Geometrical and structural properties are proved for Riemannian manifolds which are equipped with a T-parallel exterior recurren connection

Defever, Filip

Rosca, Radu

ESE - Salento University Publishing

Note di Matematica 21, n. 1, 2002, 59–70.Riemannian manifolds structured bya T -parallel exterior recurrent connectionFilip DefeverDepartement Industrie¨le Wetenschappen en Technologie,Katholieke Hogeschool Brugge-Oostende,Zeedijk 101, 8400 Oostende, BelgiumRadu Rosca59 Avenue Emile Zola, 75015 Paris, FranceReceived: 22 February 2001; accepted: 8 November 2001.Abstract. Geometrical and structural properties are proved for Riemannian manifolds whichare equipped with a T -parallel exterior recurrent connection.Keywords: Exterior recurrent forms, T -parallel connection.MSC 2000 classification: 53B20.IntroductionRiemannian manifolds structured by a T -parallel connection have been de-ﬁned in [9] and have also been studied in [6]. Let M be a 2m-dimensionalC∞-manifold and ∇ be the Levi-Civita connection. We recall that if M carriesa globally deﬁned vector ﬁeld T (T A) and the connection forms satisfyθAB =< T , eB ∧ eA >, (1)where ∧ denotes the wedge product of vector ﬁelds, then one says that Mis structured by a T -parallel connection. In the present paper we assume inaddition that θAB are exterior recurrent forms [2], which means thatdθAB = 2α ∧ θAB, where α = T 8, (2)having T 8 as recurrence form. This implies that the curvature forms ΘAB arealso exterior recurrent. In consequence of this fact, we adopt the terminologythat M is structured by a T -parallel exterior recurrent connection.For the above mentioned structures, we prove the following properties:(i) T is a concurrent vector ﬁeld and deﬁnes an inﬁnitesimal conformal trans-formation of θAB and ΘAB and the diﬀerential system ∇eA correspondingto the vector basis O = {eA} admits an inﬁnitesimal transformation withgenerator T ;60 F. Defever, R. Rosca(ii) ‖T ‖2 is an isoparametric function [13], and an eigenfunction of ∆ having4(2m+ ‖T ‖2) as eigenvalue;(iii) if V is any parallel vector ﬁeld, one has by the Weitzenbock formula that(∆T 8)V = −4m‖T ‖2g(T , V );(iv) ifΘ(p)u1,...,u2p= Θu2u1 ∧Θu3u2 ∧ · · · ∧Θ2p2p−1denotes the Bianchi forms (in the sense of Tachibana [12]), these formsare exterior recurrent with 3(2m− 1)α as recurrence form;(v) any vector ﬁeld X such that∇X = X ∧ Tis a skew symmetric Killing vector ﬁeld [11] and X deﬁnes an inﬁnitesimaltransformation of the conformal symplectic form Ω, i.e.LXΩ = −2g(X, T )Ω.In Section 4 we consider some properties of the tangent bundle manifold TMhaving the manifold M , studied in Section 3, as basis. On TM the canonicalvector ﬁeld V (V A) (A = 1, · · · 2m) is called the Liouville vector ﬁeld [3], andthe complete lift [14] ΩC of the structure 2-form of rank 4m on TM is given byΩC =∑dV a ∧ ωa∗ + ωa∧ = dV a∗ , a = 1, · · ·m; a∗ = a+m. (3)In Section 3, the following relation will be derived (see formula (24)):dωA = α ∧ ωA.By exterior diﬀerentiation of (3), and taking into account the above formula,one getsdΩC = α ∧ ΩC , (4)andLV ΩC = ΩC . (5)The above equations express that the 2-form ΩC is a homogeneous 2-form ofclass 1 [4] on TM . Next, the Liouville form µ (i.e. µ = V 8) is expressed byµ =∑V AωA A = 1, · · · 2m (6)On manifolds with T -parallel exterior recurrent connection 61and one ﬁnds by exterior diﬀerentiation thatdµ = α ∧ µ+ ψ, (7)where we have setψ =∑dV A ∧ ωA. (8)One also derives thatLV ψ = ψ, (9)and this shows that, like ΩC , the form ψ is a homogeneous 2-form of class 1.Moreover, making use of the vertical operator iv of Godbillon [3], one calculatesthativψ = 0, (10)which together with (9) proves that ψ is a Finslerian form. In addition, if T Vdenotes the vertical lift of T , one also ﬁnds thatLT V ψ = 0,which shows that T V deﬁnes an inﬁnitesimal automorphism of ψ. Some otherproperties regarding the principal almost symplectic form II = ‖T ‖2ψ are alsodiscussed.1 PreliminariesLet (M, g) be a Riemannian C∞-manifold and let ∇ be the covariant dif-ferential operator with respect to the metric tensor g. We assume that M isoriented and ∇ is the Levi-Civita connection of g. Let ΓTM = Ξ(M) be the setof sections of the tangent bundle, andO : TM 8→ T ∗M and L : TM 9← T ∗M (11)the classical isomorphisms deﬁned by g (i.e. 8 is the index lowering operator,and 9 is the index raising operator).Following [8], we denote byAq(M,TM) = ΓHom(ΛqTM, TM), (12)the set of vector valued q-forms (q < dimM), and we write for the covariantderivative operator with respect to ∇d∇ : Aq(M,TM)→ Aq+1(M,TM). (13)62 F. Defever, R. RoscaIt should be noticed that in general d∇2 = d∇ ◦ d∇ = 0, unlike d2 = d ◦ d = 0.We denote by I ∈ A1(M,TM) the canonical vector valued 1-form of M , whichis also called the soldering form of M [2]. Since ∇ is symmetric one has thatd∇(I) = 0.A vector ﬁeld Z ∈ Ξ(M) which satisﬁesd∇(∇Z) = ∇2Z = π ∧ I ∈ A2(M,TM); π ∈ Λ1M (14)is deﬁned to be an exterior concurrent vector ﬁeld [9] (see also [6]). The 1-formπ in (14) is called the concurrence form and is deﬁned byπ = λZ8, λ ∈ Λ0M. (15)Let O = {eA | A = 1, · · · 2m} be a local ﬁeld of orthonormal frames over Mand let O∗ = covect{ωA} be its associated coframe. Then E. Cartan’s structureequations can be written in indexless manner as∇e = θ ⊗ e, (16)dω = −θ ∧ ω, (17)dθ = −θ ∧ θ +Θ. (18)In the above equations θ (respectively Θ) are the local connection forms in thetangent bundle TM (respectively the curvature 2-forms on M).2 Manifolds with T -parallel exterior recurrent con-nectionLetM(Ω, T , g) be a 2m-dimensional manifold with almost symplectic 2-formΩ and with structure vector ﬁeld T (T A) (A = 1, · · · 2m). Now, by reference to [9](see also [6]), we assume that (M, g) is structured by a T -parallel connection,which means that the connection forms satisfyθAB =< T , eB ∧ eA >, (19)where ∧ stands for the wedge product of vector ﬁelds. In addition, we alsoassume that the connection forms θAB are exterior recurrent [2] with 2T 8 asrecurrence forms, which means thatdθAB = 2T 8 ∧ θAB. (20)SinceθAB = T BωA − T AωB,On manifolds with T -parallel exterior recurrent connection 63it follows thatdT A = T Aα, (21)where we have set α := T 8. Now, in view of the structure equations (17) andinvoking the curvature forms ΘAB, one derivesΘAB = ‖T ‖2ωB ∧ ωA + α ∧ θAB. (22)Since one hasd‖T ‖2 = 2‖T ‖2α, (23)then by (21) one getsdωA = α ∧ ωA. (24)By exterior diﬀerentiation of (22), one derives thatdΘAB = 3α ∧ΘAB. (25)The above equation expresses the fact that the connection forms being exteriorrecurrent implies the same property for the curvature forms ΘAB also. Takingmoreover the Lie derivatives of θAB and ΘAB with respect to the structure vectorﬁeld T , and using (23), one ﬁndsLT θAB = 2‖T ‖2θAB,LT ΘAB = 3‖T ‖2ΘAB.(26)Hence, T deﬁnes an inﬁnitesimal conformal transformation of both the connec-tion forms and the curvature forms.On the other hand, by (19) one ﬁnds that∇eA = T AI − ωA ⊗ T , (27)and in this way one gets by (21) also that∇T = ‖T ‖2I. (28)This shows that T is a concurrent vector ﬁeld (it is well known [1] that concur-rency is of conformal nature). From (27) and (28) it follows that[T , eA] = −‖T ‖2eA, (29)and this proves that the diﬀerential system {eA} corresponding to the vectorbasis admits an inﬁnitesimal transformation with generator T . We also noticethat operating on (28) with ∇ (the operator ∇ acts inductively) one gets∇(∇T ) = ∇2T = ‖T ‖4α ∧ I. (30)64 F. Defever, R. RoscaThis shows that T is an exterior concurrent vector ﬁeld [10] (see also [7]). Inconsequence of (30) one may now also writeR(T , Z) = −(2m− 1)‖T ‖4g(T , Z), Z ∈ Ξ(M), (31)whereRmeans the Ricci tensor ﬁeld of∇. In the same way one can also calculatethat∇3eA = ‖T ‖4(α ∧ ωA) ∧ I, (32)and consequently one can conclude that the elements of the vector basis {eA}are exterior concurrent vector ﬁelds; in the sequel we will use the terminologyof a 2-exterior vector basis for this case.We recall that a function f : R2m → R is called isoparametric [13] if both‖gradf‖2 and div(gradf) are functions of f . In the case under discussion, onehas ﬁrst of all thatgrad‖T ‖2 = ‖T ‖2T , (33)from which there follows that‖ grad‖T ‖2 ‖2 = ‖T ‖4. (34)Next, one also derives thatdiv grad‖T ‖2 = 4 (2m+ ‖T ‖2) ‖T ‖2, (35)from which one may conclude that ‖T ‖2 is an isoparametric function. Next, bythe general formula∆µ = −div∇µ, µ ∈ Λ0M,where ∆ denotes the Laplacian, and in virtue of (33), we see that ‖T ‖2 is aneigenfunction of ∆, having 4(2m+ ‖T ‖2) as eigenvalue of ∆. Recall now that ifZ is any vector ﬁeld, one hastr∇2Z =∑∇eA(∇eAZ).Then, by (30) one derivestr∇2T = 2‖T ‖2T . (36)With R denoting the Ricci tensor ﬁeld, one now hasR(T , V ) = −2(2m− 1)‖T ‖2g(T , V ), V ∈ Ξ(M). (37)On manifolds with T -parallel exterior recurrent connection 65Then, by reference to [8], if V is a parallel vector ﬁeld, one has the Weitzenbockformula:(∆T 8)V = R(V, T )− < tr∇2T , V >= −4m‖T ‖2g(T , V ). (38)On the other hand, regarding the almost symplectic form Ω, one writes withstandard notationΩ =∑ωa ∧ ωa∗ , a = 1, · · ·m, a∗ = a+m. (39)Taking the exterior derivative of Ω, and in view of (24), one ﬁnds thatdΩ = 2α ∧ Ω, α = T 8. (40)This aﬃrms the fact that Ω deﬁnes a locally conformal symplectic structure onM having α as covector of Lee. Then, as is known from [5], calling the mappingZ → −iZΩ = 8Z the symplectic isomorphism, one has− 8T = iT Ω =∑(T aωa∗ − T a∗ωa), (41)and by (21) and (24) one ﬁnds thatLT Ω = 2‖T ‖2Ω. (42)Hence, following a known deﬁnition [5], the above equation means that T deﬁnesa inﬁnitesimal conformal transformation of Ω. On the other hand, regarding thecurvature forms, we recall that the Bianchi forms in the sense of Tachibana [12]are deﬁned byΘ(p)u1,...,u2p= Θu2u1 ∧Θu3u2 ∧ · · · ∧Θ2p2p−1. (43)Then, by exterior diﬀerentiation one gets from (43)d(Θ(p)u1,...,u2p)= 3(2m− 1)α ∧Θ(p)u1,...,u2p, (44)and we may consequently observe that the Bianchi forms Θ(p)u1,...,u2pare exteriorrecurrent, with 3(2m− 1)α as recurrence form.In another perspective, let X be any vector ﬁeld on M ; if the covariantdiﬀerential of X is the wedge product of X with the structure vector ﬁeld T ,this means that X is a skew symmetric Killing vector ﬁeld (in the sense of [11]),i.e.∇X = X ∧ T = α⊗X −X8 ⊗ T . (45)66 F. Defever, R. RoscaOne may also remark that the above relation is indeed in correspondence withRosca’s lemma [11] concerning skew-symmetric Killing and conformal skew-symmetric Killing vector ﬁelds.dX8 = 2X ∧X8.In this case, the diﬀerentials of the components of X, i.e. dXA satisfydXA = −g(X, T )ωA +XAα. (46)In view of the mentioned facts, and taking the Lie derivative of Ω with respectto X, one calculates thatLXΩ = −2g(X, T )Ω. (47)This proves the property that any skew symmetric Killing vector ﬁeld X, havingthe structure vector ﬁeld T as generative, deﬁnes an inﬁnitesimal conformaltransformation of the conformal symplectic form Ω.Summing up, we state the followingTheorem 1. Let M(Ω, T , α) be a 2m-dimensional Riemannian manifoldstructured by a T -parallel exterior recurrent connection. In this case, the struc-ture vector ﬁeld T is concurrent and deﬁnes an inﬁnitesimal conformal trans-formation of the connection forms θAB, of the curvature forms ΘAB and of theconformal symplectic form Ω. In addition, one has the following properties:(i) ‖T ‖2 is an isoparametric function;(ii) the diﬀerential system {eA} admits an inﬁnitesimal transformation withgenerator T , i.e.[T , eA] = ‖T ‖2eA;(iii) all the basis vector ﬁelds eA are 2-exterior concurrent vector ﬁelds, i.e.∇3eA = 2‖T ‖2(α ∧ ωA) ∧ I, α = T 8.(iv) ‖T ‖2 is an eigenfunction of ∆ having 4(2m+ ‖T ‖2) as eigenvalue of ∆;(v) if V denotes any parallel vector ﬁeld, then one has the Weitzenbock formula∆α(V ) = R(V, T )− < tr∇2T , V >= −4m‖T ‖2g(T , V );(vi) if Θ(p)u1,...,u2p = Θu2u1 ∧ Θu3u2 ∧ · · · ∧ Θ2p2p−1 means the Bianchi form of type(2p, 2p), in the sense of Tachibana, then Θ(p)u1,...,u2p is exterior recurrentwith 3(2m− 1)α as recurrence form;On manifolds with T -parallel exterior recurrent connection 67(vii) any skew symmetric Killing vector ﬁeld X, having T as generative, deﬁnesan inﬁnitesimal conformal transformation of Ω, i.e.LXΩ = −2g(X, T )Ω.3 Geometry of the tangent bundleIn this section we will discuss some properties of the tangent bundle manifoldTM having as basis manifold M studied in Section 3. Denote by V (V A) (A =1, · · · 2m) the Liouville vector ﬁeld (or the canonical vector ﬁeld on TM [4]).Accordingly, one may consider the setB∗ = {ωA, dV A | A = 1, · · · 2m}as an adapted cobasis in TM (see also [6]). Let T rs be the set of all tensor ﬁeldsof type (r, s) on M . It is well known [14] that the vertical and complete lifts arelinear mappings of T rsM into Trs (TM), and one has(T1 ⊗ T2)C = T V1 ⊗ T C2 + T C1 ⊗ T V2 . (48)Hence, in the case under discussion we may deﬁne the complete lift ΩC of thestructure conformal 2-form Ω of M to be the 2-form of rank 4m on TM givenbyΩC =∑(dV a ∧ ωa∗ + ωa ∧ dV a∗), a = 1, · · ·m; a∗ = a+m. (49)On the other hand, the Liouville vector ﬁeld V is expressed byV =∑V A∂∂V A; (50)it is also known that the associated basic 1-formµ =∑V AωA (51)is called the Liouville form. (Alternatively, one can also write that µ = V 8.)Next, taking the Lie diﬀerential of ΩC with respect to the Liouville vectorﬁeld V and taking into account (24), one ﬁnds thatLV ΩC = ΩC . (52)Hence, with reference to [4], the above equation proves that ΩC is a homogeneous2-form of class 1 on TM .68 F. Defever, R. RoscaTaking moreover the Lie diﬀerential of ΩC with respect to the structurevector ﬁeld T , one also derives thatLT ΩC = ‖T ‖2ΩC . (53)The above equation shows that T deﬁnes also for ΩC an inﬁnitesimal conformaltransformation.By exterior derivation of the Liouville form µ deﬁned by (51), and takinginto account (24), one gets thatdµ = α ∧ µ+ dV A ∧ ωA. (54)Introducing the notationψ =∑dV a ∧ ωa, (55)and by reference to (24), it follows thatdψ = α ∧ ψ, (56)which shows that ψ is an exterior recurrent form with α as recurrence form.Then, since one ﬁrst calculates thatiV ψ = µ, α(V ) = 0, (57)one ﬁnally gets thatLV ψ = ψ, (58)which shows that, as Ωc, the form ψ is also a homogeneous 2-form of class 1.We remind that the vertical operator iv in the sense of [3] possesses bydeﬁnition the following properties:ivλ = 0, ivωA = 0, ivdV A = ωA, (59)from which one calculates thativψ = 0. (60)On behalf of (58) and (60) we conclude from this that ψ is a Finslerian form [3].In another order of ideas, we recall that the vertical lift ZV [14] of any vectorﬁeld Z on M with components ZA is expressed byZV =(0ZA)= ZA∂∂vA, (A = 1, · · · 2m).On manifolds with T -parallel exterior recurrent connection 69Therefore, in the case under consideration, the vertical lift T V of T is given byT V =∑T A ∂∂V A, A ∈ {1, · · · 2m}, (61)and by (55) one ﬁnds respectively thatiT V ψ = α, and LT V ψ = 0. (62)On behalf of the above, one may conclude that T V deﬁnes an inﬁnitesimalautomorphism of the 2-form ψ.Finally, consider the 2-formII = fψ; (63)following [4], f is called the energy scalar. Now, in view of (23), one hasdII = f(dff+d‖T ‖22‖T ‖2)∧ II. (64)By reference to [4] and in case thatdff+d‖T ‖22‖T ‖2 = 0,this shows that II can then be seen as the canonical symplectic form of the4m-dimensional manifold TM . Finally, we setr = fv,where v = 12∑(V A)2 denotes the Liouville function; then, by reference to [4],the pair (r, II) deﬁnes a regular mechanical system (in the sense of Klein) havingr as kinetic energy.Theorem 2. Let TM be the tangent bundle manifold having as basis theconformal symplectic manifold M(Ω, T , α) structured by a T -parallel connectionand having α = T 8 as covector of Lee. Let V , µ, and v, be the Liouville vectorﬁeld, the Liouville form, and the Liouville function of TM respectively. One hasthe following properties:(i) the complete lift ΩC on TM of the conformally symplectic form Ω of M , isa homogeneous 2-form of class 1, i.e.LV ΩC = ΩC ;70 F. Defever, R. Rosca(ii) the vertical lift T V of T deﬁnes an inﬁnitesimal automorphism of the 2-form ψ =∑dV A ∧ ωA, (A = 1, · · · 2m);(iii) if f stands for the energy function of M , then the 2-form II = fψ is thecanonical symplectic form on TM(dff +d‖T ‖22‖T ‖2 = 0), and the pair (r, II),consisting of the scalar r = fv and the 2-form fψ, deﬁnes a regular me-chanical system (in the sense of Klein) on TM .References[1] B. Y. Chen: Geometry of Submanifolds, M. Dekker, New York (1973)[2] D. K. Datta: Exterior recurrent forms on manifolds, Tensor NS 36 (1982) 115–120.[3] C. Godbillon: Ge´ometrie diﬀe´rentielle et me´canique analytique, Herman, Paris (1969).[4] J. Klein: Espaces variationnels et me´canique, Ann. Inst. Fourier 4 (1962) 1–124.[5] P. Libermann, C. M. Marle: Ge´ome´trie Symplectique, Bases The´ore´tiques de laMe´canique, U.E.R. Math. du C.N.R.S. 7 (1986).[6] I. Mihai, R. Rosca, L. Verstraelen: On a class of exact locally conformal cosymplecticmanifolds, Internat. J. Math. and Math. Sci. 19 (1996) 267–278.[7] M. Petrovic, R. Rosca, L. Verstraelen: Exterior concurrent vector fields on Rie-mannian manifolds, Soochow J. Math. 15 (1989) 179–187.[8] W. A. Poor: Diﬀerential Geometric Structures, Mc Graw Hill, New York (1981).[9] R. Rosca: On parallel conformal connections, Kodai Math. J. 2 (1979) 1–9.[10] R. Rosca: Exterior concurrent vectorfields on a conformal cosymplectic manifold admit-ting a Sasakian structure, Libertas Math. (Univ. Arlington, Texas) 6 (1986) 167–174.[11] R. Rosca: On exterior concurrent skew symmetric Killing vector fields, Rend. Mat.Messina 2 (1993) 131–145.[12] S. Tachibana: On contact structures of real and complex manifolds, Tohoku Math. J. 5(1963) 123–130.[13] A. West: Isoparametric systems, Geometry and Topology of Submanifolds 1 (1989) 222–230.[14] K. Yano, S. Ishihara: Diﬀerential Geometry of Tangent and Cotangent Bundles, M.Dekker, New York (1973).

Riemannian manifolds structured by a T-parallel exterior recurrent connection

Università del Salento: ESE - Salento University Publishing

Note di Matematica 21, n. 1, 2002, 59–70.Riemannian manifolds structured bya T -parallel exterior recurrent connectionFilip DefeverDepartement Industrie¨le Wetenschappen en Technologie,Katholieke Hogeschool Brugge-Oostende,Zeedijk 101, 8400 Oostende, BelgiumRadu Rosca59 Avenue Emile Zola, 75015 Paris, FranceReceived: 22 February 2001; accepted: 8 November 2001.Abstract. Geometrical and structural properties are proved for Riemannian manifolds whichare equipped with a T -parallel exterior recurrent connection.Keywords: Exterior recurrent forms, T -parallel connection.MSC 2000 classification: 53B20.IntroductionRiemannian manifolds structured by a T -parallel connection have been de-fined in [9] and have also been studied in [6]. Let M be a 2m-dimensionalC∞-manifold and ∇ be the Levi-Civita connection. We recall that if M carriesa globally defined vector field T (T A) and the connection forms satisfyθAB =< T , eB ∧ eA >, (1)where ∧ denotes the wedge product of vector fields, then one says that Mis structured by a T -parallel connection. In the present paper we assume inaddition that θAB are exterior recurrent forms [2], which means thatdθAB = 2α ∧ θAB, where α = T 8, (2)having T 8 as recurrence form. This implies that the curvature forms ΘAB arealso exterior recurrent. In consequence of this fact, we adopt the terminologythat M is structured by a T -parallel exterior recurrent connection.For the above mentioned structures, we prove the following properties:(i) T is a concurrent vector field and defines an infinitesimal conformal trans-formation of θAB and ΘAB and the differential system ∇eA correspondingto the vector basis O = {eA} admits an infinitesimal transformation withgenerator T ;60 F. Defever, R. Rosca(ii) ‖T ‖2 is an isoparametric function [13], and an eigenfunction of ∆ having4(2m+ ‖T ‖2) as eigenvalue;(iii) if V is any parallel vector field, one has by the Weitzenbock formula that(∆T 8)V = −4m‖T ‖2g(T , V );(iv) ifΘ(p)u1,...,u2p= Θu2u1 ∧Θu3u2 ∧ · · · ∧Θ2p2p−1denotes the Bianchi forms (in the sense of Tachibana [12]), these formsare exterior recurrent with 3(2m− 1)α as recurrence form;(v) any vector field X such that∇X = X ∧ Tis a skew symmetric Killing vector field [11] and X defines an infinitesimaltransformation of the conformal symplectic form Ω, i.e.LXΩ = −2g(X, T )Ω.In Section 4 we consider some properties of the tangent bundle manifold TMhaving the manifold M , studied in Section 3, as basis. On TM the canonicalvector field V (V A) (A = 1, · · · 2m) is called the Liouville vector field [3], andthe complete lift [14] ΩC of the structure 2-form of rank 4m on TM is given byΩC =∑dV a ∧ ωa∗ + ωa∧ = dV a∗ , a = 1, · · ·m; a∗ = a+m. (3)In Section 3, the following relation will be derived (see formula (24)):dωA = α ∧ ωA.By exterior differentiation of (3), and taking into account the above formula,one getsdΩC = α ∧ ΩC , (4)andLV ΩC = ΩC . (5)The above equations express that the 2-form ΩC is a homogeneous 2-form ofclass 1 [4] on TM . Next, the Liouville form µ (i.e. µ = V 8) is expressed byµ =∑V AωA A = 1, · · · 2m (6)On manifolds with T -parallel exterior recurrent connection 61and one finds by exterior differentiation thatdµ = α ∧ µ+ ψ, (7)where we have setψ =∑dV A ∧ ωA. (8)One also derives thatLV ψ = ψ, (9)and this shows that, like ΩC , the form ψ is a homogeneous 2-form of class 1.Moreover, making use of the vertical operator iv of Godbillon [3], one calculatesthativψ = 0, (10)which together with (9) proves that ψ is a Finslerian form. In addition, if T Vdenotes the vertical lift of T , one also finds thatLT V ψ = 0,which shows that T V defines an infinitesimal automorphism of ψ. Some otherproperties regarding the principal almost symplectic form II = ‖T ‖2ψ are alsodiscussed.1 PreliminariesLet (M, g) be a Riemannian C∞-manifold and let ∇ be the covariant dif-ferential operator with respect to the metric tensor g. We assume that M isoriented and ∇ is the Levi-Civita connection of g. Let ΓTM = Ξ(M) be the setof sections of the tangent bundle, andO : TM 8→ T ∗M and L : TM 9← T ∗M (11)the classical isomorphisms defined by g (i.e. 8 is the index lowering operator,and 9 is the index raising operator).Following [8], we denote byAq(M,TM) = ΓHom(ΛqTM, TM), (12)the set of vector valued q-forms (q < dimM), and we write for the covariantderivative operator with respect to ∇d∇ : Aq(M,TM)→ Aq+1(M,TM). (13)62 F. Defever, R. RoscaIt should be noticed that in general d∇2 = d∇ ◦ d∇ = 0, unlike d2 = d ◦ d = 0.We denote by I ∈ A1(M,TM) the canonical vector valued 1-form of M , whichis also called the soldering form of M [2]. Since ∇ is symmetric one has thatd∇(I) = 0.A vector field Z ∈ Ξ(M) which satisfiesd∇(∇Z) = ∇2Z = π ∧ I ∈ A2(M,TM); π ∈ Λ1M (14)is defined to be an exterior concurrent vector field [9] (see also [6]). The 1-formπ in (14) is called the concurrence form and is defined byπ = λZ8, λ ∈ Λ0M. (15)Let O = {eA | A = 1, · · · 2m} be a local field of orthonormal frames over Mand let O∗ = covect{ωA} be its associated coframe. Then E. Cartan’s structureequations can be written in indexless manner as∇e = θ ⊗ e, (16)dω = −θ ∧ ω, (17)dθ = −θ ∧ θ +Θ. (18)In the above equations θ (respectively Θ) are the local connection forms in thetangent bundle TM (respectively the curvature 2-forms on M).2 Manifolds with T -parallel exterior recurrent con-nectionLetM(Ω, T , g) be a 2m-dimensional manifold with almost symplectic 2-formΩ and with structure vector field T (T A) (A = 1, · · · 2m). Now, by reference to [9](see also [6]), we assume that (M, g) is structured by a T -parallel connection,which means that the connection forms satisfyθAB =< T , eB ∧ eA >, (19)where ∧ stands for the wedge product of vector fields. In addition, we alsoassume that the connection forms θAB are exterior recurrent [2] with 2T 8 asrecurrence forms, which means thatdθAB = 2T 8 ∧ θAB. (20)SinceθAB = T BωA − T AωB,On manifolds with T -parallel exterior recurrent connection 63it follows thatdT A = T Aα, (21)where we have set α := T 8. Now, in view of the structure equations (17) andinvoking the curvature forms ΘAB, one derivesΘAB = ‖T ‖2ωB ∧ ωA + α ∧ θAB. (22)Since one hasd‖T ‖2 = 2‖T ‖2α, (23)then by (21) one getsdωA = α ∧ ωA. (24)By exterior differentiation of (22), one derives thatdΘAB = 3α ∧ΘAB. (25)The above equation expresses the fact that the connection forms being exteriorrecurrent implies the same property for the curvature forms ΘAB also. Takingmoreover the Lie derivatives of θAB and ΘAB with respect to the structure vectorfield T , and using (23), one findsLT θAB = 2‖T ‖2θAB,LT ΘAB = 3‖T ‖2ΘAB.(26)Hence, T defines an infinitesimal conformal transformation of both the connec-tion forms and the curvature forms.On the other hand, by (19) one finds that∇eA = T AI − ωA ⊗ T , (27)and in this way one gets by (21) also that∇T = ‖T ‖2I. (28)This shows that T is a concurrent vector field (it is well known [1] that concur-rency is of conformal nature). From (27) and (28) it follows that[T , eA] = −‖T ‖2eA, (29)and this proves that the differential system {eA} corresponding to the vectorbasis admits an infinitesimal transformation with generator T . We also noticethat operating on (28) with ∇ (the operator ∇ acts inductively) one gets∇(∇T ) = ∇2T = ‖T ‖4α ∧ I. (30)64 F. Defever, R. RoscaThis shows that T is an exterior concurrent vector field [10] (see also [7]). Inconsequence of (30) one may now also writeR(T , Z) = −(2m− 1)‖T ‖4g(T , Z), Z ∈ Ξ(M), (31)whereRmeans the Ricci tensor field of∇. In the same way one can also calculatethat∇3eA = ‖T ‖4(α ∧ ωA) ∧ I, (32)and consequently one can conclude that the elements of the vector basis {eA}are exterior concurrent vector fields; in the sequel we will use the terminologyof a 2-exterior vector basis for this case.We recall that a function f : R2m → R is called isoparametric [13] if both‖gradf‖2 and div(gradf) are functions of f . In the case under discussion, onehas first of all thatgrad‖T ‖2 = ‖T ‖2T , (33)from which there follows that‖ grad‖T ‖2 ‖2 = ‖T ‖4. (34)Next, one also derives thatdiv grad‖T ‖2 = 4 (2m+ ‖T ‖2) ‖T ‖2, (35)from which one may conclude that ‖T ‖2 is an isoparametric function. Next, bythe general formula∆µ = −div∇µ, µ ∈ Λ0M,where ∆ denotes the Laplacian, and in virtue of (33), we see that ‖T ‖2 is aneigenfunction of ∆, having 4(2m+ ‖T ‖2) as eigenvalue of ∆. Recall now that ifZ is any vector field, one hastr∇2Z =∑∇eA(∇eAZ).Then, by (30) one derivestr∇2T = 2‖T ‖2T . (36)With R denoting the Ricci tensor field, one now hasR(T , V ) = −2(2m− 1)‖T ‖2g(T , V ), V ∈ Ξ(M). (37)On manifolds with T -parallel exterior recurrent connection 65Then, by reference to [8], if V is a parallel vector field, one has the Weitzenbockformula:(∆T 8)V = R(V, T )− < tr∇2T , V >= −4m‖T ‖2g(T , V ). (38)On the other hand, regarding the almost symplectic form Ω, one writes withstandard notationΩ =∑ωa ∧ ωa∗ , a = 1, · · ·m, a∗ = a+m. (39)Taking the exterior derivative of Ω, and in view of (24), one finds thatdΩ = 2α ∧ Ω, α = T 8. (40)This affirms the fact that Ω defines a locally conformal symplectic structure onM having α as covector of Lee. Then, as is known from [5], calling the mappingZ → −iZΩ = 8Z the symplectic isomorphism, one has− 8T = iT Ω =∑(T aωa∗ − T a∗ωa), (41)and by (21) and (24) one finds thatLT Ω = 2‖T ‖2Ω. (42)Hence, following a known definition [5], the above equation means that T definesa infinitesimal conformal transformation of Ω. On the other hand, regarding thecurvature forms, we recall that the Bianchi forms in the sense of Tachibana [12]are defined byΘ(p)u1,...,u2p= Θu2u1 ∧Θu3u2 ∧ · · · ∧Θ2p2p−1. (43)Then, by exterior differentiation one gets from (43)d(Θ(p)u1,...,u2p)= 3(2m− 1)α ∧Θ(p)u1,...,u2p, (44)and we may consequently observe that the Bianchi forms Θ(p)u1,...,u2pare exteriorrecurrent, with 3(2m− 1)α as recurrence form.In another perspective, let X be any vector field on M ; if the covariantdifferential of X is the wedge product of X with the structure vector field T ,this means that X is a skew symmetric Killing vector field (in the sense of [11]),i.e.∇X = X ∧ T = α⊗X −X8 ⊗ T . (45)66 F. Defever, R. RoscaOne may also remark that the above relation is indeed in correspondence withRosca’s lemma [11] concerning skew-symmetric Killing and conformal skew-symmetric Killing vector fields.dX8 = 2X ∧X8.In this case, the differentials of the components of X, i.e. dXA satisfydXA = −g(X, T )ωA +XAα. (46)In view of the mentioned facts, and taking the Lie derivative of Ω with respectto X, one calculates thatLXΩ = −2g(X, T )Ω. (47)This proves the property that any skew symmetric Killing vector field X, havingthe structure vector field T as generative, defines an infinitesimal conformaltransformation of the conformal symplectic form Ω.Summing up, we state the followingTheorem 1. Let M(Ω, T , α) be a 2m-dimensional Riemannian manifoldstructured by a T -parallel exterior recurrent connection. In this case, the struc-ture vector field T is concurrent and defines an infinitesimal conformal trans-formation of the connection forms θAB, of the curvature forms ΘAB and of theconformal symplectic form Ω. In addition, one has the following properties:(i) ‖T ‖2 is an isoparametric function;(ii) the differential system {eA} admits an infinitesimal transformation withgenerator T , i.e.[T , eA] = ‖T ‖2eA;(iii) all the basis vector fields eA are 2-exterior concurrent vector fields, i.e.∇3eA = 2‖T ‖2(α ∧ ωA) ∧ I, α = T 8.(iv) ‖T ‖2 is an eigenfunction of ∆ having 4(2m+ ‖T ‖2) as eigenvalue of ∆;(v) if V denotes any parallel vector field, then one has the Weitzenbock formula∆α(V ) = R(V, T )− < tr∇2T , V >= −4m‖T ‖2g(T , V );(vi) if Θ(p)u1,...,u2p = Θu2u1 ∧ Θu3u2 ∧ · · · ∧ Θ2p2p−1 means the Bianchi form of type(2p, 2p), in the sense of Tachibana, then Θ(p)u1,...,u2p is exterior recurrentwith 3(2m− 1)α as recurrence form;On manifolds with T -parallel exterior recurrent connection 67(vii) any skew symmetric Killing vector field X, having T as generative, definesan infinitesimal conformal transformation of Ω, i.e.LXΩ = −2g(X, T )Ω.3 Geometry of the tangent bundleIn this section we will discuss some properties of the tangent bundle manifoldTM having as basis manifold M studied in Section 3. Denote by V (V A) (A =1, · · · 2m) the Liouville vector field (or the canonical vector field on TM [4]).Accordingly, one may consider the setB∗ = {ωA, dV A | A = 1, · · · 2m}as an adapted cobasis in TM (see also [6]). Let T rs be the set of all tensor fieldsof type (r, s) on M . It is well known [14] that the vertical and complete lifts arelinear mappings of T rsM into Trs (TM), and one has(T1 ⊗ T2)C = T V1 ⊗ T C2 + T C1 ⊗ T V2 . (48)Hence, in the case under discussion we may define the complete lift ΩC of thestructure conformal 2-form Ω of M to be the 2-form of rank 4m on TM givenbyΩC =∑(dV a ∧ ωa∗ + ωa ∧ dV a∗), a = 1, · · ·m; a∗ = a+m. (49)On the other hand, the Liouville vector field V is expressed byV =∑V A∂∂V A; (50)it is also known that the associated basic 1-formµ =∑V AωA (51)is called the Liouville form. (Alternatively, one can also write that µ = V 8.)Next, taking the Lie differential of ΩC with respect to the Liouville vectorfield V and taking into account (24), one finds thatLV ΩC = ΩC . (52)Hence, with reference to [4], the above equation proves that ΩC is a homogeneous2-form of class 1 on TM .68 F. Defever, R. RoscaTaking moreover the Lie differential of ΩC with respect to the structurevector field T , one also derives thatLT ΩC = ‖T ‖2ΩC . (53)The above equation shows that T defines also for ΩC an infinitesimal conformaltransformation.By exterior derivation of the Liouville form µ defined by (51), and takinginto account (24), one gets thatdµ = α ∧ µ+ dV A ∧ ωA. (54)Introducing the notationψ =∑dV a ∧ ωa, (55)and by reference to (24), it follows thatdψ = α ∧ ψ, (56)which shows that ψ is an exterior recurrent form with α as recurrence form.Then, since one first calculates thatiV ψ = µ, α(V ) = 0, (57)one finally gets thatLV ψ = ψ, (58)which shows that, as Ωc, the form ψ is also a homogeneous 2-form of class 1.We remind that the vertical operator iv in the sense of [3] possesses bydefinition the following properties:ivλ = 0, ivωA = 0, ivdV A = ωA, (59)from which one calculates thativψ = 0. (60)On behalf of (58) and (60) we conclude from this that ψ is a Finslerian form [3].In another order of ideas, we recall that the vertical lift ZV [14] of any vectorfield Z on M with components ZA is expressed byZV =(0ZA)= ZA∂∂vA, (A = 1, · · · 2m).On manifolds with T -parallel exterior recurrent connection 69Therefore, in the case under consideration, the vertical lift T V of T is given byT V =∑T A ∂∂V A, A ∈ {1, · · · 2m}, (61)and by (55) one finds respectively thatiT V ψ = α, and LT V ψ = 0. (62)On behalf of the above, one may conclude that T V defines an infinitesimalautomorphism of the 2-form ψ.Finally, consider the 2-formII = fψ; (63)following [4], f is called the energy scalar. Now, in view of (23), one hasdII = f(dff+d‖T ‖22‖T ‖2)∧ II. (64)By reference to [4] and in case thatdff+d‖T ‖22‖T ‖2 = 0,this shows that II can then be seen as the canonical symplectic form of the4m-dimensional manifold TM . Finally, we setr = fv,where v = 12∑(V A)2 denotes the Liouville function; then, by reference to [4],the pair (r, II) defines a regular mechanical system (in the sense of Klein) havingr as kinetic energy.Theorem 2. Let TM be the tangent bundle manifold having as basis theconformal symplectic manifold M(Ω, T , α) structured by a T -parallel connectionand having α = T 8 as covector of Lee. Let V , µ, and v, be the Liouville vectorfield, the Liouville form, and the Liouville function of TM respectively. One hasthe following properties:(i) the complete lift ΩC on TM of the conformally symplectic form Ω of M , isa homogeneous 2-form of class 1, i.e.LV ΩC = ΩC ;70 F. Defever, R. Rosca(ii) the vertical lift T V of T defines an infinitesimal automorphism of the 2-form ψ =∑dV A ∧ ωA, (A = 1, · · · 2m);(iii) if f stands for the energy function of M , then the 2-form II = fψ is thecanonical symplectic form on TM(dff +d‖T ‖22‖T ‖2 = 0), and the pair (r, II),consisting of the scalar r = fv and the 2-form fψ, defines a regular me-chanical system (in the sense of Klein) on TM .References[1] B. Y. Chen: Geometry of Submanifolds, M. Dekker, New York (1973)[2] D. K. Datta: Exterior recurrent forms on manifolds, Tensor NS 36 (1982) 115–120.[3] C. Godbillon: Ge´ometrie diffe´rentielle et me´canique analytique, Herman, Paris (1969).[4] J. Klein: Espaces variationnels et me´canique, Ann. Inst. Fourier 4 (1962) 1–124.[5] P. Libermann, C. M. Marle: Ge´ome´trie Symplectique, Bases The´ore´tiques de laMe´canique, U.E.R. Math. du C.N.R.S. 7 (1986).[6] I. Mihai, R. Rosca, L. Verstraelen: On a class of exact locally conformal cosymplecticmanifolds, Internat. J. Math. and Math. Sci. 19 (1996) 267–278.[7] M. Petrovic, R. Rosca, L. Verstraelen: Exterior concurrent vector fields on Rie-mannian manifolds, Soochow J. Math. 15 (1989) 179–187.[8] W. A. Poor: Differential Geometric Structures, Mc Graw Hill, New York (1981).[9] R. Rosca: On parallel conformal connections, Kodai Math. J. 2 (1979) 1–9.[10] R. Rosca: Exterior concurrent vectorfields on a conformal cosymplectic manifold admit-ting a Sasakian structure, Libertas Math. (Univ. Arlington, Texas) 6 (1986) 167–174.[11] R. Rosca: On exterior concurrent skew symmetric Killing vector fields, Rend. Mat.Messina 2 (1993) 131–145.[12] S. Tachibana: On contact structures of real and complex manifolds, Tohoku Math. J. 5(1963) 123–130.[13] A. West: Isoparametric systems, Geometry and Topology of Submanifolds 1 (1989) 222–230.[14] K. Yano, S. Ishihara: Differential Geometry of Tangent and Cotangent Bundles, M.Dekker, New York (1973).

English

http://siba-ese.unisalento.it/index.php/notemat/article/download/739/583

Riemannian manifolds structured by a T-parallel exterior recurrent connection

Abstract

Similar works

Full text

Available Versions

ESE - Salento University Publishing

Università del Salento: ESE - Salento University Publishing