aa r X i v : . [ m a t h . DG ] J a n SEMI-RIEMANNIAN CONES
THOMAS LEISTNER
Abstract.
Due to a result by Gallot a Riemannian cone over a complete Riemannianmanifold is either flat or has an irreducible holonomy representation. This is false ingeneral for indefinite cones but the structures induced on the cone by holonomy invariantsubspaces can be used to study the geometry on the base of the cone. The purpose of thispaper is twofold: first we will give a survey of general results about semi-Riemannian coneswith non irreducible holonomy representation and then, as the main result, we will deriveimproved versions of these general statements in the case when the cone admits a parallelvector field. We will show that if the base manifold is complete and the fibre of the coneand the parallel vector field have the same causal character, then the cone is flat, and thatotherwise, the base manifold admits a certain global warped product structure. We willuse these results to give a new proof of the classification results for Riemannian manifoldswith imaginary Killing spinors and Lorentzian manifolds with real Killing spinors whichare due to Baum and Bohle.
Contents
1. Introduction 12. Preliminaries 33. Survey of general results 64. Semi-Riemannian cones with parallel vector fields 125. Lorentzian cones and applications to Killing spinors 19References 221.
Introduction
Given a semi-Riemannian manifold p M, g q , the (space-like or time-like) semi-Riemanniancone over p M, g q is the manifold x M “ R ą ˆ M together with the metric(1.1) p g ǫ “ ǫ d r ` r g, where ǫ “ ǫ “ ´ p M, g q is then called the base of the cone. One reason for consideringsemi-Riemannian cones is that some systems of PDE on the base correspond to PDE onthe cone where they sometimes are easier to study. The key example is the equation for Mathematics Subject Classification.
Primary 53C50; Secondary 53C29, 53B30, 53C27.
Key words and phrases.
Lorentzian manifolds, pseudo-Riemannian manifolds, metric cones, special holo-nomy, geodesic completeness, Killing spinors.This work was supported by the Niels Henrik Abel Memorial Fund in relation to the 2019 Abel Symposium“Geometry, Lie Theory and Applications” and by the Australian Research Council (Discovery ProgramDP190102360). a Killing spinor field , which is an overdetermined system of PDE. A solution to this PDEcorresponds to a spinor field on the cone that is parallel for the Levi-Civita connection of thecone metric, which is a closed system of PDE that can be understood as the prolongation ofthe original PDE and that is easier to analyse, for example, by using tools from holonomytheory. Another example is the existence of a Sasaki structure on the base, which correspondto a K¨ahler structure on the cone and hence to a holonomy reduction to the unitary group.Semi-Riemannian cones play also an important role in conformal geometry as conformalambient metrics for conformal structures containing an Einstein metric.As mentioned, the most prominent application is the classification of complete Riemann-ian manifolds with real Killing spinors by C. B¨ar in [3]. He showed that the cone over suchmanifold admits a parallel spinor. By a fundamental theorem of Gallot, the cone is eitherirreducible or flat. With Gallot’s result, the holonomy of the cone is one of the irreducibleholonomy groups from Berger’s list [8] that admit invariant spinors [13]. This leads to ashort list of structures on the cone which correspond to certain structures on the base, allof which had been shown to admit Killing spinors [7].In an attempt to apply this method to Killing spinor (and related) equations on manifoldswith indefinite metrics, in [1, 2] possible generalisations of Gallot’s theorem in the semi-Riemannian context were studied, yielding a comprehensive analysis of the case when thecone admits an invariant subspace under its holonomy representation. In the first part ofthis paper, in Section 3 we will give a brief survey of these results, including a result from[12]. However instead of providing all the details, we will then focus on the special case whenthe cone admits a parallel vector field. The focus to this this case enables us to show theessential steps in the proofs of the general result without too much technical detail and atthe same time give self contained proofs. More importantly, we will be able to improve someof the general results in this special case, in particular in regards to their global character.In Section 4 we will prove the main result of the paper:
Theorem 1.1.
Let p M, g q be a geodesically complete semi-Riemannian manifold and let p x M , p g ǫ q be the (time-like or space-like) cone over p M, g q . Assume that p x M , p g ǫ q admits aparallel vector field V .(1) If p g p V, V q “ ǫ , then the cone is flat and p M, g q is of constant curvature ǫ .(2) If p g p V, V q “ ´ ǫ , then p M, g q is globally isometric to p R ˆ N, ´ ǫ d s ` cosh p s q g N q , where p N, g N q is a complete semi-Riemannian manifold.(3) If p g p V, V q “ , then M is a disjoint union M “ M ´ Y M Y M ` with M ˘ open andsuch that M is either empty (in which case one of M ˘ is also empty) or a smoothtotally geodesic hypersurface and p M ˘ , g q is globally isometric to p R ˆ N ˘ , ´ ǫ d s ` e s g N ˘ q , where p N ˘ , g N ˘ q are complete semi-Riemannian manifolds. Moreover, M “ H ifand only if p M, g q is Riemannian or negative definite. Note that the cases (2) and (3) also include the possibility that p M, g q has constantcurvature: in (2) g N has constant curvature ǫ if and only if g also has constant curvature ǫ ,whereas in (3), g N is flat, if and only if g has constant curvature ǫ (see [1, Section 2]). EMI-RIEMANNIAN CONES 3
The improvements in this theorem over the of the general result will allow us to givean alternative proof of the classification of complete Riemannian manifolds with imaginaryKilling spinors by Baum [5] and of Lorentzian manifolds with real Killing spinors [9]. Inboth cases, the parallel spinor on the cone induces a parallel vector field on the Lorentziancone. In fact, the results in this paper will be applicable to the classification of Killingspinors whenever the parallel spinor on the cone induces a parallel vector field. Workingout the detail of this is however beyond the scope of this paper.
Acknowledgements.
This paper originated from a talk given at the Abel Symposium“Geometry, Lie Theory and Applications” in June 2019. The author would like to thankthe organisers for their hospitality and the Niels Henrik Abel Memorial Fund for financialsupport. 2.
Preliminaries
Curvature and geodesics of semi-Riemannian cones.
Let p g ǫ “ ǫ d r ` r g with ǫ “ ˘ x M “ R ą ˆ M , where p M, g q is a pseudo-Riemannian manifold.The cone is called space-like if ǫ “ ǫ “ ´
1. We denote by B r “ BB r theradial unit vector field. The Levi-Civita connection of the cone p x M , p g ǫ q is given by(2.1) p ∇ B r B r “ , p ∇ X B r “ r X, p ∇ X Y “ ∇ X Y ´ ǫg p X, Y qB r , for all vector fields X, Y P Γ p T x M q orthogonal to B r . The curvature p R of the cone is givenby the following formulas including the curvature R of the base metric g :(2.2) B r p R “ , p R p X, Y q Z “ R p X, Y q Z ´ ǫ p g p Y, Z q X ´ g p X, Z q Y q , for X, Y, Z, U P T M . This implies that if p M, g q is a space of constant curvature κ , i.e., R p X, Y, Z, U q “ κ p g p X, U q g p Y, Z q ´ g p X, Z q g p Y, U qq , then the cone has the curvature r p κ ´ ǫ q p g p X, U q g p Y, Z q ´ g p X, Z q g p Y, U qq . In particular,if κ “ ǫ , then the cone is flat, as it is the case for the ǫ “ ǫ “ ´ p γ “ p ρ, γ q : I Ñ x M “ R ą ˆ M be a geodesic of p x M , p g ǫ q starting at ˆ p and with p γ p q “ a B r ` X . The geodesic equations are easily checked to be(2.3) 0 “ ρ p t q ´ ǫr p t q g ` γ p t q , γ p t q ˘ , “ ρ p t q γ p t q ` ρ p t q ∇ γ p t q γ p t q . Let γ be a reparametrisation of a geodesic β of p M, g q ,(2.4) γ p t q “ β p f p t qq , with β p q “ p and β p q “ X ,implying the initial conditions f p q “ f p q “ f .Now let g p X, X q “ cL with c P t , ˘ u and L ą
0. Hence, from (2.3) we get(2.5) 0 “ ρ p t q ´ cǫρ p t q f p t q L , “ ρ p t q f p t q ` ρ p t q f p t q with initial conditions ρ p q “ r, f p q “ , ρ p q “ a, f p q “ . THOMAS LEISTNER
If the initial speed X satisfies cL “ g p X, X q “
0, i.e., if it is zero or light-like, then theequations become 0 “ ρ p t q , “ ρf p t q ` p ρt ` ρ q f p t q , i.e., with solutions(2.6) ρ p t q “ at ` r, f p t q “ rtat ` r . This implies that f and thus ˆ γ is defined for t P r , ´ ra q if a ă
0, and for t ě cL “
0, the solutions to equations (2.5) are then given by(2.7) ρ p t q “ a p at ` r q ` cǫL r t ,f p t q “ $&% L artan ´ Lrtat ` r ¯ , if cǫ “ L artanh ´ Lrtat ` r ¯ , if cǫ “ ´ p M, g q iscomplete: in case of cǫ “
1, in particular if the cone is Riemannian, all geodesics are definedon R if a ě t P r , ´ ra q if a ă
0. Otherwise, if the functions ρ and f are defined onan interval r , T q , where T is the first positive zero of the polynomial ´ Lrtat ` r ´ ¯ ´ Lrtat ` r ` ¯ p at ` r q “ L r t ´ p at ` r q “ pp Lr ´ a q t ´ r qpp Lr ` a q t ` r q , or T “ 8 if the polynomial has no positive zero. More explicitly, T “ rLr ´ a if a ă Lr and T “ 8 if a ě Lr . We summarise this: Proposition 2.1.
Let p M, g q be a complete semi-Riemannian manifold and p x M , p g ǫ q be thecone. Let ˆ p “ p r, p q P x M and p X “ a B r | ˆ p ` X P T ˆ p x M with g p X, X q “ cL with c P t , ˘ u and L ą . Then there is a geodesic p γ : r , T q Ñ x M of p x M , p g q starting at ˆ p with p γ p q “ p X and where (2.8) T “ $’’&’’% , if cǫ P t , u and a ě , or if cǫ “ ´ and a ě Lr , ´ ra , if cǫ P t , u and a ă , rLr ´ a , if cǫ “ ´ and a ă Lr .This geodesic is given by (2.4) together with (2.6) or (2.7). Completeness of certain warped products.
In this section we are going to studythe completeness of warped products of the form p M “ R ˆ N, g “ ´ ǫ d s ` f p s q g N q , where p N, g N q is a semi-Riemannian manifold, f is a positive function on N and ǫ “ ˘ ∇ B s B s “ , ∇ X B s “ f p s q f p s q X, ∇ X Y “ ∇ NX Y ` ǫf p s q f p s q g N p X, Y qB s , EMI-RIEMANNIAN CONES 5
Proposition 2.2.
Let f : R Ñ R ą be a smooth function and p N, g N q be a semi-Riemannianmanifold and define p M “ R ˆ N, g “ ´ ǫ d s ` f p s q g N q .(1) If all geodesics of p M, g q with initial velocity tangent to N are defined on R , then p N, g N q is complete. In particular, if p M, g q is complete, then p N, g N q is complete.(2) If f “ cosh , then p M, g q is complete if p N, g N q is complete.(3) If f p s q “ e s , then p M, g q is complete if and only if p N, g N q is complete and p M, g q is definite, i.e., if ´ ǫg N is a complete Riemannian metric.Proof. (1) Let p σ, γ q : R Ñ be a geodesic of p M, g q with σ p q “
0. Then the geodesicequations are(2.10) σ ` ǫf p σ q f p σ q g p γ , γ q “ , ∇ Nγ γ ` f p σ q f p σ q σ γ “ , in particular, γ is a pre-geodesic for g N . The first equation shows that, if γ p t q “ t , then σ p t q “ at ` b and γ p t q ” γ p t q constant. Hence, if γ p q “
0, then γ p t q “ t , and so we can parametrise γ by arc-length. The second geodesic equation showsthat the reparametrised curve is a geodesic for g N . Hence, p N, g N q is complete.(2) Assume that p M, g q is incomplete. Hence there is a maximal geodesic p σ, γ q : p a, b q Ñ M with b P R . Then the first geodesic equation in (2.10) and the equation that the geodesicis of constant length, ´ ǫ p σ q ` f p σ q g p γ , γ q “ c, for a constant c , imply thatcosh p σ q σ ` sinh p σ qp σ q ` ǫc sinh p σ q “ . Then, with substituting ξ “ sinh p σ q , this equation becomes ξ ` ǫcξ “ . This is a linear ODE for ξ and hence we can extend ξ and also σ beyond b and in fact to R . Moreover, the second geodesic equation in (2.10) implies that γ “ β ˝ τ , where β is ageodesic equation and σ and τ satisfy the equations τ ` σ τ tanh p σ q “ . With σ : R Ñ R , this is a linear ODE for τ and hence can be extended beyond b . Thisyields a contradiction to the incompleteness of p M, g q .(3) Assume that g is complete but indefinite. With g indefinite we can consider a light-likegeodesic p σ, γ q , i.e., with 0 “ ´ ǫ p σ q ` e σ g N p γ , γ q . Moreover, from the first geodesic equation we obtain0 “ ` p σ q ` σ ˘ “ ξ ´ ξ , where we substitute ξ “ e σ ą
0. This however yields the equation ξ “
0, so ξ is affine and,since p M, g q is complete, defined on R . This contradicts ξ “ e σ ą
0, so p M, g q cannot havelight-like geodesics and hence g is definite.Conversely, assume that p N, g N q is a complete Riemannian manifold. If p M, g q is notcomplete, there is a maximal geodesic γ “ p σ, β q : r , b q Ñ M that leaves every compact setin M . For such a geodesic we have1 “ p σ q ` e σ g N p β , β q . THOMAS LEISTNER
Hence, with g N Riemannian, we have 0 ď p σ q ď σ is bounded on r , b q .This implies that σ remains in a compact set, which implies that β leaves every compactset in N . It also implies that e σ is bounded away from zero and so g N p β , β q is boundedon r , b q say by c . Then we have that β p t q is contained in the geodesic ball around β p q ofradius bc since dist g N p β p q , β p t qq ď length g N p β | r ,t s q ď bc. Since p N, g N q is complete, its geodesic balls are compact, which gives a contradiction. Hence p M, g q is complete. (cid:3) Survey of general results
Holonomy groups and Gallot’s Theorem.
Let p M, g q be a semi-Riemannian con-nected manifold. The holonomy group Hol p p M, g q of p M, g q at p P M is defined as thegroup of parallel transports, with respect to the Levi-Civita connection of g , along piecewisesmooth loops that are closed at p . Since the Levi-Civita connection preserves the metric,the holonomy group is a subgroup of the orthogonal groupp O p T p M, g | p q acting on T p M .By fixing a basis of T p M , it can be identified with a subgroup of O p r, s q , where p r, s q is thesignature of p M, g q . The holonomy groups at different points in M are conjugated withinO p r, s q . Hence the holonomy group as a subgroup in O p r, s q is well defined up to conjugationand we refer to this as the holonomy group Hol p M, g q . If G Ă O p r, s q is a subgroup andHol p r, s q Ă G we say that the holonomy reduces to G .The holonomy group is a Lie group. Its connected component is given by parallel transportalong contractible loops. Its Lie algebra is denoted by hol p p M, g q , the holonomy algebra . Onecan show that the holonomy algebra contains all curvature endomorphisms R | p p X, Y q at p ,with X, Y P T p M and all derivatives of curvature endomorphisms. Moreover, the Ambrose-Singer holonomy Theorem states that the holonomy algebra at p is spanned as a vectorspace by the following linear maps, P ´ γ ˝ R | q p X, Y q ˝ P γ , where q P M , γ is a path from p to q , P γ the parallel transport along γ and X, Y P T q M .The importance of the holonomy group arises from the well-known holonomy principles.First, parallel sections (with respect to the Levi-Civita connection of g ) of T M or of anytensor bundle over M are in one-to-one correspondence with vectors (or tensors) that arefixed under the holonomy representation. For example, the existence of parallel vector fieldreduces the holonomy to a the stabiliser in O p r, s q of a vector in R r,s . Another example is theexistence of a parallel complex structure, which reduces the holonomy to the unitary group U p r { , s { q . The other principle is that subspaces in T p M , or in R r,s , that are invariantunder the holonomy group are in one-to-one correspondence with vector distributions thatare invariant under parallel transport, or for short, a parallel distribution . The paralleldistribution V Ă T M is obtained from an holonomy invariant subspace E Ă T p M byparallel transport: the fibre V | q is defined as by the parallel transport of E Ă T p M by anycurve from p to q . Because of the holonomy invariance of E , this is a well defined procedureand V | q does not depend on the chosen loop. A parallel distribution V is involutive anddefines a foliation of M into totally geodesic leaves of V .The holonomy group acts irreducibly if it does not admit any invariant subspace. In thiscase we also say that p M, g q is irreducible. If Hol p M, g q does admit an invariant subspace E , since Hol p M, g q Ă O p r, s q , the orthogonal space E K is also invariant under Hol p M, g q . EMI-RIEMANNIAN CONES 7
Hence, every holonomy invariant subspace defines two parallel distributions V and V K .If g is indefinite and E is a degenerate subspace, i.e., E X E K “ t u , there is a totallylight-like distribution V X V K with totally geodesic leaves. If E is non-degenerate, i.e.,if T p M “ E ‘ E K , then we also have T M “ V ‘ V K . In this case we say that theholonomy group acts decomposably , or for short that p M, g q is decomposable. If there is nonon-degenerate subspace that is invariant under Hol p M, g q we say that the holonomy acts indecomposably , or that p M, g q is indecomposable. If g is indefinite, the holonomy groupmay act indecomposably without acting irreducibly. This is the case if the holonomy groupadmits a totally light-like invariant subspace, but no non-degenerate invariant subspace.If the holonomy group acts decomposably, not just the tangent space decomposes intoholonomy invariant subspaces, but under certain global assumptions also the manifold de-composes into a semi-Riemannian product. This is due to the splitting theorems of deRham [10] and Wu [14]: if p M, g q is complete and simply connected and the holonomygroup acts decomposably, then p M, g q is isometric to a global semi-Riemannian product p M , g q ˆ p M , g q and the holonomy representation of p M, g q is isomorphic to the productof the holonomy representations of p M i , g i q . The manifolds M i correspond to the totallygeodesic foliations of M into the leaves of the parallel complementary distributions V and V K .The notions of irreducibility and (in-)decomposability can also be formulated for theholonomy algebras hol p M, g q , depending on wether the holonomy algebra admits a (non-degenerate) invariant subspace. Note that if M is not simply connected, the holonomyalgebra acting decomposably does not imply that the holonomy group does act decompos-ably. In particular, the existence of a non-degenerate subspace that is invariant under theholonomy algebra does not necessarily imply the existence of a globally defined paralleldistribution.In regards to the holonomy algebra of a Riemannian cone, Gallot proved the followingresult: Theorem 3.1 (S. Gallot, [11]) . Let p M, g q be a complete Riemannian manifold of dimension ě such that the holonomy algebra of the cone p x M , p g ` q does not act irreducibly. Then p x M , p g ` q is flat and hence p M, g q has constant curvature . If, in addition, p M, g q is simplyconnected, then p M, g q is isometric to the standard sphere. We will present Gallot’s proof of this theorem in Section 4. Here we will only explainits first step, which is needed in order to understand possible generalisations and which isbased on the aforementioned holonomy principle : since the aim is to show that the coneis flat, we can pass to the universal cover, which is the cone over the universal cover of M , and assume that the holonomy group of this cone does admit an invariant subspace E Ă T p x M . Hence, this invariant subspace defines a vector distribution V Ă T x M that isinvariant under parallel transport. With E holonomy invariant, its orthogonal space E K isalso holonomy invariant and defines a parallel distribution V K . If the cone is Riemannian, V and V K are non-degenerate and hence the tangent space splits into a direct sum of parallelvector distributions T M “ V ‘ V K . Both distributions are parallel and hence involutiveand define totally geodesic leaves. This splitting and the induced foliation is then used inGallot’s proof.If the cone metric is indefinite, for example by considering time-like cones over Riemannianmanifolds or because already p M, g q is indefinite, a holonomy invariant subspace may be THOMAS LEISTNER degenerate, i.e., E X E K “
0, and hence V X V K “ t u , so that the resulting paralleldistributions are not complementary. The following example shows that Gallot’s Theoremis false in this case. Example 3.2.
Consider the semi-Riemannian manifold(3.1) ` M “ R ˆ N, g “ d s ` e ´ s g N ˘ , where p N, g N q is a semi-Riemannian manifold. Then the light-like vector field V “ e ´ s pB r ` r B s q on the time-like cone p x M , p g ´ q is parallel. The manifold p M, g q has constant negative curva-ture only if g N is flat. If we now assume that p N, g N q is a complete Riemannian manifold,then, by Proposition 2.2, p M, g q is a complete Riemannian manifold whose time-like cone p x M , p g ´ q admits a parallel light-like vector field and hence has a non irreducible holonomygroup. However, unless g N is flat, the cone p g ´ is not flat. This shows that Gallot’s Theoremcannot hold when the cone has a parallel light-like vector field.This example suggests that one has to strengthen the assumptions in Gallot’s Theoremin the indefinite setting. To get Gallot’s proof started, instead of assuming the existenceof some holonomy invariant subspace, on should require the existence of a non-degenerate invariant subspace, that gives complementary parallel distributions T x M “ V ‘ V K . However,the following example shows that such a modification of Gallot’s Theorem also fails. Example 3.3.
Let p N, g N q be a complete semi-Riemannian manifold of dimension at least 2and which is not of constant curvature 1. Then the semi-Riemannian manifold(3.2) p M “ R ˆ N, g “ ds ` cosh p s q g N q is complete by Proposition 2.2. Using equation (2.9) it is easily established that the spacelikevector field V “ ´ sinh p s qB r ` cosh p s q r B s on the time-like cone p x M , p g ´ q is parallel. For the curvature tensor R of p M, g q we have R p X, Y q Z “ R N p X, Y q Z ` tanh p s q p g N p Y, Z q X ´ g N p X, Z q Y q , where X, Y, Z, U P T F and R N is the curvature tensor of p N, g N q . This shows that p M, g q cannot have constant sectional curvature, unless N has constant curvature 1. Thus, ingeneral the cone p x M , p g q over the complete manifolds p M, g q is decomposable but not flat.3.2. Decomposable cones over complete and over compact manifolds.
In this sec-tion we will review a few results that show to which extent Gallot’s Theorem generalises tothe semi-Riemannian context, having in mind the counter examples of the previous section.We will mainly focus on space-like cones, as the corresponding results for time-like can beobtained by multiplying the cone metric by ´ EMI-RIEMANNIAN CONES 9
Theorem 3.4 ([1]) . Let p M, g q be a complete semi-Riemannian manifold of dimension ě and assume that the holonomy algebra of the cone p x M , p g ` q acts decomposably. Then thereexists an open dense submanifold M Ă M such that each connected component of M isisometric to a pseudo-Riemannian manifold of the form (1) a pseudo-Riemannian manifold M of constant sectional curvature , or (2) a pseudo-Riemannian manifold M “ R ą ˆ N ˆ N with the metric ´ d s ` cosh p s q g ` sinh p s q g , where p N , g q and p N , g q are semi-Riemannian manifolds and p N , g q has con-stant sectional curvature ´ or dim N ď .Moreover, the cone x M is isometric to the open subset t r ą r u in the productof the space-like cone p R ą ˆ N , d r ` r g q over p N , g q and the time-like cone p R ą ˆ N , ´ d r ` r g q over p N , g q . Note that Example 3.3 shows that this theorem is sharp.Next we consider cones over closed semi-Riemannian manifolds p M, g q , i.e., when M compact without boundary. Recall that for indefinite metrics compactness of M does notimply the geodesic completeness of p M, g q , so we have to assume it, in order to get a versionof Gallot’s Theorem under these strengthened assumptions. Theorem 3.5 ([1]) . Let p M, g q be a closed and geodesically complete semi-Riemannianmanifold of dimension ě . If the cone p x M , p g ` q is decomposable, then it is flat and hence p M, g q has constant curvature . Since there is no simply connected compact indefinite pseudo-Riemannian manifold ofconstant curvature 1, we obtain the following corollary.
Corollary 3.6 ([1]) . If p M, g q is a simply connected compact and complete indefinite pseudo-Riemannian manifold, then the holonomy group of the cone p x M , p g q is indecomposable. Theorem 3.5 was strengthened by Matveev in [12].
Theorem 3.7 (V. Matveev [12]) . Let M be a closed manifold.(1) If g is a light-like complete indefinite semi-Riemannian metric on M , then the cone p x M , p g ` q is indecomposable.(2) If g is a Riemannian metric on M , then the cone p x M , p g ´ q is indecomposable. Note that (2) in the Theorem 3.7 implies that even though the time-like cone over acompact quotient M “ H n { Γ of hyperbolic space H n is flat, its holonomy group acts inde-composably.3.3. Local structure of non irreducible cones.
In this section we will review someresults about the local structure of non irreducible cones. We start with decomposablecones.
Theorem 3.8 ([1]) . Let p M, g q be a semi-Riemannian manifold such that the holonomyalgebra of the cone p x M , p g ` q acts decomposably. Then there exists an open dense submanifold M Ă M such that any point p P M has a neighborhood U that is isometric to a semi-Riemannian manifold of the form p a, b q ˆ N ˆ N with the metric given either by (3.3) g ` “ d s ` cos p s q g ` sin p s q g or g ´ “ ´ d s ` cosh p s q g ` sinh p s q g , where g and g are metrics on N and N respectively.Moreover, R ą ˆ U Ă x M with the metric g ˘ in (3.3) is locally isometric to the productof cone metrics p d r ` r g q ` p˘ d r ` r g q . Note that this theorem also applies to the Riemannian context. The cone over the incom-plete Riemannian metric g ` in (3.3), with g and g Riemannian, is decomposable withoutbeing flat.Next we consider the case when the holonomy of the cone admits an invariant degeneratesubspace E . This implies the existence of an invariant subspace E X E K that is totallylight-like. We restrict ourselves to the case when the dimension of E X E K is 1 or 2. In thiscase we have a parallel distribution of totally light-like lines or planes. Theorem 3.9 ([2]) . Let p x M , p g ´ q be the time-like cone over a semi-Riemannian manifold p M, g q . If the cone admits a parallel light-like line field L , then locally there is a paralleltrivializing section of L . Moreover, on a dense open subset x M reg Ă x M , the metric p g is locallyisometric to a warped product of the form (3.4) r g “ u d v ` u g , with a semi-Riemannian metric g , and the metric g is locally of the form g “ d s ` e s g . The results in the case when E X E K is of dimenion 2 are more technical and related tothe existence of s shearfree, geodesic, light-like congruence on the base: Theorem 3.10 ([2]) . The time-like cone p x M , p g q over a semi-Riemannian manifold p M, g q admits a parallel, totally light-like -plane field if and only if, locally over an open densesubset, the base p M, g q admits two vector fields V and Z satisfying (3.5) g p V, V q “ , g p Z, Z q “ , g p V, Z q “ , and such that ∇ X V “ α p X q V ` g p X, V q Z, ∇ X Z “ ´ X ` β p X q V ` g p X, Z q Z, (3.6) with -forms α and β on M . In particular, the base p M, g q admits a geodesic, shearfreelight-like congruence defined by V . Note that the first equation in equation (3.6) implies that V K is integrable. This allowsus to determine the local form of the metrics with vector fields V and Z satisfying equations(3.5) and (3.6): Proposition 3.11 ([2]) . A semi-Riemannian metric p M, g q admits vector fields V and Z with (3.5) and (3.6) if and only if p M, g q is locally of the form M “ M ˆ R and g “ d s ` e ´ s g p u q ` u η, for a family of metrics g p u q on M depending on u and a -form η on M such that η pB t q is nowhere vanishing satisfying the following system of first order PDEs: (3.7) B t η t “ B s η t “ Xη t “ B t p η p X qq “ , B t η s “ η t , B s η p X q ´ X η s “ ´ η p X q EMI-RIEMANNIAN CONES 11 for all X P Γ p T M q and where we denote η t “ η pB t q and η s “ η pB s q . One can solve explicitly the system (3.7) in the following way: Let f “ f p u q be anarbitrary nowhere vanishing smooth function on the real line equipped with the coordinate u and f “ f p x, s, u q an arbitrary smooth function on M which does not depend on t . Let h i “ h i p x, s, u q be a ( t -independent) solution of the ordinary differential equation B s h i ` h i “ B i f for all i “ , . . . , n , where B i “ B{B x i . Then η t : “ f p u q , η s : “ tf p u q ` f p x, s, u q , η pB i q : “ h i p x, s, u q solves (3.7) and every solution is of this form.This provides us with a construction method of metrics whose cone admits a totallylight-like 2-plane. Remark 3.12.
For completeness we should mention further results in [1] for the casewhen the cone admits a holonomy invariant maximal isotropic subspace V “ V K and aninvariant maximally isotropic complement. This is equivalent to the existence of a para-K¨ahler structure on the cone. In [1, Section 8] we have shown that the existence of apara-K¨ahler structure on the cone over p M, g q is equivalent to the existence of a para-Sasakistructure on p M, g q and a similar correspondence for para-hyper-K¨ahler structures on thecone and para-3-Sasakian structures on p M, g q .3.4. Holonomy of cones.
In the last part of this survey section we are going to reviewresults about the possible holonomy groups of cones. We will consider the fundamentalcases when the holonomy group acts irreducibly or not irreducibly but indecomposably.3.4.1.
Irreducible cone holonomies.
In the case when the holonomy algebra of the cone isirreducible, we can use Berger’s list and single those out that can be cone holonomies.They key here is to observe that B r p R “ Theorem 3.13 ([2]) . If p x M , p g q is a time-like cone with irreducible holonomy algebra g , then g is isomorphic to one of the following Lie algebras (3.8) so p t, s q , u p p, q q , su p p, q q Ă so p p, q q , sp p p, q q Ă so p p, q q , so p n, C q Ă so p n, n q , g C Ă so p , q , spin p , C q Ă so p , q , g Ă so p q , spin p q Ă so p q , g p q Ă so p , q , spin p , q Ă so p , q . Holonomy of non irreducible, indecomposable cones.
In general the classification ofnon irreducible, indecomposable holonomy groups is widely open and only solved in Lorentzianand in some special cases in signature p , n q and p n, n q . We will focus here in the case wherethe invariant totally light-like subspace has dimension 1. The key here is the result inTheorem 3.9, where it was shown that that a cone that admits a parallel light-like linedistribution is locally isometric to a metric of the form (3.4). Theorem 3.14 ([2]) . Let p N, g q be a semi-Riemannian manifold in dimension n and r g the metric defined in (3.4). If the holonomy of r g acts indecomposably, then (3.9) hol p r g q Ă hol p g q ˙ R t,s where hol p g q ˙ R t,s is a subalgebra of the stabiliser algebra of B v in so p t ` , s ` q , i.e., in so p t, s q ˙ R t,s “ so p t ` , s ` q B v , and pr so p t,s q p hol p r g qq “ hol p g q . There is an equality in (3.9) whenever p N, g q is one of the following:(1) an irreducible locally symmetric space, or a product thereof;(2) a Riemannian manifold;(3) a Lorentzian manifold without a parallel light-like vector field. Semi-Riemannian cones with parallel vector fields
In this section we will consider the special case when the invariant subspace under theholonomy group of the cone is given by a parallel vector field, that is, the rank of theinvariant distribution is one and the distribution admits a global parallel section. For thisspecial case we will prove versions of the theorems in the previous section that are slightlystronger and more specific, and we will prove Theorem 1.1. Before we do this we will reviewGallot’s original proof of his theorem in order to see when we can generalise it to the caseof a parallel vector field. We will see that this can be done when the radial vector field andthe parallel vector field have the same causal character.4.1.
The proof of Gallot’s Theorem.
Gallot’s proof of Theorem 3.1 uses the followingfundamental observation, which holds not only for Riemannian cones.
Lemma 4.1.
Let p x M , p g ǫ q be semi-Riemannian cone and let V Ă T x M be a non degenerate,parallel distribution. Let p P M such that B r | p P V | p and N K | p the leaf of V K through p .Then the image in N K p under the exponential map restricted to V K p P T p x M is flat.Proof. Let p γ “ p ρ, γ q : I Ñ x M “ R ą ˆ M be a geodesic in p x M , p g q with p γ p q “ p and p γ p q P V K | p . It is easy to check using (2.1) that the vector field(4.1) F p t q “ ρ p t qB r ´ t p γ p t q is parallel transported along ˆ γ . Then with F p q “ r p p qB r | ˆ p P V | ˆ p we have that F p t q P V | p γ p t q for all t . Since the curvature tensor leave parallel distributions invariant and because of B r p R “
0, we have that p R p X, Y q F p t q “ p R p X, Y q p γ p t q P R ¨ V | p γ p t q for all t . On the other hand we have that p γ p t q P V K ˆ γ p t q for all t . Hence, with V X V K “ t u this implies that p R p X, Y q p γ p t q “ , for all vector fields X and Y along p γ and all t P I . From this we see that the Jacobi fieldsalong p γ are those of a flat manifold, which implies that N is flat. (cid:3) Using this lemma, we can now proceed with the proof of Gallot’s Theorem.
EMI-RIEMANNIAN CONES 13
Proof of Theorem 3.1.
By passing to the universal cover of the cone, which is the cone overthe universal cover of M , we can assume that x M is simply connected. Let V be a paralleldistribution in T x M and V K the orthogonal distribution that are induced by the subspacethat is invariant under the holonomy group. If we assume that V is non degenerate, as wecan in the case of a Riemannian manifold p M, g q , we have T x M “ V ‘ V K . For a given point p P x M denote by N p and N K p the totally geodesic leaves of V and V K . Moreover, denote C “ t p P x M | B r | p P V | p u , C K “ t q P x M | B r | q P V K | q u . Note that p ∇ X B r “ r X for X P T M implies that neither C nor C K can contain an open setand hence that x M “ x M zp C Y C K q is dense in x M . Lemma 4.2.
Let p x M , p g q be a Riemannian cone over a complete Riemannian manifold p M, g q . Then for each point x P x M zp C Y C K q there is a p P C and a q P C K such that x lies in the image of the exponential map exp p restricted to V K | p and in the image of exp q restricted to V | q .Proof. Let x P M and assume that x R C Y C K . Let B r | x “ V ` W with V “ pr V | x pB r | x q P V | x and W “ pr V K | x pB r | x q P V K | x “
0. Then p g pB r , W q “ p g p V ` W, W q “ p g p W, W q and p g p V, V q “ p g pB r ´ W, B r ´ W q “ ´ p g p W, W q , which implies that(4.2) 0 ă p g pB r , W q ă . Let ˆ γ “ p ρ, γ q be the maximal geodesic starting at x with ρ p x q “ r , satisfying the initialcondition p γ p q “ ´ rW “ ´ r pr V K | x pB r | x q . Now we have a “ ρ p q “ ´ r p g pB r , W q , and hence, by the previous section, the maximalgeodesic is defined for t ă T with T “ ´ ra “ p g pB r , W q ą , by (4.2). Let F p t q be the parallel transported vector field defined in (4.1) along ˆ γ . Then F p q ` p γ p q “ r B r | x ´ r pr V K | x pB r | x q P V | x . The parallel transport of this vector up to t “ F p q ` ˆ γ p q “ r p ˆ γ p qqB r | ˆ γ p q , which is in V | ˆ γ p q as V is a parallel distribution. This implies that ˆ γ p q P C .The argument for C K works completely analogously. (cid:3) Both lemmas imply that each point in x M zp C Y C K q lies in the intersection of two flatleaves of V and V K and hence has a flat neighbourhood. This implies that p g on x M zp C Y C K q is flat. Since x M zp C Y C K q is dense in x M , this implies that p x M , p g q is flat. This finishes theproof of Theorem 3.1. (cid:3) A generalisation of Gallot’s Theorem.
Let p x M , p g ǫ q be a time-like or space-likecone over a semi-Riemannian p M, g q . From now on we restrict to the case when V “ R ¨ V ,where V is a parallel vector field, normalised such that p g p V, V q “ ν P t´ , , u . Since V is assumed to be parallel, the leaves of R ¨ V are flat, so in order to generaliseGallot’s Theorem we would need to show that the leaves of V K are also flat. In order showthis using Gallot’s method, we need that the set C “ t p P x M | B r | p P R V | p u is not empty. This however can only be the case when B r and V have the same causalcharacter, i.e., only when ǫ “ ν , i.e., C “ H implies ǫ “ ν. We have already seen Examples 3.2 and 3.3, which show that Gallot’s Theorem does notgeneralise when this condition is not satisfied, i.e., when ǫ “ ν . We will deal with this casein the next section. Here we consider the case when ν “ ǫ . In this special case we obtain ageneralisation of Gallots Theorem as a stronger version of Theorem 3.4. Theorem 4.3.
Let p M, g q be a complete semi-Riemannian manifold and let p x M , p g ǫ q be thecone over p M, g q . If p x M , p g q admits a parallel vector field V with p g p V, V q “ ǫ , then the coneis flat and p M, g q is of constant curvature ǫ .Proof. Let V be the parallel vector field on p x M , p g ǫ q with p g p V, V q “ ǫ . As in the proof ofTheorem 3.1 we consider the set C “ t p P x M | B r | p “ R ¨ V | p u and show that each q P x M z C admits a flat neighbourhood. Let B r | q “ αV ` W with W P V K and, since p g p V, V q “ ǫ , with α “ ǫ p g p V, B r q . Again we have p g pB r , W q “ p g p W, W q “ w “ , and α ǫ “ p g pB r ´ W, B r ´ W q “ ǫ ´ p g p W, W q “ ǫ ´ w. Hence we obtain(4.3) 0 ă α “ ´ ǫw. On the other hand we write W “ ǫ p g pB r , W qB r ` W “ ǫw B r ` W , with a W P T q M . Hence, p g p W, W q “ p g p W, W q ǫ ` p g p W , W q , and hence(4.4) p g p W , W q “ w p ´ ǫw q . Now let ˆ γ be a geodesic starting at q with r p q q “ r and with ˆ γ p q “ ´ rW . We will showthat ˆ γ is defined on r , s . We have ˆ γ p q “ a B r ´ rW with (as in Lemma 2.1) a “ ´ ǫwr, cL “ r g p W , W q “ p g p W , W q “ w p ´ ǫw q , with c “ ˘
1. We now consider the cases cǫ “ cǫ “ ´ c “ EMI-RIEMANNIAN CONES 15 If cǫ “
1, then 0 ă L “ ǫw p ´ ǫw q , which, together with 4.3 implies that ǫw ą a “ ´ ǫwr ă
0. By Lemma 2.1, ˆ γ isdefined for t ă T with T “ ´ ra “ ǫw ą , because of (4.3).If cǫ “ ´ ă L “ ´ ǫw p ´ ǫw q , and hence that a “ ´ ǫwr ą r L “ a p a ` r q “ a ` ar ą a . Hence, we are in the case a ă rL in Lemma 2.1, and ˆ γ is defined for t ă T with with T “ rLr ´ a . We show now that T ą
1. For this note that by the previous displayed equationwe have L r ´ p r ` a q “ a ` ra ´ p r ` a q “ ´ r p r ` a q ă a ą
0. This shows that Lr ă r ` a and therefore T “ rLr ´ a ą c “ w “ a “
0, so ˆ γ is defined on r , .Now we proceed in the proof of Theorem 3.1: the vector field F p t q along ˆ γ satisfies F p q ` ˆ γ p q “ rαV | q whose parallel transport is given by F p q ´ ˆ γ p q “ r p ˆ γ p qqB r | ˆ γ p q . Thisimplies that ˆ γ p q P C and by Lemma 4.1 the leaf of V K though q is flat. Since V is a parallelvector field, this implies that q has a flat neighbourhood and hence, since x M z C is dense,that p x M , p g q is flat. (cid:3) Non flat cones with parallel vector field.
Recall the two Examples 3.2 and 3.3. Wewill now show that cone with parallel vector fields satisfying the condition p g p V, V q “ ǫ arealways of the form as in these examples and thus obtain a stronger version of Theorem 3.4in the case of a parallel vector field on the cone.First we define the function u “ p g p V, B r q and observe: Lemma 4.4.
Let V be a parallel vector field on the cone p x M , p g ǫ q over a (not necessarilycomplete) semi-Riemannian manifold p M, g q . Then u “ p g p V, B r q is a smooth function on M , u P C p M q , that satisfies (4.5) V “ ǫu B r ` r ∇ u, where ∇ u is the gradient of u with respect to g , that satisfies (4.6) ∇ d u “ ´ ǫug. Proof.
With u “ p g p V, B r q , we split V as V “ ´ u B r ` W where W is a section of T M Ñ x M .Since V is parallel, we use (reflem1) to get 0 “ ˆ ∇ B r V which implies that B r p u q “ rB r , W s ` r W “
0. The latter implies that W “ r U with U P Γ p T M q is a vector field on M . The equation ∇ V | T M “ ∇ u “ U , where ∇ u denotes the gradient of u with respect to g , and ∇∇ u “ ´ ǫu Id, i.e., that ∇ d u “ ǫug . (cid:3) Recall that in the case when ν “ p g p V, V q “ ν “ ´ ǫ we have that C “ t p P x M | B r | p P R ¨ V | p u “ H and also that the set of critical points of u is empty,(4.7) C “ t p P M | ∇ u | p “ u “ H . Moreover we have(4.8) g p ∇ u, ∇ u q “ " ´ ǫu , if ν “ , ´ ǫ p ` u q , if ν “ ´ ǫ. Then we can show:
Theorem 4.5.
Let p M, g q be a complete semi-Riemannian manifold and p x M , p g ǫ q be the coneover p M, g q . If p x M , p g q admits a parallel vector field V with p g p V, V q “ ´ ǫ , then p M, g q isglobally isometric to p R ˆ N, ´ ǫ d s ` cosh p s q g N q , where p N, g N q is a complete semi-Riemannian manifold.Proof. The idea is to rescale the gradient ∇ u in a way that the rescaled vector field is ageodesic gradient vector field. To this end consider the function s “ ´ ǫ arcsinh ˝ u on M ,i.e., u p p q “ sinh p´ ǫs p p qq , for which we write u “ sinh p´ ǫs q . Then we have ∇ u | p “ ´ ǫ cosh p s p p qq ∇ s | p , and hence g p ∇ u, ∇ u q “ cosh p s q g p ∇ s, ∇ s q “ p ` sinh p s qq g p ∇ s, ∇ s q “ p ` u q g p ∇ s, ∇ s q . Hence, from (4.8) we get g p ∇ s, ∇ s q “ ´ ǫ , so S “ ∇ s is a unit gradient vector field.Moreover, from (4.6) we get ´ ǫ sinh p´ ǫs q X “ ∇ X ∇ u “ sinh p s q g p X, S q S ´ ǫ cosh p s q ∇ X S, and hence(4.9) ∇ X S “ tanh p´ ǫs q p X ` ǫg p X, S q S q . This implies that S is a geodesic vector field. Since p M, g q is assumed to be complete, theflow φ of S is defined on R ˆ M . By the above observation (4.7) we have ∇ u “ s in direction of S vanishes, L S d s p X q “ d s p X q ` X p d s p S qq “ X p g p S, S qq “ . This implies that the flow of S maps each level set of s to a level set of s .For a fixed p P M we define the function σ p t q “ s p φ t p p q . Since S “ ∇ s is complete, σ isdefined on R and satisfies the differential equation σ p t q “ d s | φ t p p q p S q “ g φ t p p q p S, S q ” ´ ǫ. Hence σ p t q “ ´ ǫt ` s p p q , which shows that(4.10) φ t p N c q “ N ´ ǫt ` c , EMI-RIEMANNIAN CONES 17 where N c “ s ´ p c q denotes the level set of s , Now set N “ N “ t u “ u “ t s “ u , whichis a smooth hypersurface and denote by g N the restriction of g to N . We define a smoothmap Φ : R ˆ N Q p t, p q ÞÑ φ t p p q P M. which, because of (4.10), has the inverseΦ ´ p q q “ ` s p q q , φ ´ s p q q p q q ˘ P R ˆ N. This shows that φ is a diffeomorphism.Finally, equation (4.9) implies that L S g p X, Y q “ p´ ǫs q g p X, Y q , for all X, Y P S K , i.e., al X, Y tangent to the level sets of s . This shows thatΦ ˚ g “ ´ ǫ d s ` p cosh p s qq g N . Since p M, g q was assumed to be complete p N, g N q has to be complete by Proposition 2.2. (cid:3) Now, let p M, g q be a semi-Riemannian manifold and p x M , p g ǫ q be the cone over p M, g q . Weconsider the case that V is a parallel light-like vector field. Recall that in this case we have,in addition to Lemma 4.4, that g p ∇ u, ∇ u q “ ´ ǫu . In this situation we observe: Lemma 4.6. If γ : I Ñ M is a geodesic on p M, g q with g p γ p q , γ p qq “ ´ ǫ and f “ u ˝ γ ,then f “ f , i.e., (4.11) f p t q “ u p γ p qq cosh p t q ` g p ∇ u | γ p q , γ p qq sinh p t q . In particular, if p M, g q is complete, then the image of u contains p , if t u ą u “ H and p´8 , q if t u ă u “ H .Proof. With f “ u ˝ γ we have f “ g | γ p ∇ u | γ , γ q and hence by Lemma 4.4, f “ g | γ p ∇ γ ∇ u, γ q “ ´ ǫf g | γ p γ , γ q “ f. The general solution to this equation is given by (4.11). If p M, g q is complete, the maximalgeodesics through a point with u p p q “ R and hence, by choosing a geodesicwith γ p q “ u p p q ∇ u | p , i.e., with g p ∇ u, γ p qq “ ´ ǫu p p q , we get(4.12) f p t q “ u p p q ` p ´ ǫ q e t ` p ` ǫ q e ´ t ˘ “ u p p q e ´ ǫt . This implies the statement about the image of u . (cid:3) Theorem 4.7.
Let p M, g q be a complete semi-Riemannian manifold and p x M , p g ǫ q be thecone over p M, g q . If p x M , p g q admits a parallel light-like vector field V , then M is a disjointunion M “ M ´ Y M Y M ` with M ˘ open and such that M is either empty (in which caseone of M ˘ is also empty) or a smooth totally geodesic hypersurface and p M ˘ , g q is globallyisometric to p R ˆ N ˘ , ´ ǫ d s ` e s g N ˘ q , where p N ˘ , g N ˘ q are complete semi-Riemannian manifolds. Moreover, M “ H if and onlyif p M, g q is Riemannian. Proof.
Recall that for in the case of V being light-like we have that g p ∇ u, ∇ u q “ ´ ǫu . Theproof is analogous to the previous proof, with a difficulty arising from the possibility thatthe set M “ t p P M | g p ∇ u | p , ∇ u | p q “ u “ t p P M | u p p q “ u may be non empty, so that the geodesic gradient vector field S from the previous proofmay not be defined on all of M . However, since V is light-like, we have ∇ u “
0, and so M is either empty or a smooth hypersurface. In fact, if M “ H , it is totally geodesic:if X P T M “ ∇ u K | M , then Lemma 4.6 shows that f p t q ”
0, so the geodesics starting indirection of M remain in M .We set M ˘ “ t˘ u ą u . Without loss of generality, we assume that M ` “ H , in whichcase we get that N ` “ t u “ u “ H by the previous lemma.We consider the function s “ ´ ǫ ln ˝p˘ u q on M ˘ , i.e., u “ ˘ e ´ ǫs . Then we have ∇ u “ ¯ ǫ e ´ ǫs ∇ s, and hence, for S “ ∇ s , g p ∇ u, ∇ u q “ e ´ ǫs g p S, S q “ u g p S, S q , and so g p ∇ s, ∇ s q “ ´ ǫ by (4.8). Next we get from (4.6) that ¯ ǫ e ´ ǫs X “ ∇ X ∇ u “ ¯ ǫ e ´ ǫs p´ ǫg p X, S q S ` ∇ X S q , and hence(4.13) ∇ X S “ p X ` ǫg p X, S q S q . Again, this shows that S is a geodesic vector field on M ˘ . Equation 4.12 in the proof ofLemma 4.6 then shows that the geodesics with initial speed given by S | p for p P M ˘ remainin M ˘ for all t P R . Hence S is a complete vector field on M ˘ with its flow defined on R ˆ M ˘ ,so we can continue with the proof as for the previous theorem yielding a diffeomorphismΦ ˘ : R ˆ N ˘ Q p t, p q ÞÑ φ t p p q P M, where N ˘ “ t p P M | u p p q “ ˘ u “ t p P M ˘ | s p p q “ u , with the inverseΦ ´ ˘ p q q “ ` s p q q , φ ´ s p q q p q q ˘ P R ˆ N ˘ . Now equation (4.13) implies that L S g p X, Y q “ g p X, Y q , for all X, Y P S K , i.e., al X, Y tangent to the level sets of s . This shows thatΦ ˚˘ g “ ´ ǫ d s ` e s g N ˘ with a semi-Riemannian manifold p N ˘ , g N ˘ q . In order to conclude that p N ˘ , g N ˘ q arecomplete, we observe that (4.11) in Lemma 4.6 shows that geodesics of p M, g q with initialspeed tangent to N ˘ , i.e., with initial speed orthogonal to ∇ u | N ˘ , remain in M ˘ and hence,because p M, g q is complete, are defined on R . With this, Proposition 2.2 implies that p N ˘ , g N ˘ q are complete.For the last statement, first note that if p M, g q is Riemannian, then, since ∇ u “
0, weget that M “ H . On the other hand assume that M “ H and without loss of generalitythat M ` “ M , so that globally p M “ R ˆ N, g “ ´ ǫ d s ` e s g N qq . By Proposition 2.2, themetric g is only complete if it is definite and g N is complete. (cid:3) EMI-RIEMANNIAN CONES 19
As a corollary we obtain a global version of Theorem 3.9.
Corollary 4.8.
Let p M, g q be a complete semi-Riemannian manifold and p x M , p g ǫ q be thecone over p M, g q . If p x M , p g q admits a parallel light-like vector field V , then x M is a disjointunion x M “ x M ´ Y x M Y x M ` with x M ˘ “ t p P x M | ˘ p g p V, B r q ą u , x M “ t p P x M | p g p V, B r q “ u “ R ą ˆ M and such that p x M ˘ , p g q is globally isometric to p R ` ˆ R ǫ ˆ N ˘ , r g “ u d v ` u g N ˘ q , where p N ˘ , g N ˘ q are a complete semi-Riemannian manifolds and where R ˘ “ t x P R | ˘ x ą u . The isometry is given by Ψ ˘ : x M Q p r, s, p q ÞÑ p u “ r e s , v “ ǫ r e ´ s , p q P p R ` ˆ R ǫ ˆ N ˘ q . Proof.
We have u “ r e s and2d u d v “ ǫ p e s d r ˘ r e s ds q ` e ´ s d r ¯ r e ´ s d s ˘ “ ǫ p d r ´ r d s q . Hence, by the previous theorem, Ψ ˚˘ r g “ p g . (cid:3) Lorentzian cones and applications to Killing spinors
Parallel spinors and Killing spinors.
Let p M, g q be a semi-Riemannian spin ma-nifold, i.e., a space and time oriented semi-Riemannian manifold with a spin structure, andlet Σ its complex spinor bundle. This is a complex vector bundle that is equipped with thefollowing structures:(1) the Clifford multiplication T M b Σ Q X b ϕ ÞÑ X ¨ ϕ P Σ , (2) a hermitian bundle metric x ., . y P Γ p Σ ˚ b Σ ˚ q on Σ, conjugate-linear in the secondcomponent, that is positive definite if g is Riemannian and of neutral signature if g is indefinite,(3) the lift ∇ Σ of the Levi-Civita connection to Σ,that satisfy the following properties, where r is the number of negative eigenvalues of g ,(5.1) p X ¨ Y ` Y ¨ X q ¨ ϕ “ ´ g p X, Y q ϕ, x X ¨ ϕ, ψ y Σ “ p´ q r ` x ϕ, X ¨ ψ y Σ , ∇ Σ Y p X ¨ ϕ q “ p ∇ Y X q ¨ ϕ ` X ¨ ∇ Σ Y ϕ,X px ϕ, ψ y Σ q “ x ∇ Σ X ϕ, ψ y Σ ` x ϕ, ∇ Σ X ψ y Σ . The second of these relations together with x ., . y being Hermitian shows that to each spinorfield ϕ one can assign a (real) vector field V ϕ P Γ p T M q g p V ϕ , X q : “ i r ` x ϕ, X ¨ ϕ y Σ for all X P T M .
This vector field is sometimes called the
Dirac current of ϕ . The above relations also showthat ∇ V ϕ “ ϕ is a parallel a parallel spinor field, i.e., if ∇ Σ ϕ “
0. However, V ϕ beidentically zero even if ϕ is not. This happens for example for parallel spinors on Riemannianmanifolds. Moreover, the Ricci tensor of a semi-Riemannian manifold with parallel spinor satisfies g p Ric p X q , Ric p X qq “
0. In particular, Riemannian manifolds with parallel spinors areRicci-flat.A
Killing spinor with Killing number z P C is a spinor field ϕ P Γ p Σ q that satisfies theequation ∇ Σ X ϕ “ z X ¨ ϕ. Using the above formula one can show that the scalar curvature of a semi-Riemannianmanifold with a Killing spinor is equal to 4 n p n ´ q z . This implies that z is either realor imaginary and hence the scalar curvature is a positive or negative constant. A Killingspinor with Killing number z “ ˘ is called real Killing spinor and with z “ ˘ i2 , ϕ an imaginary Killing spinor . Moreover, Riemannian manifolds with Killing spinor are Einstein,so Riemannian manifolds with real/imaginary Killing spinor provide examples of Einsteinmanifolds with positive/negative scalar curvature. The question which Einstein manifolds(or constant scalar curvature manifolds) can be constructed in this way lead to the problemof classifying manifolds with Killing spinors. The fundamental observation for solving thisproblem is the relation to semi-Riemannian cones: Theorem 5.1 ([3, 9]) . Let p M, g q be a semi-Riemannian spin manifold that admits a Killingspinor with Killing number ˘ ? ǫ if and only if the semi-Riemannian cone p x M , p g ǫ q admits aparallel spinor field. Remark 5.2 ([9]) . In [9] Bohle proved a more general result: Let p M, g q be a semi-Riemannian spin manifold and f : I Ñ R be a smooth function. Then the warped productmetric g ǫ,f “ ǫ d s ` f p s q g on I ˆ M admits a Killing spinor with Killing number ˆ λ P t , ˘ , ˘ i2 u if and only if(1) The warping function satisfies the ODE f “ ´ ǫ ˆ λ f , and(2) p M, g q admits a Killing spinor with Killing number ˘ λ , where λ “ ˆ λ f ` ǫ p f q .Theorem 5.1 together with Gallot’s Theorem 3.1 was used by B¨ar [3] to derive a classifi-cation of complete Riemannian manifolds with real Killing spinors: if p M, g q admits a realKilling spinor, the cone admits a parallel spinor and under the assumption of completeness,by Gallot’s theorem, the cone is irreducible. Then by Berger’s classification of irreducibleholonomy groups [8], Wangs classification of those admitting an invariant spinor [13] undertheir spin representation, and the correspondence between holonomy groups and geometricstructures, B¨ar arrived at the following classification: Theorem 5.3 (C. B¨ar [3]) . Let M be a complete, simply connected Riemannian spin ma-nifold with a real Killing spinor. Then M is isometric to round sphere, or ta a compactEinstein space with one of the following structures: Sasaki, -Sasaki, -dimensional nearly-K¨ahler, or nearly parallel G . Baum gave a classification of Riemannian manifolds with imaginary Killing spinors [5].Baum’s proof does not use the cone construction of Theorem 5.1 explicitly. In other signa-tures the classification of semi-Riemannian manifolds with Killing spinors is only known inspecial cases: for example, Bohle and Baum classified Lorentzian manifolds with real Killingspinors [9, Section 5], with an addition made in [6, Proposition 7.1], again without using the
EMI-RIEMANNIAN CONES 21 cone construction explicitly. In the next section we will use our results from the previoussection to obtain Baum’s and Bohle’s classification results.5.2.
Lorentzian cones and Killing spinors.
In this section we will use our results ofSection 4 to derive the classification of complete Riemannian manifolds with imaginaryKilling spinors and of complete Lorentzian manifolds with real Killing spinors. In bothcases Theorem 5.1 yields a parallel spinor on a Lorentzian cone and hence a parallel Diraccurrent by the observations in Section 5.1. In Lorentzian signature one can show that theDirac current is a causal vector field:
Lemma 5.4.
Let ϕ be a parallel spinor field on a spin Lorentzian manifold p M, g q . Then V ϕ is a causal parallel vector field, i.e, V ϕ “ , ∇ V ϕ “ and g p V ϕ , V ϕ q ď .Proof. We have already seen that V ϕ is parallel, so it is either identically zero or non van-ishing and we have to verify its causal character. Since p M, g q is time orientable we fix atime-like unit vector field T and split V ϕ “ V ϕ “ ´ g p T, V ϕ q T ` g p N, V ϕ q N, where N is a spacelike unit normal field orthogonal to T . Then we have by (5.1) that g p V ϕ , V ϕ q “ ´ g p T, V ϕ q ` g p N, V ϕ q “ ´x T ¨ ϕ, ϕ y ` x N ¨ ϕ, ϕ y , and we have to show that this is not positive. For this observe that the endomorphism T ¨ N on Σ squares to the identity by the defining relation for the Clifford algebra in (5.1), T ¨ N ¨ T ¨ N “ ´ N ¨ T ¨ T ¨ N “ ´ N ¨ N “ . Hence T ¨ N has eigenvalues ˘ ϕ “ ϕ ` ` ϕ ´ into its components in thecorresponding eigenspaces. Note that T ¨ ϕ ˘ “ ˘ N ¨ ϕ ˘ , which, together with (5.1), implies that x T ¨ ϕ ` , ϕ ´ y “ x ϕ ` , T ¨ ϕ ´ y “ ´x ϕ ` , N ¨ ϕ ´ y “ ´x N ¨ ϕ ` , ϕ ´ y “ ´x T ¨ ϕ ` , ϕ ´ y , so that x T ¨ ϕ ` , ϕ ´ y “
0. Then we use the fact (see [4] for a proof) that the hermitianform p φ, ψ q T “ x T ¨ φ, ψ y on Σ is positive definite. The last equation then shows that p ϕ ` , ϕ ´ q T “ g p V ϕ , V ϕ q “ ´p ϕ, ϕ q T ` p T ¨ N ¨ ϕ, ϕ q T “ ´ p ϕ ` , ϕ ` q T p ϕ ´ , ϕ ´ q T ď . This shows that V ϕ is either time-like or light-like. (cid:3) In fact, on a Lorentzian manifold the Dirac current of spinor field is always causal even ifthe spinor is not parallel, but it may change its causal character from light-like to time-like.The proof of this has to take into account that V ϕ may have zeros so that N may not bewell defined.The following theorem gives a classification of Riemannian manifolds with imaginaryKilling spinors. Theorem 5.5 ([5]) . Let p M, g q be a complete Riemannian manifold with an imaginaryKilling spinor. Then p M, g q is globally isometric to hyperbolic space or to a warped productof the form (5.2) ` R ˆ N, d s ` e s g N ˘ , where p N, g N q is a complete Riemannian manifold with a parallel spinor field.Proof. Let p M, g q be a complete Riemannian manifold with an imaginary Killing spinor field.Then, by Theorem 5.1, the Lorentzian cone p x M , g ´ q admits a parallel spinor field ϕ , whichby Lemma 5.4 provides us with a parallel vector field V ϕ that is either light-like or time-like.In case it is time-like, Theorem 4.3 yields that p M, g q has constant sectional curvature ´ V ϕ is light-like, we can apply Theorem 4.7 toget the desired warped product in (5.2) with a complete Riemannian manifold p N, g N q . Toget that p N, g N q admits a parallel spinor field we can either use the result in Remark 5.2or recall Corollary 4.8 and Theorem 3.14 to obtain that the holonomy algebra of p x M , p g ´ q isequal to hol p N, g N q ˙ R dim p N q . This is an indecomposable holonomy algebra that admits aninvariant spinor under its spin representation if and only if hol p N, g N q admits an invariantspinor. (cid:3) The next theorem provides a classification of Lorentzian manifolds with real Killingspinors.
Theorem 5.6 ([9, 6]) . Let p M, g q be a complete Lorentzian manifold with a real Killingspinor. Then(1) either p M, g q is globally isometric to de Sitter space or space or to a warped productof the form (5.3) ` R ˆ N, ´ d s ` cosh p s q g N ˘ , where p N, g N q is a complete Riemannian manifold with a real Killing spinor (i.e.,with one of the structures in Theorem 5.3), or(2) M is a disjoint union M “ M ´ Y M Y M ` with M a smooth totally geodesichypersurface and M ˘ and such that p M ˘ , g q are globally isometric to p R ˆ N ˘ , ´ d s ` e s g N ˘ q , where p N ˘ , g N ˘ q are complete Riemannian manifolds with parallel spinors.Proof. If p M, g q admits a real Killing spinor, then the cone p x M , p g ` q admits a parallel spinorand hence a parallel causal vector field V .If V is time-like, then we apply Theorem 4.5, to get that p M, g q is isometric to theLorentzian manifolds in (5.3) with a complete Riemannian manifold p N, g N q . If p N, g N q isthe round metric on the sphere then p M, g q is de Sitter space. The result in Remark 5.2shows that p M, g q admits a real Killing spinor if and only if p N, g N q does.If V is light-like, Theorem 4.7 shows that (2) holds with complete Riemannian manifolds p N ˘ , g N ˘ q . To obtain that p N, g N q admits a parallel spinor, we use again Remark 5.2 orrecall Corollary 4.8 and Theorem 3.14, as for the proof of Theorem 5.5. (cid:3) References [1] D. Alekseevsky, V. Cort´es, A. Galaev, and T. Leistner. Cones over pseudo-Riemannian manifolds andtheir holonomy.
J. Reine Angew. Math. , 635:23–69, 2009.[2] D. Alekseevsky, V. Cort´es, and T. Leistner. Geometry and holonomy of indecomposable cones, Feb2019. Preprint arXiv:1902.02493.[3] C. B¨ar. Real Killing spinors and holonomy.
Commun. Math. Phys. , 154(3):509–521, 1993.[4] H. Baum.
Spin-Strukturen und Dirac-Operatoren ¨uber pseudoriemannschen Mannigfaltigkeiten , vol-ume 41 of
Teubner-Texte zur Mathematik . Teubner-Verlagsgesellschaft, Leipzig, 1981.
EMI-RIEMANNIAN CONES 23 [5] H. Baum. Complete Riemannian manifolds with imaginary Killing spinors.
Ann. Global Anal. Geom. ,7(3):205–226, 1989.[6] H. Baum. Twistor and Killing spinors in Lorentzian geometry. In
Global analysis and harmonic analysis(Marseille-Luminy, 1999) , volume 4 of
S´emin. Congr. , pages 35–52. Soc. Math. France, Paris, 2000.[7] H. Baum, T. Friedrich, R. Grunewald, and I. Kath.
Twistors and Killing spinors on Riemannian man-ifolds , volume 124 of
Teubner-Texte zur Mathematik [Teubner Texts in Mathematics] . B. G. TeubnerVerlagsgesellschaft mbH, Stuttgart, 1991.[8] M. Berger. Sur les groupes d’holonomie homog`ene des vari´et´es `a connexion affine et des vari´et´es rieman-niennes.
Bull. Soc. Math. France , 83:279–330, 1955.[9] C. Bohle. Killing spinors on Lorentzian manifolds.
J. Geom. Phys. , 45(3-4):285–308, 2003.[10] G. de Rham. Sur la reductibilit´e d’un espace de Riemann.
Comment. Math. Helv. , 26:328–344, 1952.[11] S. Gallot. ´Equations diff´erentielles caract´eristiques de la sph`ere.
Ann. Sci. ´Ecole Norm. Sup. (4) ,12(2):235–267, 1979.[12] V. S. Matveev. Gallot-Tanno theorem for pseudo-Riemannian metrics and a proof that decompos-able cones over closed complete pseudo-Riemannian manifolds do not exist.
Differential Geom. Appl. ,28(2):236–240, 2010.[13] M. Y. Wang. Parallel spinors and parallel forms.
Ann. Global Anal. Geom. , 7(1):59–68, 1989.[14] H. Wu. On the de Rham decomposition theorem.
Illinois J. Math. , 8:291–311, 1964.
School of Mathematical Sciences, University of Adelaide, SA 5005, Australia
E-mail address ::