating(27) with respect to t and evaluating at t = 0 yields

(28) ∇XΨ = A(X) ·Ψ + 12X · Ψ.

Taking the covariant derivative in this equation and skew-symmetrizing gives

RY,XΨ = −[(∇XA)Y − (∇Y A)X] ·Ψ + [A(X) ∧ Y +X ∧ A(Y )] ·Ψ + 12X ∧ Y · Ψ.

Using the fact that the spinorial curvature on the sphere satisfies RY,XΦ = 12X ∧ Y · Φ for

every spinor Φ, the previous equation reads

(29) [(∇XA)Y − (∇Y A)X] ·Ψ = [A(X) ∧ Y +X ∧ A(Y )] ·Ψ.

On the other hand, differentiating at t = 0 the equation (8) satisfied by At yields

0 = 2(trA)(trA)− 2tr(AA) = (n− 1)trA,


whence A is trace-free at t = 0. Moreover, from (7) we also get δ∇A = 0.

We now use the fact that |X · Φ|2 = |X|2 for every X ∈ TM and for every unit spinor Φ,whereas |ω ·Φ|2 ≤ |ω|2 for ω ∈ Λ2M . From (29) we thus get (using a local orthonormal basisei of the tangent bundle):

|d∇A|2 = 12

n∑i,j=1

|(∇eiA)ej − (∇ej A)ei|2 ≤ 12

n∑i,j=1

|A(ei) ∧ ej + ei ∧ A(ej)|2

= (n− 1)|A|2 +n∑

i,j=1

g(A(ei) ∧ ej, ei ∧ A(ej)) = (n− 2)|A|2.

Recall now the Weitzenbock formula for trace-free symmetric tensors h (cf. [7, Prop. 4.1]):

(30) (d∇δ∇ + δ∇d∇)h = ∇∗∇h+ h Ric− R(h),

where

R(h)(X) :=n∑i=1

RX,h(ei)ei

(note that there is a sign change between Bourguignon’s and our curvature convention). On

the round sphere Sn we have Ric = (n − 1)id and R(h)(X) = −h(X). Applying (30) toh := A and using the relation above δ∇A = 0, we get

δ∇d∇A = ∇∗∇A+ nA.

Taking the scalar product with A and integrating over Sn (whose volume element is denotedby vol) yields ∫

Sn|d∇A|2vol =

∫Sn

(|∇A|2 + n|A|2

)vol,

which together with the previous inequality |d∇A|2 ≤ (n− 2)|A|2 implies A = 0.

Going back to (28) we thus see that Ψ is a Killing spinor. In other words, we have shownthat for every Ψ ∈ K 1

2(Sn), the tangent space TΨM is contained in K 1

2(Sn). This shows that

M = K 12(Sn). The proof of the statement for K− 1

2(Sn) is similar.

6. Appendix. The canonical spinor on 3-Sasakian manifolds of dimension 7

We give here an alternative definition of the canonical spinor on 3-Sasakian 7-dimensionalmanifolds discovered by Agricola and Friedrich [1]. This approach makes use of the Riemann-ian cone construction which we now recall.

The Riemannian cone over (M, g) is the Riemannian manifold (M, g) := (R+×M,dt2+t2g).The radial vector ξ := t ∂

∂tsatisfies the equation

(31) ∇Xξ = X, ∀ X ∈ TM,


where ∇ denotes the Levi-Civita covariant derivative of g. Assume now that M is 3-Sasakian.It is well known (and is nowadays the standard definition of 3-Sasakian structures) that Mhas a hyperkahler structure J1, J2, J3, such that the vector fields ξi := Ji(ξ) on M are Killingand tangent to the hypersurfaces Mt := t ×M . When restricted to M = M1, ξi are unitKilling vector fields satisfying the 3-Sasakian relations.

Suppose now that M has dimension 7. The real spin bundle of M is canonically identifiedwith the positive spin bundle Σ+M restricted to M1 = M . With respect to this identification,if ψ ∈ Γ(ΣM) is the restriction to M of a spinor Ψ ∈ Γ(Σ+M) and X is any vector field onM identified with a vector field on M along M1, then

(32) X · ψ = X · ξ ·Ψ

and

(33) ∇Xψ = ∇XΨ + 12X · ξ ·Ψ.

Recall now that the restriction to Sp(2) of the half-spin representation Σ+8 has a 3-dimensional

trivial summand. Correspondingly, on M there exist three linearly independent ∇-parallelspinor fields on which every 2-form from sp(2) (i.e. commuting with J1, J2, J3) acts triviallyby Clifford multiplication. Moreover, there exists exactly one such unit spinor Ψ1 (up to sign)on which the Clifford action of Ω1 (the Kahler form of J1) is also trivial (cf. [19]).

Lemma 6.1. The spinor Ψ0 := 1|ξ|2 ξ · ξ1 ·Ψ1 satisfies

(34) ∇XΨ0 = A(X) · ξ ·Ψ0,

where

A(X) :=

0 if X belongs to the distribution D := 〈ξ, ξ1, ξ2, ξ3〉

−2X if X ∈ D⊥.

Proof. Since Ji are ∇-parallel, (31) yields ∇Xξi = Ji(X) for all X ∈ TM . We thus have

(35) ∇XΨ0 = 1|ξ|2 (X · ξ1 ·Ψ1 + ξ · J1(X) ·Ψ1)− 2

|ξ|4 g(ξ,X)ξ · ξ1 ·Ψ1.

This relation gives immediately ∇ξΨ0 = 0 and ∇ξ1Ψ0 = 0. Moreover, since the 2-formξ∧ ξ1− ξ2∧ ξ3 commutes with J1, J2, J3, it belongs to sp(2) and thus acts trivially by Cliffordmultiplication on Ψ1. We then obtain ξ · ξ1 ·Ψ0 = ξ2 · ξ3 ·Ψ0, which together with (35) yields∇ξ2Ψ0 = ∇ξ3Ψ0 = 0.

It remains to treat the case where X is orthogonal to 〈ξ, ξ1, ξ2, ξ3〉. Assume that X isscaled to have unit norm. We consider the orthonormal basis of TM at some point x ∈ M1

given by e1 = ξ, e2 = ξ1, e3 = ξ2, e4 = ξ3, e5 = X, e6 = J1(X), e7 = J2(X), e8 = J3(X). SinceΩ1 · Ψ1 = 0 where Ω1 = e1 · e2 + e3 · e4 + e5 · e6 + e7 · e8, we obtain Ω1 · Ψ0 = 0. Now, the2-form e5 ∧ e6 − e7 ∧ e8 belongs to sp(2) and its Clifford action commutes with e1 · e2, thus


e5 · e6 ·Ψ0 = e7 · e8 ·Ψ0. Together with the relation e1 · e2 ·Ψ0 = e3 · e4 ·Ψ0 proved above andthe fact that Ω1 ·Ψ0 = 0, we get

(36) (e1 · e2 + e5 · e6) ·Ψ0 = 0.

Using (35) we then compute at x:

∇XΨ0 = (X · ξ1 ·Ψ1 + ξ · J1(X) ·Ψ1) = (e5 · e2 + e1 · e6) · (−e1 · e2 ·Ψ0)

= (e1 · e5 + e2 · e6) ·Ψ0 = e5 · e2 · (e1 · e2 + e5 · e6) ·Ψ0 + 2e1 · e5 ·Ψ0

= 2e1 · e5 ·Ψ0 = −2X · ξ ·Ψ0,

thus proving the lemma.

As a direct consequence of this result, together with (32)–(33), we obtain the following:

Corollary 6.2 ([1], Thm. 4.1). The spinor ψ0 := Ψ0|M is a generalized Killing spinor on Msatisfying

(37) ∇Xψ0 = A(X) · ψ0,

where

A(X) :=

12X if X belongs to the distribution D := 〈ξ1, ξ2, ξ3〉

−32X if X ∈ D⊥.

References

[1] I. Agricola, Th. Friedrich, 3-Sasakian manifolds in dimension seven, their spinors and G2-structures, J.Geom. Phys. 60 (2010), no. 2, 326–332.

[2] B. Ammann, A. Moroianu, S. Moroianu, The Cauchy problems for Einstein metrics and parallel spinors,Commun. Math. Phys. 320 (2013), 173–198.

[3] C. Bar, Real Killing spinors and holonomy, Commun. Math. Phys. 154 (1993), 509–521.[4] C. Bar, P. Gauduchon, A. Moroianu, Generalized Cylinders in Semi-Riemannian and Spin Geometry,

Math. Z. 249 (2005), 545–580.[5] H. Baum, Th. Friedrich, R. Grunewald, I. Kath, Twistor and Killing Spinors on Riemannian Manifolds,

Teubner-Verlag, Stuttgart-Leipzig, 1991.[6] A. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 10 Springer-Verlag,

Berlin, 1987.[7] J.-P. Bourguignon, Les varietes de dimension 4 a signature non nulle dont la courbure est harmonique

sont d’Einstein, Invent. Math. 63 (1981), 263–286.[8] S. Chiossi, S. Salamon, The intrinsic torsion of SU(3) and G2 structures, Differential geometry, Valencia,

2001, 115–133, World Sci. Publ., River Edge, NJ, 2002.[9] D. Conti, S. Salamon, Reduced holonomy, hypersurfaces and extensions, Int. J. Geom. Methods Mod.

Phys. 3 (2006), no. 5-6, 899–912.[10] M. Fernandez, A. Gray, Riemannian manifolds with structure group G2, Ann. Mat. Pura Appl. (4) 132

(1982), 19–45.[11] Th. Friedrich, On the spinor representation of surfaces in Euclidean 3-space, J. Geom. Phys. 28 (1998),

143–157.[12] Th. Friedrich, E.C. Kim, Some remarks on the Hijazi inequality and generalizations of the Killing equation

for spinors J. Geom. Phys. 37 (2001), no. 1-2, 1–14.


[13] N. Hitchin, Stable forms and special metrics, Global differential geometry: the mathematical legacy ofAlfred Gray (Bilbao, 2000), 70–89, Contemp. Math. 288, Amer. Math. Soc. Providence, RI, 2001.

[14] B. Lawson, M.-L. Michelson, Spin Geometry, Princeton University Press, Princeton 1989.[15] T.B. Madsen, S. Salamon, Half-flat structures on S3×S3, Ann. Global Anal. Geom. doi:10.1007/s10455-

013-9371-3.[16] B. Morel, The energy-momentum tensor as a second fundamental form, math.DG/0302205.[17] A. Moroianu, U. Semmelmann, Generalized Killing spinors on Einstein manifolds, arXiv:1303.6179.[18] F. Schulte-Hengesbach, Half-flat structures on Lie groups, thesis, Universitat Hamburg, 2010,

http://www.math.uni-hamburg.de/home/schulte-hengesbach/diss.pdf.[19] M.Y. Wang, Parallel Spinors and Parallel Forms, Ann. Global Anal. Geom. 7 (1989), 59–68.

Andrei Moroianu, Universite de Versailles-St Quentin, Laboratoire de Mathematiques,UMR 8100 du CNRS, 45 avenue des Etats-Unis, 78035 Versailles, France

E-mail address: [email protected]

Uwe Semmelmann, Institut fur Geometrie und Topologie, Fachbereich Mathematik, Uni-versitat Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

E-mail address: [email protected]

Andrei MoroianuE-Mail: [email protected]

Uwe SemmelmannUniversitat StuttgartFachbereich MathematikPfaffenwaldring 5770569 StuttgartGermanyE-Mail: [email protected]



Erschienene Preprints ab Nummer 2007/2007-001Komplette Liste: http://www.mathematik.uni-stuttgart.de/preprints

2013-013 Moroianu, A.; Semmelmann, U.: Generalized Killling spinors on Einstein manifolds

2013-012 Moroianu, A.; Semmelmann, U.: Generalized Killing Spinors on Spheres

2013-011 Kohls, K; Rosch, A.; Siebert, K.G.: Convergence of Adaptive Finite Elements forControl Constrained Optimal Control Problems

2013-010 Corli, A.; Rohde, C.; Schleper, V.: Parabolic Approximations of Diffusive-DispersiveEquations

2013-009 Nava-Yazdani, E.; Polthier, K.: De Casteljau’s Algorithm on Manifolds

2013-008 Bachle, A.; Margolis, L.: Rational conjugacy of torsion units in integral group ringsof non-solvable groups

2013-007 Knarr, N.; Stroppel, M.J.: Heisenberg groups over composition algebras

2013-006 Knarr, N.; Stroppel, M.J.: Heisenberg groups, semifields, and translation planes

2013-005 Eck, C.; Kutter, M.; Sandig, A.-M.; Rohde, C.: A Two Scale Model for Liquid PhaseEpitaxy with Elasticity: An Iterative Procedure

2013-004 Griesemer, M.; Wellig, D.: The Strong-Coupling Polaron in Electromagnetic Fields

2013-003 Kabil, B.; Rohde, C.: The Influence of Surface Tension and Configurational Forceson the Stability of Liquid-Vapor Interfaces

2013-002 Devroye, L.; Ferrario, P.G.; Gyorfi, L.; Walk, H.: Strong universal consistent estimateof the minimum mean squared error

2013-001 Kohls, K.; Rosch, A.; Siebert, K.G.: A Posteriori Error Analysis of Optimal ControlProblems with Control Constraints

2012-018 Kimmerle, W.; Konovalov, A.: On the Prime Graph of the Unit Group of IntegralGroup Rings of Finite Groups II

2012-017 Stroppel, B.; Stroppel, M.: Desargues, Doily, Dualities, and ExceptionalIsomorphisms

2012-016 Moroianu, A.; Pilca, M.; Semmelmann, U.: Homogeneous almostquaternion-Hermitian manifolds

2012-015 Steinke, G.F.; Stroppel, M.J.: Simple groups acting two-transitively on the set ofgenerators of a finite elation Laguerre plane

2012-014 Steinke, G.F.; Stroppel, M.J.: Finite elation Laguerre planes admitting atwo-transitive group on their set of generators

2012-013 Diaz Ramos, J.C.; Dominguez Vazquez, M.; Kollross, A.: Polar actions on complexhyperbolic spaces

2012-012 Moroianu; A.; Semmelmann, U.: Weakly complex homogeneous spaces

2012-011 Moroianu; A.; Semmelmann, U.: Invariant four-forms and symmetric pairs

2012-010 Hamilton, M.J.D.: The closure of the symplectic cone of elliptic surfaces

2012-009 Hamilton, M.J.D.: Iterated fibre sums of algebraic Lefschetz fibrations

2012-008 Hamilton, M.J.D.: The minimal genus problem for elliptic surfaces

2012-007 Ferrario, P.: Partitioning estimation of local variance based on nearest neighborsunder censoring

2012-006 Stroppel, M.: Buttons, Holes and Loops of String: Lacing the Doily

2012-005 Hantsch, F.: Existence of Minimizers in Restricted Hartree-Fock Theory

2012-004 Grundhofer, T.; Stroppel, M.; Van Maldeghem, H.: Unitals admitting all translations

2012-003 Hamilton, M.J.D.: Representing homology classes by symplectic surfaces

2012-002 Hamilton, M.J.D.: On certain exotic 4-manifolds of Akhmedov and Park

2012-001 Jentsch, T.: Parallel submanifolds of the real 2-Grassmannian

2011-028 Spreer, J.: Combinatorial 3-manifolds with cyclic automorphism group

2011-027 Griesemer, M.; Hantsch, F.; Wellig, D.: On the Magnetic Pekar Functional and theExistence of Bipolarons

2011-026 Muller, S.: Bootstrapping for Bandwidth Selection in Functional Data Regression

2011-025 Felber, T.; Jones, D.; Kohler, M.; Walk, H.: Weakly universally consistent staticforecasting of stationary and ergodic time series via local averaging and leastsquares estimates

2011-024 Jones, D.; Kohler, M.; Walk, H.: Weakly universally consistent forecasting ofstationary and ergodic time series

2011-023 Gyorfi, L.; Walk, H.: Strongly consistent nonparametric tests of conditionalindependence

2011-022 Ferrario, P.G.; Walk, H.: Nonparametric partitioning estimation of residual and localvariance based on first and second nearest neighbors

2011-021 Eberts, M.; Steinwart, I.: Optimal regression rates for SVMs using Gaussian kernels

2011-020 Frank, R.L.; Geisinger, L.: Refined Semiclassical Asymptotics for Fractional Powersof the Laplace Operator

2011-019 Frank, R.L.; Geisinger, L.: Two-term spectral asymptotics for the Dirichlet Laplacianon a bounded domain

2011-018 Hanel, A.; Schulz, C.; Wirth, J.: Embedded eigenvalues for the elastic strip withcracks

2011-017 Wirth, J.: Thermo-elasticity for anisotropic media in higher dimensions

2011-016 Hollig, K.; Horner, J.: Programming Multigrid Methods with B-Splines

2011-015 Ferrario, P.: Nonparametric Local Averaging Estimation of the Local VarianceFunction

2011-014 Muller, S.; Dippon, J.: k-NN Kernel Estimate for Nonparametric FunctionalRegression in Time Series Analysis

2011-013 Knarr, N.; Stroppel, M.: Unitals over composition algebras

2011-012 Knarr, N.; Stroppel, M.: Baer involutions and polarities in Moufang planes ofcharacteristic two

2011-011 Knarr, N.; Stroppel, M.: Polarities and planar collineations of Moufang planes

2011-010 Jentsch, T.; Moroianu, A.; Semmelmann, U.: Extrinsic hyperspheres in manifoldswith special holonomy

2011-009 Wirth, J.: Asymptotic Behaviour of Solutions to Hyperbolic Partial DifferentialEquations

2011-008 Stroppel, M.: Orthogonal polar spaces and unitals

2011-007 Nagl, M.: Charakterisierung der Symmetrischen Gruppen durch ihre komplexeGruppenalgebra

2011-006 Solanes, G.; Teufel, E.: Horo-tightness and total (absolute) curvatures in hyperbolicspaces

2011-005 Ginoux, N.; Semmelmann, U.: Imaginary Kahlerian Killing spinors I

2011-004 Scherer, C.W.; Kose, I.E.: Control Synthesis using Dynamic D-Scales: Part II —Gain-Scheduled Control

2011-003 Scherer, C.W.; Kose, I.E.: Control Synthesis using Dynamic D-Scales: Part I —Robust Control

2011-002 Alexandrov, B.; Semmelmann, U.: Deformations of nearly parallel G2-structures

2011-001 Geisinger, L.; Weidl, T.: Sharp spectral estimates in domains of infinite volume

2010-018 Kimmerle, W.; Konovalov, A.: On integral-like units of modular group rings

2010-017 Gauduchon, P.; Moroianu, A.; Semmelmann, U.: Almost complex structures onquaternion-Kahler manifolds and inner symmetric spaces

2010-016 Moroianu, A.; Semmelmann,U.: Clifford structures on Riemannian manifolds

2010-015 Grafarend, E.W.; Kuhnel, W.: A minimal atlas for the rotation group SO(3)

2010-014 Weidl, T.: Semiclassical Spectral Bounds and Beyond

2010-013 Stroppel, M.: Early explicit examples of non-desarguesian plane geometries

2010-012 Effenberger, F.: Stacked polytopes and tight triangulations of manifolds

2010-011 Gyorfi, L.; Walk, H.: Empirical portfolio selection strategies with proportionaltransaction costs

2010-010 Kohler, M.; Krzyzak, A.; Walk, H.: Estimation of the essential supremum of aregression function

2010-009 Geisinger, L.; Laptev, A.; Weidl, T.: Geometrical Versions of improvedBerezin-Li-Yau Inequalities

2010-008 Poppitz, S.; Stroppel, M.: Polarities of Schellhammer Planes

2010-007 Grundhofer, T.; Krinn, B.; Stroppel, M.: Non-existence of isomorphisms betweencertain unitals

2010-006 Hollig, K.; Horner, J.; Hoffacker, A.: Finite Element Analysis with B-Splines:Weighted and Isogeometric Methods

2010-005 Kaltenbacher, B.; Walk, H.: On convergence of local averaging regression functionestimates for the regularization of inverse problems

2010-004 Kuhnel, W.; Solanes, G.: Tight surfaces with boundary

2010-003 Kohler, M; Walk, H.: On optimal exercising of American options in discrete time forstationary and ergodic data

2010-002 Gulde, M.; Stroppel, M.: Stabilizers of Subspaces under Similitudes of the KleinQuadric, and Automorphisms of Heisenberg Algebras

2010-001 Leitner, F.: Examples of almost Einstein structures on products and incohomogeneity one

2009-008 Griesemer, M.; Zenk, H.: On the atomic photoeffect in non-relativistic QED

2009-007 Griesemer, M.; Moeller, J.S.: Bounds on the minimal energy of translation invariantn-polaron systems

2009-006 Demirel, S.; Harrell II, E.M.: On semiclassical and universal inequalities foreigenvalues of quantum graphs

2009-005 Bachle, A, Kimmerle, W.: Torsion subgroups in integral group rings of finite groups

2009-004 Geisinger, L.; Weidl, T.: Universal bounds for traces of the Dirichlet Laplace operator

2009-003 Walk, H.: Strong laws of large numbers and nonparametric estimation

2009-002 Leitner, F.: The collapsing sphere product of Poincare-Einstein spaces

2009-001 Brehm, U.; Kuhnel, W.: Lattice triangulations of E3 and of the 3-torus

2008-006 Kohler, M.; Krzyzak, A.; Walk, H.: Upper bounds for Bermudan options onMarkovian data using nonparametric regression and a reduced number of nestedMonte Carlo steps

2008-005 Kaltenbacher, B.; Schopfer, F.; Schuster, T.: Iterative methods for nonlinear ill-posedproblems in Banach spaces: convergence and applications to parameteridentification problems

2008-004 Leitner, F.: Conformally closed Poincare-Einstein metrics with intersecting scalesingularities

2008-003 Effenberger, F.; Kuhnel, W.: Hamiltonian submanifolds of regular polytope

2008-002 Hertweck, M.; Hofert, C.R.; Kimmerle, W.: Finite groups of units and theircomposition factors in the integral group rings of the groups PSL(2, q)

2008-001 Kovarik, H.; Vugalter, S.; Weidl, T.: Two dimensional Berezin-Li-Yau inequalities witha correction term

2007-006 Weidl, T.: Improved Berezin-Li-Yau inequalities with a remainder term

2007-005 Frank, R.L.; Loss, M.; Weidl, T.: Polya’s conjecture in the presence of a constantmagnetic field

2007-004 Ekholm, T.; Frank, R.L.; Kovarik, H.: Eigenvalue estimates for Schrodingeroperators on metric trees

2007-003 Lesky, P.H.; Racke, R.: Elastic and electro-magnetic waves in infinite waveguides

2007-002 Teufel, E.: Spherical transforms and Radon transforms in Moebius geometry

2007-001 Meister, A.: Deconvolution from Fourier-oscillating error densities under decay andsmoothness restrictions

andrei moroianu, uwe semmelmann - uni-stuttgart.de...andrei moroianu, uwe semmelmann abstract. we...

Documents