We consider several transformation groups of a locally conformally K\"ahler
manifold and discuss their inter-relations. Among other results, we prove that
all conformal vector fields on a compact Vaisman manifold which is neither
locally conformally hyperk\"ahler nor a diagonal Hopf manifold are Killing,
holomorphic and that all affine vector fields with respect to the minimal Weyl
connection of a locally conformally K\"ahler manifold which is neither
Weyl-reducible nor locally conformally hyperk\"ahler are holomorphic and
conformalComment: 8 page

A. Moroianu

Andrei Moroianu

F. Belgun

L. Ornea

Liviu Ornea

P. Gauduchon

S. Gallot

S. Kobayashi

English

arXiv

Abstract. We consider several transformation groups of a locally conformally Kähler mani-fold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperkähler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally Kähler manifold which is neither Weyl-reducible nor locally conformally hyperkähler are holomorphic and conformal. 1

CiteSeerX

Transformations of locally conformally Kähler

8 pagesInternational audienceWe consider several transformation groups of a locally conformally Kähler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperkähler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally Kähler manifold which is neither Weyl-reducible nor locally conformally hyperkähler are holomorphic and conforma

Moroianu, Andrei

Ornea, Liviu

HAL-Polytechnique

Transformations of locally conformally Ka¨hler manifoldsAndrei Moroianu, Liviu OrneaTo cite this version:Andrei Moroianu, Liviu Ornea. Transformations of locally conformally Ka¨hler manifolds.manuscripta mathematica, Springer Verlag, 2009, 130 (1), pp.93-100. <10.1007/s00229-009-0278-z>. <hal-00347485v2>HAL Id: hal-00347485https://hal.archives-ouvertes.fr/hal-00347485v2Submitted on 4 Sep 2009HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.TRANSFORMATIONS OF LOCALLY CONFORMALLY KA¨HLERMANIFOLDSANDREI MOROIANU AND LIVIU ORNEAAbstract. We consider several transformation groups of a locally conformally Ka¨hlermanifold and discuss their inter-relations. Among other results, we prove that all con-formal vector fields on a compact Vaisman manifold which is neither locally conformallyhyperka¨hler nor a diagonal Hopf manifold are Killing, holomorphic and that all affinevector fields with respect to the minimal Weyl connection of a locally conformallyKa¨hler manifold which is neither Weyl-reducible nor locally conformally hyperka¨hlerare holomorphic and conformal.2000 Mathematics Subject Classification: Primary 53C15, 53C25.Keywords: Killing vector field, conformal vector field, affine transformation, locallyconformally Ka¨hler manifold, Gauduchon metric.1. IntroductionWe briefly present the necessary background for locally conformally Ka¨hler (in short,LCK) geometry.In the sequel, J will denote an integrable complex structure on a connected, smoothmanifold M2n of complex dimension n ≥ 2. For a Hermitian metric g on (M,J), wedenote by ∇g its Levi-Civita connection and by ω its fundamental two-form ω(X, Y ) :=g(JX, Y ).Traditionally, a LCK metric g on the complex manifold (M,J) is defined by thefollowing “integrability” condition satisfied by its fundamental two-form:dω = θ ∧ ω, dθ = 0. (1)The closed one-form θ is called the Lee form. Note that, on manifolds of complexdimension at least 3, the first equation implies the closedness of θ, the second conditionbeing relevant only on complex surfaces.Any other Hermitian metric efg which is conformal with g is a LCK metric too, itsLee form being θ + df . Hence, a LCK metric determines a 1-cocycle associated merelyto the conformal class c = [g].This work was accomplished in the framework of the Associated European Laboratory “MathMode”.L.O. is partially supported by CNCSIS PNII grant code 8.12 ANDREI MOROIANU AND LIVIU ORNEAClearly, on local open sets on which θ is exact, g is conformal with some local Ka¨hlermetrics which, on overlaps, are homothetic. The above 1-cocycle is associated to thissystem of scale factors.Note that on the universal cover of M , the pull-back of θ is exact (the pull-backs ofthe local Ka¨hler metrics glue together to a global Ka¨hler metric) and the fundamentalgroup ofM acts by biholomorphic homotheties with respect to this Ka¨hler metric. Thisis usually taken for definition of a LCK manifold.Returning on M , the Levi-Civita connection DU of a local Ka¨hler metric in theconformal class of g on some open set U is related to the Levi-Civita connection of gby the formula:DU = ∇g − 12(θ ⊗ id + id⊗ θ − g ⊗ θ♯). (2)It is obvious that, in fact, these connections do not depend on the particular local Ka¨hlermetric, i.e. they glue together to a global connection, here denoted D, which has thefollowing two properties:DJ = 0, Dg = θ ⊗ g.The second equation implies that D preserves the conformal class: Dc = 0. Beingtorsion free (as, locally, any Levi-Civita connection), it is a Weyl connection. Noticethat the Lee form of the Weyl connection D with respect to g defined by (2) coincideswith the Lee form of the complex structure J with respect to g defined by (1).Definition 1.1. A Hermitian-Weyl structure on M2n is a triple (c, J,D) where c isa conformal structure, J is a complex structure compatible with c and D is a Weylconnection such that DJ = 0 and Dc = 0.From the above, every LCK structure defines a Hermitian-Weyl structure on M , andconversely, every metric in the conformal class of a Hermitian-Weyl structure is LCK,provided n ≥ 3. Notice that the Weyl connection D is uniquely defined by J (see e.g.[2, Lemma 5.1]). We call D the minimal Weyl connection.In general, there is no way to choose a “canonical” metric in the conformal classof a Hermitian-Weyl manifold (M, c, J,D), unless M is compact, where there exists aunique, up to homothety, metric g0 in c such that the Lee form θ0 of D with respectto g0 is co-closed (and hence harmonic), see [5, p.502]. We call g0 the the Gauduchonmetric.A particular class of LCK manifolds are the Vaisman manifolds. These are definedby the condition ∇gθ = 0. By the uniqueness up to homotheties, a Vaisman metricis, necessarily, the Gauduchon metric in its conformal class. The prototype of Vaismanmanifolds are the Hopf manifolds Cn \ {0}/Z, with Z generated by a semi-simple en-domorphism (see [10] for the structure of compact Vaisman manifolds). On the otherhand, there exist examples of compact LCK manifolds which do not admit any Vaismanmetric: such are the Inoue surfaces and the non-diagonal Hopf surfaces, see [1].TRANSFORMATIONS OF LOCALLY CONFORMALLY KA¨HLER MANIFOLDS 3The hypercomplex version of LCK geometry is straightforward. One starts with ahypercomplex manifold (M, I, J,K) endowed with a conformal class [c] and with a Weylconnection D satisfying:Dc = 0, DI = DJ = DK = 0.One obtains the notion of hyperhermitian-Weyl structure. It can be seen that for any g ∈c, the Lee forms associated to the three complex structures coincide. As above, the Leeform is closed in real dimension at least 8, and hence, with the exception of dimension4, a hyperhermitian-Weyl manifold is locally conformally hyperka¨hler (LCHK).The local Ka¨hler metrics of a LCHK manifold are now hyperka¨hler, hence Ricci-flat. It follows that LCHK manifolds are necessarily Einstein-Weyl. But on compactEinstein-Weyl manifolds, the Gauduchon metric is parallel, see [6]. Hence, a compactLCHK manifold bears three nested Vaisman structures.For examples and properties of LCK and LCHK manifolds, we refer to [3], to morerecent papers by Ornea and Verbitsky and to the references therein.Remark 1.2. In the whole paper, we tacitly assume that the manifolds we considerare not globally conformally Ka¨hler, i.e. the Lee form is never exact. This is especiallyimportant on compact manifolds, where LCK and Ka¨hler structures impose completelydifferent topologies.We now consider the following transformation groups on a Hermitian-Weyl manifold(M, c, J,D):• Aff(M,D), the group of affine transformations, i.e. preserving the Weyl connec-tion.• H(M,J), the group of biholomorphisms with respect to J .• Conf(M, c), the group of conformal transformations.• Aut(M):= Hol(M,J) ∩ Conf(M, c), the group of automorphisms.It is well known that these are Lie groups. Their Lie algebras will be denoted,respectively by: aff(M,D), h(M,J), conf(M, c), aut(M).On Vaisman manifolds, the Lee field θ♯ is Killing and real-analytic but on non-VaismanLCK manifolds, almost no information about these groups is available. It is the purposeof this note to clarify some relations among the above groups. Essentially, we prove that:• aff(M,D) = aut(M), provided that Hol0(D) is irreducible and M is not LCHK(Corollary 2.3).• conf(M, c) = aut(M) on compact Vaisman manifolds which are neither LCHKnor diagonal Hopf manifolds (Theorem 3.2).Note that on compact LCHK manifolds and on Hopf manifolds there exist examplesof affine transformations which are not holomorphic, see Remark 2.4 (ii) below.We believe that the second equality holds on all compact LCK manifolds which areneither LCHK nor diagonal Hopf manifolds. By Lemma 2.5 below, this amounts to4 ANDREI MOROIANU AND LIVIU ORNEAprove that every conformal vector field is Killing for the Gauduchon metric, which isthe LCK counterpart of the result saying that every conformal vector field on a compactKa¨hler manifold is Killing ([8, §90]).2. Affine vector fields on LCK manifoldsOur first result is the LCK analogue of [8, §54], (see also [7]).Lemma 2.1. Let (M, c, J,D) be a LCK manifold which is not locally conformally hy-perka¨hler and such that Hol0(D) is irreducible. Then any f ∈ Aff(M,D) is holomorphicor anti-holomorphic.Proof. Let f ∈ Aff(M,D) and let J ′x := (dxf)−1 ◦ Jf(x) ◦ (dxf) denote the image by f ofthe complex structure J . Then J ′2 = −id and, as J and df commute with the paralleltransport induced by D, we easily derive that J ′ is D-parallel too.Consider the decomposition of JJ ′ into symmetric and skew-symmetric parts: JJ ′ =S + A, where {S := 12(JJ ′ + J ′J),A := 12(JJ ′ − J ′J).Since S isD-parallel, its eigenvalues are constant and the corresponding eigenbundles areD-parallel. The irreducibility assumption implies S = kid for some k ∈ R. Similarly, theD-parallel symmetric endomorphism A2 has to be a multiple of the identity: A2 = pid.If A were non-zero, A(X) 6= 0 for some X ∈ TM , so0 > −c(AX,AX) = c(A2X,X) = pc(X,X),whence p < 0. The endomorphism K := A/√−p is then a D-parallel complex structureand satisfies KJ = −JK, so (J,K) defines a LCHK structure on (M, c), which isforbidden by the hypothesis. Thus A = 0 and JJ ′ = kid, so J ′ = −kJ . UsingJ ′2 = −id we get k = ±1, so J ′ = ±J , which just means that f is holomorphic oranti-holomorphic.In a similar manner one can prove the followingLemma 2.2. Let (M, c, J,D) be a LCK manifold such that Hol0(D) is irreducible. Thenany f ∈ Aff(M,D) is conformal.Proof. Since f is affine, the pull-back by f of the conformal structure c is a D-parallelconformal structure c′. The symmetric endomorphism B of TM defined by c′(X, Y ) =c(BX, Y ) is D-parallel. The irreducibility of Hol0(D) shows as before that B is amultiple of the identity. TRANSFORMATIONS OF LOCALLY CONFORMALLY KA¨HLER MANIFOLDS 5Corollary 2.3. Let (M, c, J,D) be a LCK manifold which is not LCHK and such thatHol0(D) is irreducible. Then any infinitesimal affine transformation of the Weyl con-nection is an infinitesimal automorphism, i.e. aff(M,D) = aut(M).Remark 2.4. (i) We do not know whether the assumption of irreducibility of theWeyl connection can be relaxed or not, at least on compact M . But we can showthat a compact Vaisman manifold, which is not a (diagonal) Hopf manifold, is Weyl-irreducible. Indeed, by the structure theorem in [10],M is a mapping torus of a Sasakianisometry and its universal cover is a Ka¨hlerian cone over the compact, hence complete,Sasakian fibre. Then, as D is, locally, the Levi-Civita connection of the local Ka¨hlermetrics, if D is reducible, the Ka¨hler metric of the covering cone is reducible. Now, byProposition 3.1 in [4], a cone over a complete manifold is reducible if and only if it isflat. But a flat cone is the cone over a sphere, hence M is a Hopf manifold.(ii) Let W be a 3-Sasakian manifold. Then M :=W × S1 is LCHK (for W = S2n−1,M is a Hopf manifold) and each of the three Killing fields on W which generate theSU(2) action, induce Killing, hence affine with respect to the Weyl connection, fieldson M and each of them is holomorphic only with respect to one of the three complexstructures (see e.g. [3, Chapter 11]).We apply the above to prove:Lemma 2.5. Let (M, c, J,D) be a compact LCK manifold which is not LCHK and suchthat Hol0(D) is irreducible. Then every Killing vector field ξ of the Gauduchon metricis a holomorphic vector field.Proof. According to Corollary 2.3, it is enough to prove that ξ is affine. As the Weylconnection is uniquely defined by the Gauduchon metric and its Lee form θ0, it will beenough to show thatLξθ0 = 0. (3)But, as ξ is g0-Killing, the codifferential δ of g0 commutes with Lξ, hence δ(Lξθ0) =Lξ(δθ0) = 0, because θ0 is co-exact. Now, θ0 is closed, so d(Lξθ0) = 0, i.e. Lξθ0 isharmonic. On the other hand, by Cartan’s formula, Lξθ0 = d(ξ y θ0). On a compactRiemannian manifold, a one-form which is both exact and harmonic necessarily vanishes,so (3) is proved. 3. Conformal vector fields on Vaisman manifoldsAs mentioned in the introduction, every conformal vector field on a compact Ka¨hlermanifold is Killing ([8, §90]). We consider the LCK analogue of this statement: Isevery conformal vector field on a compact LCK manifold Killing with respect to theGauduchon metric?As no sphere S2n can bear an LCK metric for n ≥ 2 (S2n being simply connected,such a structure would be automatically Ka¨hler), by Obata’s theorem we derive thatany conformal vector field on a compact LCK manifold is Killing for some metric in the6 ANDREI MOROIANU AND LIVIU ORNEAgiven conformal class, but not necessarily the Gauduchon metric. In fact, an argumententirely similar to the above, shows that a conformal vector field is Killing with respectto the Gauduchon metric if and only if it preserves the Lee form of the metric withrespect to which it is Killing.The answer to the above question is not known to us in general. Nevertheless, wecan show that it holds on compact Vaisman manifolds, on which every conformal fieldis Killing with respect to the Gauduchon metric. This, actually, follows from a moregeneral statement:Theorem 3.1. Let ϕ be an isometry of a compact Riemannian manifold (W,h) and let(M, g) := (W,h)× R/{(x,t)∼(ϕ(x),t+1)} be the mapping torus of ϕ. Then every conformalvector field on (M, g) is Killing.Proof. In [9] it is shown that every twistor form on a Riemannian product is defined byKilling forms on the factors. We adapt the argument there to the situation of mappingtori.Every conformal vector field on (M, g) induces a conformal vector field denoted ξ onthe covering space W × R of M , endowed with the product metric g˜ := h + dt2. Wewrite ξ asξ = a∂t + η,where ∂t = ∂/∂t, η is tangent to W and a is a function. Both a and η can be viewedas objects on W indexed upon the parameter t on R. Since ξ is invariant under theisometry (x, t) 7→ (ϕ(x), t+ 1) of W × R, we get in particular a(w, t) = a(ϕ(w), t+ 1),and thus, if we write at(w) for a(w, t),at = ϕ∗at+1, a˙t = ϕ∗a˙t+1, ∀ (w, t) ∈W ×R. (4)Now, ξ being conformal, there exists a function f on W ×R such that Lξg˜ = f g˜. Thiscan be writteng˜(∇Aξ, B) + g˜(A,∇Bξ) = f g˜(A,B) (5)for every tangent vectors to W × R, A and B. Taking A = B = ∂t in (5) givesf = 2g˜(∇∂tξ, ∂t),hence, by the parallelism of ∂t, one hasf = 2∂t(a) = 2a˙. (6)Applying the same equation (5) on pairs (∂t, X), (Y, Z), with X, Y, Z tangent toW andindependent on t, we obtain, respectively:∇∂tη + da = 0,andh(∇Y η, Z) + h(Y,∇Zη) = fh(Y, Z).TRANSFORMATIONS OF LOCALLY CONFORMALLY KA¨HLER MANIFOLDS 7Notice that here da denotes the exterior derivative of a on W and should be understoodas a family of 1-forms on W indexed on t, identified by h with a family of vector fieldson W . Taking (6) into account, the two equations above become:{η˙ = −da,h(∇WY η, Z) + h(Y,∇WZ η) = 2a˙h(Y, Z)(7)As the vector fields Y, Z are independent on t, we can take the derivative with respectto t in the second equation of (7):h(∇WY η˙, Z) + h(Y,∇WZ η˙) = 2a¨h(Y, Z).Letting here Y = Z = Ei, where {Ei} is a local orthonormal basis on W , we obtainδW η˙ = −ma¨, (m = dimW ).Using the first equation in (7), we finally obtain:δWda = ma¨. (8)We use this equation to show that a is constant. To this end, we integrate ‖da‖2 on W :∫Wh(da, da) =∫W(δWda)a = m∫Waa¨ = m∫W(aa˙)′ −m∫W(a˙)2.If we let b := maa˙, then, from the above equation we have∫Wb˙ =∫Wh(da, da) +m∫W(a˙)2 ≥ 0. (9)On the other hand, taking t = 0 in (4) yields∫Wb0 =∫Wϕ∗b1 =∫Wb1,because ϕ is a diffeomorphism. We can therefore compute∫W×[0,1]b˙ =∫ 10(∫Wb˙)dt =∫Wb1 −∫Wb0 = 0.By (9), a˙ = 0 and da = 0. As M is connected, a is constant. It follows that f = 0 andthus ξ is Killing.We apply the previous result to compact Vaisman manifolds which, by the structuretheorem in [10], are mapping tori with compact, Sasakian fibre. As a direct consequenceof Remark 2.4(i), Lemma 2.5 and Theorem 3.1 we get:Theorem 3.2. Any conformal vector field on a compact Vaisman manifold which isneither LCHK nor a diagonal Hopf manifold is Killing and holomorphic.8 ANDREI MOROIANU AND LIVIU ORNEAReferences[1] F. Belgun, On the metric structure of non-Ka¨hler complex surfaces, Math. Ann. 317 (2000), 1–40.[2] F. Belgun, A. Moroianu, Weyl-parallel Forms and conformal Products, arXiv:0901.3647.[3] S. Dragomir, L. Ornea, Locally conformal Ka¨hler geometry, Progress in Math. 155, Birkha¨user,1998.[4] S. Gallot, Equations diffe´rentielles caracte´ristiques de la sphe`re, Ann. Sci. Ec. Norm. Sup. 12,235–267 (1979).[5] P. Gauduchon, La 1-forme de torsion d’une varie´te´ hermitienne compacte, Math. Ann. 267, 495–518 (1984).[6] P. Gauduchon, Structures de Weyl-Einstein, espaces de twisteurs et varie´te´s de type S1 × S3, J.Reine Angew. Math. 469 (1995), 1–50.[7] S. Kobayashi, K. Nomizu, On automorphisms of a Ka¨hlerian structure, Nagoya Math. J. 11,115–124 (1957).[8] A. Lichne´rowicz, Ge´ome´trie des groupes de transformations, Dunod, Paris, 1958.[9] A. Moroianu, U. Semmelmann, Twistor Forms on Riemannian Products, J. Geom. Phys. 58,1343–1345 (2008).[10] L. Ornea, M. Verbitsky, Structure theorem for compact Vaisman manifolds, Math. Res. Lett., 10,799–805 (2003).Centre de Mathe´mathiques, Ecole Polytechnique, 91128 Palaiseau Cedex, FranceE-mail address : am@math.polytechnique.frUniv. of Bucharest, Faculty of Mathematics, 14 Academiei str, 70109 Bucharest,Romania, and Institute of Mathematics “Simion Stoilow” of the Romanian Academy,21, Calea Grivitei str., 010702-Bucharest, Romania.E-mail address : lornea@gta.math.unibuc.ro, Liviu.Ornea@imar.ro

Transformations of locally conformally K\"ahler manifolds

Abstract

Similar works

Full text

Available Versions

CiteSeerX

HAL-Polytechnique

HAL: Hyper Article en Ligne

Crossref