ON A CLASS OF EXACT LOCALLY CONFORMAL COSYMLECTIC MANIFOLDS

Internat. J. Math. & Math. Sci.

VOL. 19 NO. 2 (1996) 267-278

267

ON A CLASS OF EXACT LOCALLY CONFORMAL COSYMLECTIC MANIFOLDS

I. MIHAlandL. VERSTRAELEN

Department Wiskunde, Katholieke Universiteit Leuven Celestijnenlaan 200 B, B 3000 Leuven, Belgium

R. ROSCA 59 Avenue Emile Zola

75015 Paris, France

(Received May 6, 1993 and in revised form Je !94)

ABSTRACT. An

almost cosymplectic manifold

M

is a

(2m

+

1)-dimensional

^{oriented Riemannian}

manifoldendowedwith a 2-form f2 ofrank 2m, a 1-formr such thatf2’"

^

^{q 0 anda vectorfield}

satisfying

if2

^{0 andq() 1.} Particular caseswere considered in

[3]

^and

[6].

Let (M, g)be anodddmensional orientedRiemannian manifoldcarryingagloballydefinedvector field

T

suchthatthe Riemannian connection isparallelwithrespect^toT. Itisshown that inthiscase

M

isahyperbolicspaceformendowed withanexactlocallyconformal cosymplecticstructure.Moreover

T

defines an infinitesimalhomothety ofthe connectionforms and arelative infinitesimal conformal transformation ofthecurvatureforms.

The existence ofa structure conformal vector field

C

on

M

is proved and their propertiesare investigated.

In

the last section,westudythegeometry ofthetangentbundleofanexactlocallyconformal cosymplecticmanifold.

KEY

WORDS AND PHRASES: Locally conformal cosymplectic manifold,T-parallelconnection, infinitesimal homothety, infinitesimal conformal transformation, Hamiltonian vector field, tangent bundle,Liouvillevectorfield,completelift,mechanicalsystem.

1991

AMS SUBJECT CLASSIFICATION CODES:

53C15, 53C05, 53C20, 58A15,58A30.

1.

INTRODUCTION

In

the last decade a series of papers ^{have been devoted to} almost cosymplectic manifolds

M (f2,

rl,

, ^{g). As}

^{iswellknown,analmost}cosymplectic^manifold

M

isanodd dimensional

(say

2m+

1)

orientedmanifold,where thetriple

(ff2,rl,)

^{oftensorfieldsis}

i)

^{a2-form of rank 2m}

ii)

^a1-form r suchthat

^

^{r 0}

iii)

a vector field

(called

^theReebvector

field)

such that

i.f2

^{0 and}

^rl()

^1.

Onehas thefollowingmore studied cases:

if2andr areboth closedforms. Then

M

is calledacosymplecticmanifold.

2 dr 0,dff2 2r

^

^{if2. Then}

^M

^iscalleda

^Kenmotsu

^manifold.

3 dr 09

^

^{rl,dff2 209}

^

^{if2. Then}

^M

^{iscalledalocallyconformal}cosymplecticmanifold

(see [3],[16]). ^In

^{this case09}and its dual vector

T b-(co)

^withrespect tog^is called the

Lee

form

(or

characteristic

form)

^andLeevectorfieldrespectively.

In

the presentpaper^weconsider an almostcosymplectic manifold

M(,rl, ,g)

carryingaglobally defined vector field

T

whose dualform

b(T)

^{is denoted}byco.

NextdenotebyO=vect{eA’A=O,X,...,2m}anorthonormalvectorbasisonMandby{OAB]the

associated connectionforms. If theconnectionforms satisfy

(T, ee ^ ^ea) ^

^isthewedge product,

thenone has

Vre

A 0

Thereforeweagreetosay that

M

isstructuredby, aT-parallelconnection.

In

thiscondition the following signtficativefactemerges: ^thealmostcosymplecticstructure x

Sp(2m, R)

of

M

movesto anexactlocallycontormalcosymplectcstructure xSp(2m, R) (abbreviatedexact

L.C.C.),

having

T

(resp.^to

=-df/]’)

^asLeevectorfield(resp. Lee

form).

Moreoverany such amanifold

M

is aspace form of curvature-2c and

f

^{is theenergyfunction}

corresponding to a Hamiltonian vector field associated with

T

_(in the sense of

[3]).

^If0 (resp.

representsthe indexless(orgeneric)connection forms(resp.curvature

ffrms)

of

M,

then

T

defines an infinitesimalhomothetyof0,t.e.

LIO

^{2c0, andarelative} infinitesimal

T

conformal transformation of (R)andV2, i.e.

d(L.r(R)

^2cto

^

^(R),

^d(Lrg2

^2cto

^

In

Section3 theexistenceof astructureconffrmaivectorfield

C

on

M

isproved,^i.e.

VzC=XZ +g(Z,T)C-g(Z,C)T. XC(R)M, Z EI-’(TM).

Moreover

C

sadivergence conformal vectorfield, i.e.grad(div

C)

isaconcurrent vectorfield andtdefinesan infinitesimalconff)rmaltransffrmationof:

) theconformal cosymplectic formQ,i.e.

Lc. ^OQ,

^p

^XL;

ii) thedualforms

o,

i.e.

LcoY toa.

iii) ^thecurvatureforms

,

^i.e.

LcO ^pOOh"

iv) ^{all the}

(2q

+

l)-forms

_(xq

b(C ^ ^{Va ’,}

^i.e.

^Lca,t

^+q

^)pct"

v)

all the functions

g(C,Z),

^i.e.

Lcg(C,Z) pg(C,Z), Z F(TM).

inthelastsection,wediscusssomepropertiesof the tangent bundlemanifold

TM

havingasbasis theexact

(L.C.C.)-manifold M. Denote

by

V,y

^andv the Liouvillevectorfield

([13]),

the Liouville l-form and the Liouviile functionrespectively,on

TM.

Thefollowing propertiesareproved:

i)

thecomplete^liftV2"of isad-"-exact 2-form

(d"

^istheeohomological operator

[11])

^andis

homogeneousof class 1,i.e.

Lv

ii) "1’ satisfies

d-"y p

and "q.,^{is a} Finslerianform,i.e.

Lvlp lp i,,lp 0

(i,,

denotes theverticaldifferentiation operator

[11]);

iii)

^{theverticallift}

T"

of

T

defines an infinitesimalautomorphismof

ap,

i.e.

L T"

^0;

iv) the functionr

fv

^{and the2-form}

fW

^{define a}regular^mechanicalsystem^9,/"

([ 13])

havingr as kineticenergyandfxpas canonicalsymplectic

(exact)

^form.

1.

PRELIMINARIES

Let (M,g)

be a RiemannianC(R)-manifold and letV be the covariant differentialoperatorwith respect to the metric tensorg. Assutnethat

M

is oriented andV is aLevi-Civita connection.

Let F(TM)--x(M)

^andb"

^TM T’M

be the setofsectionsof the tangentbundle

TM

and the musical isomorphism

([ 18])

^definedbyg,respectively. Following

[18]

^weset

A’(M, TM) F Hom(AqTM, TM)

LOCALLY CONFORMAL COSYMPLECTIC MANIFOLDS 269 and notice that elementsofAq(M,

TM)

arevectorvaluedq-forms

(q dimM).

Denote

by

dr:

A"(M,

TM)

A

I(M, TM)

the exterior covariant derivativeoperatorwithrespect to V. Itshould be noticedthatgenerally

dV’= dVo

^dV,0 unlike

d:’=

d d--0. Ifp

M,

thenthe vectorvalued 1-tormdp A

(M,

TM)isthe canonical vectorvalued1-formof

M ([5])

^andsinceVis

symmetriconehas

dV(dp)

O. Theoperator

d’=d+

e(co) (1.1)

actingon

AM,

where

e(co)

meansthe exteriorproductbytheclosed 1-form co,iscalled thecohomological operator

([11 ]).

Onehas

dod’ O.

(1.2)

Any

formuE

AM

such thatd"’u 0is saidtobed"-closedand ifcois an exactform, thenuis sad tobe ad’"-exact form.

Any

vectorfield

Z F(TM)

such that

dv(Vz) VeZ

zt

^

^{dp CA}

^{2(M, TM) (1.3)}

for some 1-formzt,is saidtobe anexterior concurrent vector field

([ 17]).

The form nwhich iscalled theconcurrence formisgivenby

rt

),.b(Z)

^),.C(R)M.

(1.4)

A

nonflat manifold of dimension m>2isanellipticorhyperbolic space-formif andonlyifevery vector fieldon

M

isan exteriorconcurrentone

([ 17]). ^On

the tangent bundle manifold

TM, d, and/,,

define the verticaldifferentiationand thevertical derivationoperators respectively

([7]).

d,,is an anti- derivationofdegree on

A(TM)

^andi,,is aderivation ofdegree0 on

V(TM).

In

ann-dimensionalRiemannian manifold

M,

denoteby 0 vect

{eA’,A

^1,...,n

a local fieldof orthonormalframes and let

O* covect

COA ;A

n beitsassociatedcoframe.

The soldering tbrmdpisexpressedby

dp

coa

^{(R) e}A

(1.5)

and

E. Cartan’s

structureequationswritten indexlessmannerare

Ve 0 (R) e

(1.6)

do.)=-0

^

^co

^(1.7)

dO -0

^

⁰⁺⁰

^(1.8)

Any

vector field

T

such that

VT

sdp+ u (R)

T

^u

AtM (1.9)

iscalleda torseforming

(K. Yano t20]).

^{Ifdu 0, then}

^T

^{isa closedtorse}forming, whichimpliesthat

T

isan exterior concurrent^vectorfield, and if u 0,then

T

^isa concurrentvectorfield

([22]).

Let

now

W

beany^{conformalvectorfield on}

M (i.e.

theconformalversionof

Killing’s equations).

As

iswell known, Wsatisfies

Lwg pg

^or

g(VzW, Z’) +g(V z,W,Z) 9g(Z,Z’) (1.10)

where the conformal scalar9^isdefinedby 2

9

--(divW). (1.11)

n

Werecall some basic formulaswhich weshalluseinthefollowing^sections.

L,, t,(Z) pt,(z)

+

t,[w,z] (Orsted lemma) (1.12)

LwK

^{(n )Ap}

^{Kp (1.13)}

2L ^S(Z,Z’) (A)pg(Z,Z’)-(n 2) (HessVP)(Z,Z’). (1.14) In

the aboveequations

L, K,

A andSdenotetheLte derivative with respect^to

W,

the scalar curvatureofM,the placian and the Riccitensorfield ofV,respectively.

One

has

(Hessvp)(Z,Z’) g(Z,HpZ’), HpZ’--- Vz.(grad p) (see

also

21).

2.

EXACT LOCALLY CONFORMAL COSYMPLECTIC MANIFOLDS

Let

(M,g)

bea

(2m

+l)-dimensionalorientedRiemannianC(R)-manifoldand let

T- Y taea

^and

A-0

oa

b(T)

^beaglobally ^{definedvectorfield}on

M

anditsdualformrespectively.

Denote

by

O

vect

{ea ^A

^{0, 2m}

(resp. )

a local fieldof orthonormalframeson

M (resp.

the associated connection

forms).

^Recallthatthe vectorialwedge product

^

^isdefinedby

(X ^ Y)Z--g(Y,Z)X-g(X,Z)Y; Z r’(TM)

i.e.

X ^ Y-b(Y)(R)X-b(X)(R)Y.

Assume

now that allthe connectionforms0satisfy

- ^{<T, en ^ ea>. (2.1)}

Thenbythestructureequations

(1.6),

^{itfollowsatonce}

t3 ^-tnco

^a

^-t%J

ⁿ

^(2.2)

It

should be noticed that if0 satisfy

(2.2)

one has

0(T)

⁰and the aboveequation showsthatall the connection forms 0 are relations of integral invariance for thevectorfield

T (in

the senseof

A.

Liehnerowicz

14]).

Next

bythestructureequations

(1.6)

^andby

(2.2)

^oneobtains

Ve_A

tAdp

^-Oaa(R)

T (2.3)

and the aboveequation implies

Vre

a-0.

(2.4)

From (2.4)

^thefollowing significativefact

emerges:

allthevectorsof the O-basis are

T-parallel.

Thereforeweagreetosaythat the Riemannianmanifold under considerationisstructuredby^a

T-parallel

connection

(abr. T.P.).

Furtheragainby

(2.2)

one derivesbythestructureequations

(1.7)

dco^a to

^

^{toa o b}

^{r taro}

^a

^(2.5)

whichby^asimple argument impliesthat the dual form^toof

T

^isclosed,i.e.

dw-0.

(2.6)

Thus intermsof

d’-cohomology, (2.5)

maybewritten as

d-’%o

^a 0

(2.7)

andO*

{to

^{a is}defined as ad-’-closedcovectorbasis.

Now

forreasonswhich willsoonappear,^{we set}

co-n, eo- ^(2.8)

LOCALLY CONFORNAL COSYHPI.ECTIC HANIFOLDS 271 and consideron

M

thegloballydefined 2-torm ofrank2m givenby

=Y-m"^o0""

a m" a*=a+m

(2.9)

Thensince Q’"

^

^{1 0,iQ (),onemaysaythat thetriple}

^(Q,q,)

defines an almostcosymplectic

structure xSp(2m,R)having asReeb’svectorleld.

Nexttaktng theexteriordfferental of if2ashort calculation giveswiththehelpof

(2.5)

rig2 2o0

^

^ff2ca,

d-"’"f2

0

(2.10)

and by

(2.5)

wemaywrite

dq o0

^

^{q =,d-+’q 0.}

^(2.11)

Weconcludethatany odd dimensional Riemannien manifold

M

structured byaT-parallel con- nection is endowed withalocallyconformal cosymplectic^{structure x}

CSp(2n,R) (abr. L.C.C.).

^We

notice that thevectorfield

T

(resp. the l-formm

b(T))

istheLeevectorfield(resp.^theLee

form)

^of

this structure.

Moreover

sinceo0

,"c@,

thenbyasimpleargumentitfollows on behalf of

(2.5)

that onemayset

dtA _f(.O

^A"

f CM (2.12)

whichbyexteriordifferentiation gives instantly

o0

-af/f ^(2.13)

Thereforesinceoisanexacttbrm,itfollows on behalf ofaknownterminology,that the manifold

M

under consideration isanexact

(L.C.C.)-manifold. We

agreetocall

f

^thedistinguishedscalar field associatedwiththe^exact

(L.C.C.)-structure.

Now

takingthecovariantdifferential of

T

onefindsby

(2.3)

^and

(2.12)

VT (f

+

2l)dp

^-o0(R)

r (2.14)

wherewehaveset

g(T,T)=21. (2.15)

Using

(2.12)

^and

(2.15),

^wehave

d

fo0

⁺

f

^{c const 0}

^(2.16)

and

(2.14)

^becomes

VT=(I

+c)dp-m(R)T.

(2.17)

Hence,

by

(1.9)

^and

(2.6) ^T

^{is a}closed^torseforming^andconsequentlyanexteriorconcurrent

(abr.

E.C.)-vector

^field.

Operatingnow onV

ea

^and

^VT

^{bytheexterior}covariant derivativeoperatord

v,

onegets by

(2.12)

and

(2.16)

dv(V ca)= V2ea

^2Cma

_^

^dp

^(2.18)

dv(V T)= V2T

2cm

^

^dp

^(2.19)

From

the aboveequationsitis seen thatanyvectorfield

Z

on

M

isE.C.with constantconformal scalar2c. Thereforeonbehalfof thegeneral propertiesof

E.C.-vector

fields

([17]),

^{wemaystatethe}

following strikingproperty: ^theexactL.C.C.-manifold

M(,q,)

under discussion is a

space-form

of curvature-2c.

As

aconsequence,itfollowsthat the

curvature

forms^(R)are

expressed

by

EP

_n=-2cm^A

^ ^{o0n (2.20)}

Nexttakingthe exterior differentialof the forms(R), one quickly^findsby

dEr

^2o0

_^ ^,, d-"’

⁰

^(2.21)

which shows that all thecurvatureforms0are

On

theotherhand takingtheLederivativesof thecovectors ofO* onederivesby

(2.12)

^and

(2.16)

L *

⁽¹⁺

^{c)co* t"*. (2.22)}

Thereforesince

L

^sat,sfiesLeibniz rule one deducesby

(2.20)

L rO;’

2(/+c

)O;]

⁺2c

Oj’,

^a

(2.23)

Similarly,weoblain

d ^2]tA

⁺

A ^(2.24)

Clearlyby

(2.12)

^onehas

Lr =fr

^{andwihhehelpof}

^(2.22)

^wededuce

L 2c’. ^(2.25)

Accordingly bytheaboveequationswemay sayhat theLie vectorfield

T

defines on infinitesimal homothety of all theconnectionforms0.

Takingnowthe exteriordifferential of theequations

(2.23),

a standard calculation gives

d(Lr

⁸

^(2.26)

which

proves

^thnl

T

definesa relativeinfinitesimal conformal transformation

([19])

^{of thecuature}

forms.

let

" ^TM

^T’M,

^{p(Z) iz}

^{be thebundle}isomorphism^definedby andset

(T),

^i.e.

ir ^(t ==" ^{t=’o =) (2.2?)}

for the dual Ibrm of

T

withrespect^to

. ^{By (2.5)}

^and

^(2.12)

^an

^easy

calculationgives

d 2f

⁺

^(2.28)

andby

(2.10)

^and

(2.13)

^onegets

andconsequently by

(2.28)

^ittbllows

Lrff2 ²⁽¹

⁺

^c)O

^+to

^

^co

^(2.29)

d(Lrf2

^2cco

^

^if2.

^(2.30)

Hence

asfor thecurvatureformsO,

T

defines a relativeconformal transformation of thestructure 2-form

.

Considernow the vectorvalued1-form

F =co" (R)e,,.-co"’(R)eo

CA

I(M, TM). (2.31)

If

Z

is

any

vectorfield, asimplecalculationgives

(F,Z) Z"e,,. Z"’e,, ² (2.32)

whichimplies

g(Z,Z’)

+g(Z,Z’)--

O, Z,Z’ F(TM)

and

(F,

dp 2.

On

theother hand since

co(T)

0onegetsby

(2.27)

Lr--

^2cw

thatis

T

defines aninfinitesimal homothetyof co

(la b)T.

Next

by

(2.12)

^and

(2.13)

one easily gets

(2.33)

(2.34)

LOCALLY CONFORMAL COSYMPLECTIC MANIFOLDS 273

Thereforebyreferenceto 3 onemaycall

T

thec()symplecticHamiltonianvectorfieldof

M

and thedstnguishedscalar

ftuns

^{outtobe theenergyfunction}correspondingtoT.

Moreover by

(2.35)

one derives

L ^i(T)I

^A(o

^d(LQ)

⁰

^(2.36)

whichshowsthat

T

dehnesarelative nflntesmal autonorphsm

(R.

Abraham

])

of

.

Summing up, we statethefollowing

THEOREM.

Let M be a(2m +l)-dimensional Riemannian manifold and let

T

be aglobally definedvectorfield onM. If

M

is structuredby aT-parallelconnection, then

M

isendowedwithan exactlocallyconformal cosymplecticstructure xCSp(2m,R),having

T

(resp.w

b(T))

as

e

vector (resp.

Lee

form)^andanysuch an

M

is aspace-formofcuature-2c.

Moreover

onehas thetbllowing properties:

i)

T

defines an infinitesimal homothety of theconnectionforms 0 and of the 1-form

a(T),

i.e.

LrO

^2c0,

^{Lr(T 2c(T)}

ii)

T

definesa relativeinfinitesimalcontbrmal transformation of thecuatureforms

O

andof the structure2-form

,

i.e.

d(LrO )=8cO, d(L)=2c

iii)

^{the vectorfield}

T

(b

- ^{) T}

^{(resp. is the}cosymplecticHamiltonian 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

^beaconformaldiffeomorphism

(abr. C.D.)

^thatis

"geg=g" oCM.

Onealsosaythatgand

g

areconformally equivalent^metricsandsettinge v

,

^{weagreetocall}

thefunction vthe argument of^the

C.D.

As

isshownone has for

Z, Z’ F(TM)

Z VZ

+

b(grad o)@Z b(Z)

@ grado +

g(Z,grad oMp (2.37)

orequivalently

z,Z VzZ

⁺

^Z’(o

⁺

^Z(o’ g(Z,Z’)grado (2.38)

and if

K

and denotethescalarcuatureof

M

and respectivelythenonehas

([8])

e-{K

⁺

^{2(n 1)(n} z)ll

grad

oll } ^(2.39)

(n

-dimM).

If

M

isanexact

.C.C.)--manifold,

itsRiccitensorfield

S

satisfies

S(Z,Z’) ^-4mc

g(Z,Z’) Z,Z’ F(TM) (2.40)

and the scalarcuature

K

isgiven by

K =-4m(2m

+

1)c. (2.41)

Perfo now a conformal transformation of

M

havingasargument e theenergyfunction

It

is obviousthat

(2.42

^o

df/f . ^(2.42)

Thenwehavegrado

=-T,

which implies

Ao div

T

(2m ⁺l)c+

(2m I)/. (2.43)

|fenceby(2.41)and(2.43)^wederiveatonce from

(2.39),/

0,that is’/isaflat manifold.

We

noticethatthisfactsn accordance wth theknown

PROPOSITION.

A

Remannan manfl)id ofconstant curvature isconformaily fiat, provided

I>3.

Umng (2.37)

^onemayprovethat allvectors 6_aareparallel(theconnection forms

(

vanish,i.e.

’

s afiatconnection). Thuswehave

PROPOSITION. II M

s anexact

(L.C.C.)-manifold

wth metric tensorgandenergyfunction

f,

thenthe metric

f2g

sfiat.

3.

STRUCTURE CONFORMAL VECTOR FIELDS ON AN EXACT (L.C.C.)-MANIFOLD In consequence

of some conformal properties induced by the

T-parallel

connectionwhich structures

M(Q,q,_,g)

wearenaturallyledtosee if the manifold

M

under consideration carries a structure con- formalvectorfieldCinthe senseof

I6], 15].

^{Thereforethecovariant}differentialof

C

isexpressed by

VC=kdp

+C

^ ^T=.dp

+oo(DC-c(R)T.

.C(R)M, ct-b(C). (3.1) Put

C

Caea

^:=:,

^b(C)

^{a cAoja}

(3.2)

ands

g(C, T).

Thenby

(2.3)

^and

(3.1)

^onequicklygets

dC^a

(.- s)to

^a+

Caco _(3.3)

da 2o0

^

^a

= ^d-2"ct

^0.

(3.4)

Nextsinceds

(VC, T)

+

(VT, C),a

short calculationgives

ds

Z.co- (/- c)et (3.5)

ds d.

(3.6)

By (3.4), (3.5)

^and

(3.6)

^{it isseen}that the existence of

C

isassuredbyan exteriordifferentialsystem Ywhose characteristic numbersare

r--3,

s0=2,

Sl=1.

Thengis in involution in thesense ofE.

Cartan (i.e.

^r

s,

+

s).

Accordinglyone

may

saythat the existenceof

C

dependson2arbitraryfunctionsof one argument

(E. ^Cartan’s test).

^Theeonformal

scalar

p

associatedwith

C(Lcg 9g)

isgiven by

O

^2k.

(3.7)

By

a short calculationonehas

[C, T]--

-%.T-

(l c)C" ]:

^Liebracket

(3.8)

andfrom

(3.5)

itfollows

Lcco=ds

^),.co-

^{(l c)ct.} (3.9)

This equation matchesbyOrsted’s lemma

(1.12)the

expression of

[C,T].

On

the other handsince

C

isnecessarilyan

E. C.

vectorfield

(M

^{is a}

space-form),

thenoperating

(3.1)

byd^vand takingaccountof

(3.4) and.(3.5),

^onederives

dV(vc)- V:C

2cc

^dp. (3.10)

Theaboveequationiscoherentwiththe propertiesobtainedin Section 2.

Settingnow

= ^tcq2 Y(C%o""- C"’eo") (3.11)

LOCALLY CONFORHAL COSYNPLECT[C HANIFOLDS 275 oneget,,,by

(3.4)and (2.5)

d(z

2(.- s)

+200

^

⁽

^(3.12)

and onefollows

L

^f2

O. ^(3.13)

Hence

(3.13)

reveals thatCdefinesan infinitesimalconformaltransformation

(abr. I.C.T.)of

^the

cont{rmalcosymplecticform

.

By

^similarmethods,onegetsby

(2.5), (2.24), (2.20)

^and

(2.21)

P ’ ^{Lc =p (3.14)}

Lc LcO

₂ B

Therefore onemay saythat

C

definesan

I.C.T.

ofthe 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} Lu u A

;u AtM,

and consideron

M

the

(

+l)-fos

Lqa=q =

^AQq

^(3.15)

Snce

byOrsted’s lemma one has

Lca=pa (3.16)

thenby

(3.13)

^andastandardcalculation onederives

Lca ^=(q

⁺

1)p%. ^(3.17)

Hence Cdefinesan

(I.C.T.)

^{ofall the}

(

+

1)-forms aq.

NextsinceCisaconformalvectorfield,then as is

own _{(see (1.11))}

onehas

dry

C (p/2)(2m

+

1) (3.18)

andsincep 2kitfollowsby

(3.5)

^and

(3.6)

^that

gradp

pT

+

2(c I)C. (3.19)

Furtherby

(2.16)

^andtakingaccountof

(2.14)

^and

(3.1)

^{it is}easily deduced

V grad p ^2cpdp.

(3.20)

Thusonemaystatethefollowing relevant property: the gradient of the associated scalar

p

of

C

is aconcurrentvectorfield

. ^Yano

^andB.

^Y.

^Chen

^{[22]). We}

^{agreetocallaconformalvectorfield such}

that the gradient ofitsconformal scalarpis a concurrent vectorfield,adivergenceconformalvector field. Suchasituationoccursalso whenstudyingconformalvectorfieldsonrentzianP.S.manifolds

(see I.

^Mihaiand

R. Rosca [15]).

On

theother handfrom

(2.14)

^onederives

div

T (2m 1)l

+

(2m

+

)c (3.21)

andsince div

C (2m

+

1)K

^onegetsonbehalf of

(3.20)

Ap

-div(grad

p) -2(2m

+

1)cp (3.22)

which shows that

p

isaneigenfunctionofA.

C

being an

E.C.

vectorfieldsatisfying

(3.10),

one has

([ 17])

S(C,Z)

-4mc

g(C,Z), Z F(TM) (3.23)

whereSdenotes the RiccitensorfieldofV.

Now makinguseof

(1.14)

andcaringoutthe calculations, one findsby

(3.19)

^and

(3.22)

Lcg(C,Z) pg(C,Z). (3.24)

HencethevectorfieldCdefines anI.C.T.of all thefunctions

g(C,Z),

where

Z

C

F(TM).

Concuding,wehaveprovedthe following

THEOREM.

Let

M

be theexact

(L.C.C.)

^manifolddefined in Section2and

C

a structureconformal vectorfield on

M (which

^existenceisproved),i.e.

VC=dp+CAT" Lcg=pg

ThenCisadivergencecontormalvectorteld(i.e. grad(div C)isaconcurrent vector

field)

^andit

defines thetbllownginfinitesimal contbrmaltransformations

p L

’ ^{m LcO;=vO);}

Lc

_z

L, @ ^{=p@, Lc(t,}

⁼⁽¹

^+q)p(t,,, Lcg(C,Z)=pg(C,Z)(Z F(TM)

where

,

^cd

, ft), Er

^and (,,

b(C) ^

^{q are the} conformal symplectic 2-form, the dual forms, the connectionforms,thecurvatureforms and the

(2q

+

1)-torms

^definedbythe

(1,1)-operator L,

respec- tivelyonM.

4. GEOMETRY’

OF THE

TANGENT

BUNDLE OF AN EXACT (L.C.C.)-MANIFOLD

Let now

TM

bethe tangent bundle manifold havingtheexact

(L.C.C.)-manifold M

discussed in Section2as a basis.

Denote by

V(va)(A

^=0,

2m)

the Liouville vectorfield

(or

^the canonical vectorfield

[7]).

Accordinglywemayconsider thesetB

{toA,dva

^{as an}adaptedcobasisin

TM.

FollowingGodbillon

([ 7])

^{we denote}by d,,and

4

^theverticaldifferentiation and theverticalderivativeoperatorswithrespect

toB*,respectively

(d,,

sanantiderivationofdegree on

A(TM)

and

4

is a derivationof degree 0^on

A(TM)). ^Let TM

^{bethesetof alltensorfields oftype}

^(r,s)

^on

^M.

In

generalasisknown

([23 ])

^{theverticaland}completelifts are linearmappingsof

TfM

^into

^Tf(TM)

andonehas

(Tl

(R)

T:,)" T(R)T

⁺

T ^{(R)T. (4.1)}

In

thecaseunder discussionwe

may

define thecomplete^liftff2"of thestructure2-form of

M

by the2-form of rank 4m on

TM

ffa"=Y(dv"^to"’+to"^dv"’),

a=l m; a*=a+m.

(4.2)

Onthe other hand since theLiouvillevectorfield Visexpressed by

V

E

v

’t---0 _(4.3)

Ov^a thenas is knownthe basic1-form

y E vato

^a

_(4.4)

iscalled the Liouville form

(see

also

[13]).

Takingnowthe exteriordifferentialoff"onefindsby

(2.5)

dg2"=to

^

^Q":,

^d-’

⁼⁰

^(4.5)

whichshows that

"

^{issimilarlyasff2ad-exactform. We}recall thatingeneralconformalproperties

arenotpreservedby completelifts

([23]).

Onehas

ivf2 Y(v"m"’- v"’to") (4.6)

whichimplies

re(V)

0andsoby

(4.5)

^and

(4.6)

one gets

Lvfg

ff.

(4.7)

Accordinglyonbehalf ofaknown definition

([ 13]),

^{the above}equationshows that isof class 1, ahomogeneousformonTM. Takingnowthe exteriordifferential of the Liouville form

y

definedby

(4.4),

one getsatonce by

(2.5)

dy to

^

^{y+ ’:=:’}

^{d-’y (4.8)}

LOCALLY CONFORMAL COSYMPI.ECTIC MANIFOLDS wherewehaveset

q d

v"’

A

toa

From

(4.8)

and

(1.2)

one obtainsnstantly

d"tp

0 dtp=

.

277

(4.9)

Sinceclearlythe2-formqisofmaximalrank,weagree^tocalltpthecanonicalconformal symplectic formofM. Noticingthat one has

,,q y,

to(V)

0

(4.11)

whichimplies

Lvq

p.

(4.12)

Hence p

isasf2’ ahomogeneousof class 1,2-form.

Next

making use of the vertical operatori,.definedby

i

^{k 0,i,,dva co}

a,

i,,oJ

0(L C(R)M)

one quicklyfindsby

(4.9)

i,3p⁼⁰

(4.13)

andthe aboveequationtogether^with

(4.12)

^{provesthat is aFinslerianform}

([7]).

We

recall that the vertical lift

Z" ([23])

ofa vectorfield

Z F(TM)

withcomponents

Z

^ain

M,

has

ascomponents

Z"(0 __Za

^O

Z

^a Ov^A Henceinthecaseunder considerationonehas

T Z

^{A 0}

--" ^A

^{=0,1 2m}

andby

(4.9)

^onegets

Thereforeby

(4.10)

one derives

(4.14)

Lr, ^ap

⁰

^(4.16)

andonemay say^that

T"

defines aninfinitesimalautomorphismof

ap.

Finallywe set

where

denotes the Liouville function on

M ([9]).

r

:fv ^(4.17)

)2

v

Z(v

^A

^(4.18)

Operating^onrbythe vertical differentiation operator

d,, ([7])

onegets

dvr f Y vto ft ^(4.19)

A

andtakingtheexteriordifferentialof

(4.19)

^{we obtain}by

(2.13)

and

(4.9)

d(dr) f ^Z

^dvA

^ ^{toA ----lap. (4.20)}

Next

putting

H

--fapitfollowsby

(2.13)

dH=0.

(2.21)

Therefore the exactsymplecticform//can be viewed asthe canonical symplectic form of the

(4m

+2)-dimensionalmanifold

TM ([ 13]).

Finally by reference to

[13]

^onemayconsider that thepair

(r,ll)

definesaregularmechanical system 9’d

(in

^thesenseof Klein

[13])

havingthescalar r as kineticenergy.

ira ^p

^=to.

^(4.15)

THEOREM.

Let

TM

be tiletangentbundle manifoldhavingasbasis theexact

(L.C.C.)-manifold

M(O,

T,

co)discussed in Section2. Let V, yandvbe tileLouvillevectorfield,theLiouvilleform and theLiouviile functionofTM, respectively.

One

has the

following

properties:

i) thecompletelft if2’ on

TM

ot the contormal cosymplectic form of

M

isahomogeneousof class 1,2-form,.e.

LvO’

^{’,and it s}d-"-exact, .e.d--’" 0;

ii) satisfies

d-"7

^tp

d’

^()andp

,

the canonicalconformal symplectic form of

TM

^and

enJoysalso the propertytobeaFnslerlan form;

ii) ^thevertical lift

T

of

T

defines aninfinitesimal automorphism of

,

^i.e.

_Lr p O;

v) r

fv

^and

f

^{definearegularmechanicalsystem on}

TM

havingr as kineticenergyand

f’

^as

canonicalsymplectic form(where

f

^{is the energy}function of

M).

I31

[61

[71

I81 191

[10]

[12]

[13]

[5]

[16]

[71 [181

[201 [21]

I221 [23]

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