ON A CLASS OF EXACT LOCALLY CONFORMAL COSYMLECTIC MANIFOLDS

An almost cosymplectic manifold M is a ( 2 m + 1 ) -dimensional oriented Riemannian 
manifold endowed with a 2-form Ω of rank 2 m , a 1-form η such that Ω m  Λ  η ≠ 0 and a vector field ξ 
satisfying i ξ Ω = 0 and η ( ξ ) = 1 . Particular cases were considered in [3] and [6].Let ( M , g ) be an odd dimensional oriented Riemannian manifold carrying a globally defined vector 
field T such that the Riemannian connection is parallel with respect to T . It is shown that in this case 
 M is a hyperbolic space form endowed with an exact locally conformal cosymplectic structure. Moreover 
 T defines an infinitesimal homothety of the connection forms and a relative infinitesimal conformal 
transformation of the curvature forms.The existence of a structure conformal vector field C on M is proved and their properties are 
investigated. In the last section, we study the geometry of the tangent bundle of an exact locally conformal 
cosymplectic manifold.


INTRODUCTION
In the last decade a series of papers have been devoted to almost cosymplectic manifolds M (f2, rl, , g).As is well known, an almost cosymplectic manifold M is an odd dimensional (say 2m + 1) oriented manifold, where the triple (ff2,rl,) of tensor fields is i) a 2-form of rank 2m ii) a 1-form r such that ^r 0 iii) a vector field (called the Reeb vector field) such that i.f2 0 and rl() 1.
One has the following more studied cases: if2 and r are both closed forms.Then M is called a cosymplectic manifold.
2 dr 0, dff2 2r ^if2.Then M is called a Kenmotsu manifold.3 dr 09 ^rl, dff2 209 ^if2.Then M is called a locally conformal cosymplectic manifold (see [3], [16]).In this case 09 and its dual vector T b-(co) with respect to g is called the Lee form (or characteristic form) and Lee vector field respectively.
In the present paper we consider an almost cosymplectic manifold M(,rl, ,g) carrying a globally defined vector field T whose dual form b(T) is denoted by co.
NextdenotebyO=vect{eA'A=O,X,...,2m}anorthonormalvectorbasisonMandby{OAB]the associated connection forms.If the connection forms satisfy (T, ee ^ea) ^is the wedge product, then one has VreA 0 Therefore we agree to say that M is structured by, a T-parallel connection.In this condition the following signtficative fact emerges: the almost cosymplectic structure x Sp(2m, R) of M moves to an exact locally contormal cosymplectc structure x Sp(2m, R) (abbreviated exact L.C.C.), having T (resp.to =-df/]') as Lee vector field (resp.Lee form).
Moreover any such a manifold M is a space form of curvature -2c and f is the energy function corresponding to a Hamiltonian vector field associated with T (in the sense of [3]).If 0 (resp.represents the indexless (or generic) connection forms (resp.curvature ffrms) of M, then T defines an infinitesimal homothety of 0, t.e.LIO 2c0, and a relative infinitesimal T conformal transformation of (R) and V2, i.e. d(L.r(R) 2cto ^(R), d(Lrg2 2cto În Section 3 the existence of a structure conffrmai vector field C on M is proved, i.e. VzC=XZ +g(Z,T)C-g(Z,C)T.XC(R)M, Z EI-'(TM).
Moreover C s a divergence conformal vector field, i.e. grad (div C) is a concurrent vector field and t defines an infinitesimal conff)rmal transffrmation of: ) the conformal cosymplectic form Q, i.e.Lc.OQ, p XL;   ii) the dual forms o, i.e.LcoY toa.
in the last section, we discuss some properties of the tangent bundle manifold TM having as basis the exact (L.C.C.)-manifold M. Denote by V,y and v the Liouville vector field ([13]), the Liouville l-form and the Liouviile function respectively, on TM.
The following properties are proved: i) the complete lift V2" of is a d-"-exact 2-form (d" is the eohomological operator [11]) and is homogeneous of class 1, i.e.
Lvlp lp i,,lp 0 (i,, denotes the vertical differentiation operator [11]); iii) the vertical lift T" of T defines an infinitesimal automorphism of ap, i.e.L T" 0; iv) the function r fv and the 2-form fW define a regular mechanical system 9,/" ([ 13]) having r as kinetic energy and fxp as canonical symplectic (exact) form.
1. PRELIMINARIES Let (M,g) be a Riemannian C(R)-manifold and let V be the covariant differential operator with respect to the metric tensor g.Assutne that M is oriented and V is a Levi-Civita connection.Let F(TM)--x(M) and b" TM T'M be the set of sections of the tangent bundle TM and the musical isomorphism ([ 18]) defined by g, respectively.Following [18] we set A'(M, TM) F Hom(AqTM, TM) and notice that elements ofA q(M, TM) are vector valued q-forms (q dimM).
Denote by dr: A"(M, TM) A I(M, TM) the exterior covariant derivative operator with respect to V. It should be noticed that generally dV'= dVo dV, 0 unlike d:'= d d--0.Ifp M, then the vector valued 1-torm dp A (M, TM) is the canonical vector valued 1-form of M ([5]) and since V is symmetric one has dV(dp) O.The operator d'=d + e(co) (1.1) acting on AM, where e(co) means the exterior product by the closed 1-form co, is called the cohomological operator ([11 ]).One has dod' O. (1.2) Any form u E AM such that d"'u 0 is said to be d"-closed and if co is an exact form, then u is sad to be a d'"-exact form.Any vector field Z F(TM) such that dv(Vz) VeZ zt ^dp CA 2(M, TM) for some 1-form zt, is said to be an exterior concurrent vector field ([ 17]).The form n which is called the concurrence form is given by rt ),.b(Z) ),.C(R)M. (1.4) A non flat manifold of dimension m > 2 is an elliptic or hyperbolic space-form if and only if every vector field on M is an exterior concurrent one ([ 17]).On the tangent bundle manifold TM, d, and/,, define the vertical differentiation and the vertical derivation operators respectively ([7]).d,, is an anti- derivation of degree on A(TM) and i,, is a derivation of degree 0 on V(TM).
In an n-dimensional Riemannian manifold M, denote by 0 vect {eA',A 1,..., n a local field of orthonormal frames and let O* covect COA ;A n be its associated coframe.
The soldering tbrm dp is expressed by dp coa (R) e A (1.5) and E. Cartan's structure equations written indexless manner are Ve 0 (R) e (1.6) do.) =-0 ^co (1.7) dO -0 ^0 + 0 (1.8) Any vector field T such that VT s dp + u (R) T u AtM (1.9) is called a torse forming (K.Yano t20]).If du 0, then T is a closed torse forming, which implies that T is an exterior concurrent vector field, and if u 0, then T is a concurrent vector field ([22]).
(1.11) n We recall some basic formulas which we shall use in the following sections.

EXACT LOCALLY CONFORMAL COSYMPLECTIC MANIFOLDS
Let (M,g) be a (2m + l)-dimensional oriented Riemannian C(R)-manifold and let T-Y taea and A-0 oa b(T) be a globally defined vector field on M and its dual form respectively.
Denote by O vect {ea A 0, 2m (resp.) a local field of orthonormal frames on M (resp.
the associated connection forms).Recall that the vectorial wedge product ^is defined by ( Assume now that all the connection forms 0 satisfy -<T, en ^ea>. (2.1) Then by the structure equations (1.6), it follows at once t3 -tnco a -t%J n (2.2) It should be noticed that if 0 satisfy (2.2) one has 0(T) 0 and the above equation shows that all the connection forms 0 are relations of integral invariance for the vector field T (in the sense of A. Liehnerowicz 14]).
Next by the structure equations (1.6) and by (2.2) one obtains Ve A tAdp -Oa a (R) T (2.3) and the above equation implies Vrea -0. (2.4) From (2.4) the following significative fact emerges: all the vectors of the O-basis are T-parallel.
Therefore we agree to say that the Riemannian manifold under consideration is structured by a T-parallel connection (abr.T.P.).
Further again by (2.2) one derives by the structure equations (1.7) dco a to ^to a o b r taro a (2.5) which by a simple argument implies that the dual form to of T is closed, i.e. dw-0. (2.6) Thus in terms of d'-cohomology, (2.5) may be written as d-'%o a 0 (2.7) and O* {to a is defined as a d-'-closed covector basis.Now for reasons which will soon appear, we set co-n, eo- (2.8) and consider on M the globally defined 2-torm of rank 2m given by =Y-m"^o0"" a m" a*=a+m (2.9) Then since Q'" ^1 0, iQ (), one may say that the triple (Q,q,) defines an almost cosymplectic structure xSp(2m,R)having as Reeb's vector leld.
(2.11) We conclude that any odd dimensional Riemannien manifold M structured by a T-parallel con- nection is endowed with a locally conformal cosymplectic structure x CSp(2n,R) (abr.L.C.C.).We notice that the vector field T (resp.the l-form m b(T)) is the Lee vector field (resp.the Lee form) of this structure.
Moreover since o0 ,"c@, then by a simple argument it follows on behalf of (2.5) that one may set dtA f(.O A" f CM (2.12) which by exterior differentiation gives instantly o0 -af/f (2.13) Therefore since o is an exact tbrm, it follows on behalf of a known terminology, that the manifold M under consideration is an exact (L.C.C.)-manifold.We agree to call f the distinguished scalar field associated with the exact (L.C.C.)-structure.Now taking the covariant differential of T one finds by (2.3) and (2.12) VT (f + 2l)dp -o0(R) r (2.14) where we have set g(T,T)=21.
Operating now on V ea and VT by the exterior covariant derivative operator dv, one gets by (2.12) and (2.16) From the above equations it is seen that any vector field Z on M is E.C. with constant conformal scalar 2c.Therefore on behalf of the general properties of E.C.-vector fields ([17]), we may state the following striking property: the exact L.C.C.-manifold M(,q,) under discussion is a space-form of curvature -2c.
As a consequence, it follows that the curvature forms (R) are expressed by EP n =-2cm A ^o0n (2.20) Next taking the exterior differential of the forms (R), one quickly finds by dEr 2o0 ^,, d-"' 0 (2.21) which shows that all the curvature forms 0 are On the other hand taking the Le derivatives of the covectors of O* one derives by (2.12) and (2.16) L * (1 + c)co* t"*. (2.22) Therefore since L sat,sfies Leibniz rule one deduces by (2.20) L rO;' 2(/+ c )O;] + 2c Oj', a (2.23) Similarly, we oblain d 2]tA + A (2.24) Clearly by (2.12) one has Lr =fr and wih he help of (2.22) we deduce L 2c'. (2.25) Accordingly by the above equations we may say hat the Lie vector field T defines on infinitesimal homothety of all the connection forms 0.
Taking now the exterior differential of the equations (2.23), a standard calculation gives d(Lr 8 (2.26) which proves thnl T defines a relative infinitesimal conformal transformation ([19]) of the cuature forms.
Next by (2.12) and (2.13) one easily gets (2.33) (2.34) Therefore by reference to 3 one may call T the c()symplectic Hamiltonian vector field of M and the dstnguished scalar ftuns out to be the energy function corresponding to T.
Summing up, we state the following THEOREM.Let M be a (2m + l)-dimensional Riemannian manifold and let T be a globally defined vector field on M. If M is structured by a T-parallel connection, then M is endowed with an exact locally conformal cosymplectic structure x CSp(2m, R), having T (resp.w b(T)) as e vector (resp.Lee form) and any such an M is a space-form of cuature -2c.Moreover one has the tbllowing properties: i) T defines an infinitesimal homothety of the connection forms 0 and of the 1-form a(T), i.e.
LrO 2c0, Lr(T 2c(T) ii) T defines a relative infinitesimal contbrmal transformation of the cuature forms O and of the structure 2-form , i.e.

d(LrO)=8cO, d(L)=2c
iii) the vector field T (b -) T (resp. is the cosymplectic Hamiltonian associated with the Ix CSp(2m,R)-structure of M (resp.its corresponding energy function) and T defines a relative infinitesimal automorphism of .
Let now " M be a conformal diffeomorphism (abr.C.D.) that is "geg=g" oCM.
One also say that g and g are conformally equivalent metrics and setting e v , we agree to call the function v the argument of the C.D.
Umng (2.37) one may prove that all vectors 6 a are parallel (the connection forms ( vanish, i.e. ' s a fiat connection).Thus we have PROPOSITION.II M s an exact (L.C.C.)-manifold wth metric tensor g and energy function f, then the metric f2g s fiat.

STRUCTURE CONFORMAL VECTOR FIELDS ON AN EXACT (L.C.C.)-MANIFOLD
In consequence of some conformal properties induced by the T-parallel connection which structures M(Q,q,_,g) we are naturally led to see if the manifold M under consideration carries a structure con- formal vector field C in the sense of I6], 15].Therefore the covariant differential of C is expressed by VC=kdp +C ^T=.dp +oo(DC-c(R)T..C(R)M, ct-b(C).(3.6)By (3.4), (3.5) and (3.6) it is seen that the existence of C is assured by an exterior differential system Y whose characteristic numbers are r--3, s0=2, Sl =1.Then g is in involution in the sense of E. Cartan (i.e.r s, + s).Accordingly one may say that the existence of C depends on 2 arbitrary functions of one argument (E.Cartan's test).The eonformal scalar p associated with C(Lcg 9g) is given by O 2k.
(3.9)This equation matches by Orsted's lemma (1.12)the expression of [C,T].On the other hand since C is necessarily an E. C. vector field (M is a space-form), then operating (3.1) by d v and taking account of (3.4) and.(3.5), one derives dV(vc)-V:C 2cc ^dp. (3.10) The above equation is coherent with the properties obtained in Section 2. Setting now = tcq2 Y(C%o""-C"'eo") (3.13) Hence (3.13) reveals that C defines an infinitesimal conformal transformation (abr.I.C.T.)of the cont{rmal cosymplectic form .

Lc
LcO 2 B Therefore one may say that C defines an I.C.T. of the exact (L.C.C.)-structure of M. Moreover let L be the operator of type (I.I) on forms defined by S. Goldberg ([8]), that is L u u A ;u AtM, and consider on M the ( + l)-fos then by (3.13) and a standard calculation one derives Lca =(q + 1)p%.
(3.20) Thus one may state the following relevant property: the gradient of the associated scalar p of C is a concurrent vector field .Yano and B. Y. Chen [22]).We agree to call a conformal vector field such that the gradient of its conformal scalar p is a concurrent vector field, a divergence conformal vector field.Such a situation occurs also when studying conformal vector fields on rentzian P.S. manifolds (see I. Mihai and R. Rosca [15]).
C being an E.C. vector field satisfying (3.10), one has ([ 17]) S(C,Z) -4mc g(C,Z), Z F(TM) (3.23) where S denotes the Ricci tensor field of V. Now making use of (1.14) and caring out the calculations, one finds by (3.19) and (3.22) Lcg(C,Z) pg(C,Z). (3.24) Hence the vector field C defines an I.C.T. of all the functions g(C,Z), where Z C F(TM).
Concuding, we have proved the following THEOREM.Let M be the exact (L.C.C.) manifold defined in Section 2 and C a structure conformal vector field on M (which existence is proved), i.e.
Accordingly we may consider the setB {toA,dva as an adapted cobasis in TM.Following Godbillon ([ 7]) we denote by d,, and 4 the vertical differentiation and the vertical derivative operators with respect to B*, respectively (d,, s an antiderivation of degree on A(TM) and 4 is a derivation of degree 0 on A(TM)).Let TM be the set of all tensor fields of type (r,s) on M.
In general as is known ([23 ]) the vertical and complete lifts are linear mappings of TfM into Tf(TM)   and one has (Tl (R) T:,)" T(R)T + T (R)T. (4.1) In the case under discussion we may define the complete lift ff2" of the structure 2-form of M by the 2-form of rank 4m on TM ffa"=Y(dv"^to"'+to"^dv"'), a=l m; a*=a+m. (4.2) On the other hand since the Liouville vector field V is expressed by then as is known the basic 1-form y E vato a (4.4) is called the Liouville form (see also [13]).
Taking now the exterior differential of f" one finds by (2.5) dg2" =to ^Q" :, d-' =0 (4.5) which shows that " is similarly as ff2 a d-exact form.We recall that in general conformal properties are not preserved by complete lifts ([23]).
One has ivf2 Y(v"m"'-v"'to") (4.6) which implies re(V) 0 and so by (4.5) and (4.6) one gets Lvfg ff.(4.7) Accordingly on behalf of a known definition ([ 13]), the above equation shows that is of class 1, a homogeneous form on TM.Taking now the exterior differential of the Liouville form y defined by (4.4), one gets at once by (2.5) dy to ^y + ':=:' d-'y (4.8) where we have set q d v"' A toa From (4.8) and (1.2) one obtains nstantly d"tp 0 dtp= .277 (4.9)Since clearly the 2-form q is of maximal rank, we agree to call tp the canonical conformal symplectic form of M. Noticing that one has ,,q y, to(V) 0 (4.11) which implies Lvq p. (4.12) Hence p is as f2' a homogeneous of class 1, 2-form.
Next making use of the vertical operator i,.defined by i k 0, i,, dv a coa, i,, oJ 0(L C(R)M) one quickly finds by (4.9) i,3p =0 (4.13) and the above equation together with (4.12) proves that is a Finslerian form ([7]).We recall that the vertical lift Z" ([23]) of a vector field Z F(TM) with components Z a in M, has and one may say that T" defines an infinitesimal automorphism of ap.A and taking the exterior differential of (4.19) we obtain by (2.13) and (4.9) d(dr) f Z dv A ^toA ----lap. (4.20) Next putting H --fap it follows by (2.13) dH =0. (2.21) Therefore the exact symplectic form//can be viewed as the canonical symplectic form of the (4m + 2)-dimensional manifold TM ([ 13]).
Finally by reference to [13] one may consider that the pair (r,ll) defines a regular mechanical system 9'd (in the sense of Klein [13]) having the scalar r as kinetic energy.

Lqa=q
4. GEOMETRY' OF THE TANGENT BUNDLE OF AN EXACT (L.C.C.)-MANIFOLDLet now TM be the tangent bundle manifold having the exact (L.C.C.)-manifold M discussed in Section 2 as a basis.