We study a locally conformal almost cosymplectic manifold M which carries a horizontal skew-symmetric Killing vector field X. Such X denes a relative conformal cosymplectic transformation of the conformal cosymplectic 2-form Ω of M and the square of its length is both　an isoparametric function and an eigenfunction of the Laplacian.　Math. Subject Classification 2000: 53D15, 53C15.　Keywords: Locally conformal almost cosymplectic manifolds, skew-symmetric Killing vector feld, infinitesimal concircular transformation, relative conformal cosymplectic transformation

Matsumoto  Koji

Mihai  Adela

Naitza  Dorotea

論文(Article)We study a locally conformal almost cosymplectic manifold M which carries a horizontal skew-symmetric Killing vector field X. Such X denes a relative conformal cosymplectic transformation of the conformal cosymplectic 2-form Ω of M and the square of its length is both　an isoparametric function and an eigenfunction of the Laplacian.　Math. Subject Classification 2000: 53D15, 53C15.　Keywords: Locally conformal almost cosymplectic manifolds, skew-symmetric Killing vector feld, infinitesimal concircular transformation, relative conformal cosymplectic transformation.departmental bulletin pape

Matsumoto,  Koji

Mihai,  Adela

Naitza,  Dorotea

Yamagata University Academic Repository

Bull. of Yamagata Univ., Nat. Sci., Vol.15, No.4, Feb. 2004Locally Conformal Almost CosymplecticManifolds Endowed with a Skew-SymmetricKilling Vector FieldKoji MATSUMOTO†, Adela MIHAI‡ and Dorotea NAITZA††(Received, March 04, 2003)AbstractWe study a locally conformal almost cosymplectic manifold Mwhich carries a horizontal skew-symmetric Killing vector ﬁeldX. SuchX deﬁnes a relative conformal cosymplectic transformation of the con-formal cosymplectic 2-form Ω ofM and the square of its length is bothan isoparametric function and an eigenfunction of the Laplacian.Math. Subject Classiﬁcation 2000: 53D15, 53C15.Keywords: Locally conformal almost cosymplectic manifolds, skew-symmetric Killing vector ﬁeld, inﬁnitesimal concircular transforma-tion, relative conformal cosymplectic transformation.1 PreliminariesLet (M, g) be an oriented n-dimensional Riemannian C∞-manifold and∇ be the covariant diﬀerential operator with respect to the metric tensor g.Let ΓTM be the set of sections of the tangent bundle and � : TM → T ∗Mand � = �−1 the classical musical isomorphisms deﬁned by g. We denote byAq(M,TM) the set of all vector valued q-forms, q < dimM.†Department of Mathematics, Faculty of Education, Yamagata University, Yamagata,Japan.e-mail address: ej192@kdw.kj.yamagata-u.ac.jp‡Faculty of Mathematics, University of Bucharest, Str. Academiei 14, 70109 Bucharest,Romania.e-mail address: adela@geometry.math.unibuc.ro††Istituto di Matematica, Facolta di Economia, Via dei Verdi 75, 98122 Messina, Italia.e-mail address: dorothe4@inwind.it105Koji MATSUMOTO, Adela MIHAI and Dorotea NAITZAA vector ﬁeld U is said to be exterior concurrent if it satisﬁes(1.1) ∇2U = α ∧ dp ∈ A2(M,TM), α ∈ Λ1(M,TM),where α = λU � for a certain λ ∈ Λ0 and it is called a concurrence form([MRV], [PRV], [R2]).In (1.1), α is called the concurrence form and is deﬁned byα = λU �, λ ∈ Λ0M.A function f : M → R is isoparametric if �∇f� and div(∇f) are func-tions of f ([W]).Let O = {eA |A = 1, . . . , n} be a local ﬁeld of orthonormal frame over Mand let O∗ = {ωA} be its associated coframe. Then the soldering form dp isexpressed by dp = ω ⊗ e. Also, the Cartan’s structure equations written inindexless manner are(1.2) ∇e = θ ⊗ e,(1.3) dω = −θ ∧ ω,(1.4) dθ = −θ ∧ θ +Θ.In the above equations, θ (resp. Θ) are the local connection forms in thetangent bundle TM (resp. the curvature forms on M).A (2m + 1)-dimensional locally conformal almost cosymplectic manifoldM with structure (φ,Ω, ξ, η, g) is deﬁned bydΩ = 2ω ∧ Ω, η = ω ∧ η,for certain 1-form ω, where φ is an endomorphism of the tangent bundle TMof square −1, Ω is the structure 2-form, which is called a locally conformalalmost cosymplectic 2-form, Ω a conformal cosymplectic 2-form of rank 2m,ξ the Reeb vector ﬁeld and η the Reeb covector ﬁeld.It is known that the 1-form ω from the above equation is a closed 1-formwhich is called the characteristic form associated with the locally conformalalmost cosymplectic structure ([MMR]).In addition, if M is endowed with a quasi-Sasakian structure deﬁned bya ﬁeld φ of endomorphism of its tangent space and ω satisﬁes ω = −η, thenM is called an almost cosymplectic −1-manifold. Let D�p (resp. D⊥p ) be a setof all tangent vectors at p which are orthogonal to (resp. proportional to) ξp.Then we may split the tangent space TpM ofM at p ∈M as TpM = D�p ⊕D⊥p .106Locally Conformal Almost Cosymplectic Manifolds Endowed with a Skew-Symmetric Killing Vector FieldWe can construct the distribution D : p → D�p = {X ; ηp(Xp) = 0}, calledthe horizontal distribution and the distribution D⊥ : p→ D⊥p = {ξp}, calledthe vertical distribution.In almost cosymplectic −1-manifold M , one has the following (see, forinstance, [MMR], [OR])(1.5) dΩ = −2η ∧ Ω, Ω(Z,Z �) = g(φZ,Z �),(1.6) (∇Z�φ)Z = η(Z)φZ � + g(φZ,Z �)ξ,(1.7) ∇ξ = −dp+ η ⊗ ξ,(1.8) dη = 0.A vector ﬁeld X is called a horizontal skew-symmetric Killing vector ﬁeldwith generatives ξ if it satisﬁes(1.9) ∇X = ξ ∧X, η(X) = 0.Then we haveLemma 1. Let X be a horizontal skew-symmetric Killing vector ﬁeld. Ifwe put 2l = �X�2, then we have the following properties:i) 2l is an isoparametric function,ii) grad 2l deﬁnes an inﬁnitesimal concircular transformation andiii) l is an eigenfunction of the Laplacian ∆.Also, we haveLemma 2. The above vector ﬁeld X satisﬁes the following∇3X = 2(X � ∧ η) ∧ dp,i.e., by deﬁnition, X is a 2-exterior concurrent vector ﬁeld, andd(LXΩ) = −2η ∧ LXΩ,i.e., by deﬁnition, X deﬁnes a relative almost cosymplectic transformation ofΩ (LXΩ is exterior recurrent with −2η as recurrence form).Proofs of the above lemmas will be given in the next section.107Koji MATSUMOTO, Adela MIHAI and Dorotea NAITZA2 Main ResultWe assume in this paper that a vector ﬁeld X is a skew-symmetric Killingvector ﬁeld having the Reeb vector ﬁeld ξ as generative ([R2]), i.e.,(2.1) ∇X = ξ ∧X,or, equivalently,(2.2) ∇X = X� ⊗ ξ − η ⊗X.Let O = {eA |A = 1, . . . , 2m + 1} be a local ﬁeld of orthonormal frameover M and let O∗ = {ωA} be its associated coframe and we assume thate2m+1 = ξ and ω2n+1 = η.We assume that X is a horizontal vector ﬁeld (η(X) = 0). Then thevector ﬁeld X is written as(2.3) X� =2m�a=1Xaωaand(2.4) ∇X = (dXa +Xbθab )⊗ ea +X� ⊗ ξ, a, b = 1, . . . , 2m.Hence, by (2.2), one obtains by a standard calculation(2.5) dXa +Xbθab = Xaηand setting(2.6) 2l = �X�2 ,one derives from (2.5)(2.7) dl = −2lη,which is concordance with (1.8). Next, from (2.7), one has grad l = −2lξ,which imply(2.8) �grad l�2 = 4l2,and(2.9) div(grad l) = 4ml,108Locally Conformal Almost Cosymplectic Manifolds Endowed with a Skew-Symmetric Killing Vector Fieldwhich say that the length 2l of the vector ﬁeldX is an isoparametric function.In addition, one has(2.10) g(∇Z grad l, Z �) = 2lg(Z,Z �)for any Z,Z � ∈ ΓTM . This means, by deﬁnition, that grad l deﬁnes aninﬁnitesimal concircular transformation of a vector ﬁeld Z ([MRV]).In the same order of ideas, one gets(2.11) ∆l = 4ml,i.e., l is eigenfunction of the Laplacian ∆.In this way, Lemma 1 has been proved.Next, since ∇ acts inductively, one derives(2.12) ∇2X = X� ∧ dp− 2(η ∧X�)⊗ ξ.This means that the distinguished vector ﬁeld X is a quasi-exterior con-current vector ﬁeld.Further, one has(2.13) ∇(∇2X) = ∇3X = 2(X � ∧ η) ∧ dp,i.e., by deﬁnition, X is a 2-exterior concurrent vector ﬁeld ([MRV]).Finally, regarding the conformal cosymplectic form Ω, we deﬁne β(2.14) β = iXΩ =n�a=1(Xaωa∗ −Xa∗ωa).Then, since(2.15) LXΩ = d(iXΩ) + 2η ∧ iXΩ,one may write(2.16) LXΩ = dβ + 2η ∧ β,and, by exterior diﬀerentiation, one derives(2.17) d(LXΩ) = −2η ∧ LXΩ.Then, the relation (2.17) aﬃrms that the distinguished vector ﬁeld Xdeﬁnes a relative conformal cosymplectic transformation of Ω (see [R1]).109Koji MATSUMOTO, Adela MIHAI and Dorotea NAITZAIn this way, Lemma 2 has been proved.Summing up, and making use of Lemmas 1 and 2, we proved the following.Theorem. Let M(φ,Ω, ξ, η, g) be a (2m+ 1)-dimensional locally confor-mal almost cosymplectic C∞-manifold, with Reeb vector ﬁeld ξ. Then, if Mcarries a horizontal vector ﬁeld X such that X is a skew-symmetric Killingvector ﬁeld, one has the properties:i) 2l = ||X||2 is an isoparametric function; moreover, grad l is an in-ﬁnitesimal concircular transformation and l is an eigenfunction of the Lapla-cian ∆;ii) X is a closed vector ﬁeld which is 2-exterior concurrent, i.e.,∇3X = 2(X � ∧ η) ∧ dp;iii) X deﬁnes a relative conformal cosymplectic transformation of Ω, i.e.d(LXΩ) = −2η ∧ LXΩ.Acknowledgements. The authors wish to express their hearty thanksto the referee’s valuabla advices and are very obliged to Prof. Dr. Radu Roscafor useful discussions and valuable advices.Remark. This paper was written while the second author visited Yama-gata University, supported by a JSPS postdoctoral fellowship.References[D] J. Dieudonne´, Treatise of Analysis, vol. 4, Academic Press, New York(1974).[MMR] K. Matsumoto, I. Mihai and R. Rosca, A certain locally confor-mal almost cosymplectic manifold and its submanifolds, Tensor, N.S. 51(1992), 91-102.[MOR] I. Mihai, A. Oiaga˘ and R. Rosca, Lorenzian Kenmotsu manifoldshaving two skew-symmetric conformal vector ﬁelds, Bull. Math. Soc.Sci. Roumanie (N.S.) 42 (90) (1999), 237-251.[MRV] I. Mihai, R. Rosca and L. Verstraelen, Some Aspects of the Diﬀeren-tial Geometry of Vector Fields, 2, PADGE, KU Leuven, KU Brussel.110Locally Conformal Almost Cosymplectic Manifolds Endowed with a Skew-Symmetric Killing Vector Field[NO] D. Naitza and A. Oiaga˘, On φ-skew symmetric conformal vector ﬁelds,Publ. Math. Debrecen, 59 (2001), 379-387.[OR] Z. Olszak and R. Rosca, Normal locally conformal almost cosymplecticmanifolds, Publ. Math. Debrecen, 39 (1991), 3-4.[PRV] M. Petrovic, R. Rosca and L. Verstraelen, Exterior concurrent vectorﬁelds on a Riemannian manifold, Soochow J. Math., 15 (1989), 179-187.[R1] R. Rosca, Exterior concurrent vector ﬁelds on a conformal cosymplec-tic manifold endowed with a Sasakian structure, Libertas Math., (Univ.Arlington, Texas) 6 (1986), 167-174.[R2] R. Rosca, On exterior concurrent skew symmetric Killing vector ﬁelds,Rend. Sem. Mat. Messina 2 (1993), 131-145.[W] A. West, Isoparametric systems, Geometry and Topology of Submani-folds, 222-230, World Scientiﬁc, River Edge, U.S.A. (1989).111

Locally Conformal Almost Cosymplectic Manifolds Endowed with a Skew-Symmetric Killing Vector Field

https://yamagata.repo.nii.ac.jp/?action=repository_action_common_download&item_id=3030&item_no=1&attribute_id=17&file_no=1

Locally Conformal Almost Cosymplectic Manifolds Endowed with a Skew-Symmetric Killing Vector Field

Abstract

Similar works

Full text

Available Versions

Yamagata University Academic Repository