PSEUDO-SASAKIAN MANIFOLDS ENDOWED WITH A CONTACT CONFORMAL CONNECTION

ABS’I’RACI’. Pseudo-Sasakian manifolds M(U,E,,,g) endowed wlth a contact conformal connection are defined. It is proved tlat sucl manifolds are space forms M(K),K < O, and somo remarkable properttos of the 1,ie algebra of infinitesimal transformatton. of the principal vector feld U on M are discussed. Properties of tle leaves of a co-tsotroptc foliation on I’! and properties of the tangent bundle manifold TM having lq as a basis nro studied.

true integ,al 4nuargant of U UM c Further the necessa and sufficient conditions for M to be feZ{ate is that the isotroplc component of U vanishes.In thls case M is a CH product (see c Yano and  Ken [;] nd Rosc Finally using soe notions Introduced by 7ano and Ishthara [9] and 1 by KIetn [10], ue consider in Section 5 cert properties of he angent bundle antfold '1' hatng the manifold (,,,g) as n basis.
At each point p e M one has the standard decomposition (see Rosca [1]): T() (2. l) P P P where T, H, and T% are the tangent space, a (2m)-dlmensional nut vector P P space, and a t{me-l{ke llne orthogonal to 1, respectively., P Let S, S H be two 8eZ-oPthoonZ (abbrevlatlon s.o.) m-distributions P P P which define an inuolut4ve automorphlsm U of square +I (U is the p operator defined by Libeann [11]) Let e r% and e () be the pairing P which defines a contact structure on , and be the covartan tton operator defined by the metric tensor g.The if for any vector fields ' on the structure tensors (,,,g) sattsgy the manifold (, ,g) hs been elled pedo-Saoakgan mangod (ee Re [1]).
In rder to study real eo-gott,ope nd gaotopge Jgatgon n (that gml''opev terstons tn ), we consider an adapted gteld of tt ; {hA ,B,C 0, 1, 2}    { is tile cobasis associated witll W, we set q and the line ele- meut d (d is a canonical vector l-form and is independent on any connection on {) is given by d 'A ' 'A" (2.6) It [ollows fom (2.4) that tle metric tensor g Is: B are the connection forms and the curvature 2-forms on the bundle () respectively, then the structure equations (E.Caftan) may be uitten in the indexlcss form as follows: vh . ,, Referring to (2.4)  the Rfec l-oPm and the Rcc 2-olvn respectively (see Rosca [12]).

pp c
The distribution D (resp.Da) is called the hoPizontal (resp.vertcal) distribution.Such type of CR submanifolds is called CICR manod8 (see Goldberg and Rosca [2]).
As a natural generalization of the de[[,ttton given by Rosca [3], we assume tha the structure equations (2.9) are urltte in the d (u+n) A,, + (.-,,) a + ?awhere il d/2, a,ta, e C (H), and u e A () Is a closed l-fo.Note that and t, are the components of a vector field U Z (tl,a+ta, h ,) a of eonat length.
We shall say (see Rosca [3]) that In tlIs case the pseudo-Saaaklan mnlfold Is endowed with a eonae$ oonfoa (abr.c.c.) eonneotgon.e aIo agree to U the princpa vector feld associated with this connection.
Since (?,71 const, we may write by ( [4]).In the following we agree to call u the principal PJhJ'fan associated with the c.c. connectio,t.
Consider no the l-form a aktng the exterior dtfgerent(al of v, one finds tth the help og (3.1) and (3.4) that c 2. In this case we deduce d= 2 (3.) and this equation asserts that v is exe'iop PeeuPren (see atta [13] tth 2 s the 'eem'enee l-f o. gy (2.4) and (2.5) one easily ginds t ?,.
( Next if ,: + i?, T() T*() is tl,e bundle isomorphism defined by R d/2, one readily gtnds p(?) 2; (3.9) In the following we agree to call the p)'e:)!/m/,let{o form (dim ker() O) the fundammztat 2-form on Let now f + ( C=()) be a eontt eztengon of and f the Lle derivative with respect to Uf.Then by (3.9) one quickly finds f O.
Therefore according to the definition given by Roses [3], we may say that f a z'eZaive contact infinitegmaZ transformation of Denote now by S (reap.oS) the simple unit form hteh eorropond (resp. ).One has Comt,g back to reIatto,s (3.12), oe re;dlly [tnds Therefore we may say that all connection forms of the pseudo-Sasaklan manifold under consideration are ntt'a 'eZatoa of m,aganoe for the vector field UU.

PSEUDO-SASAKIAN MANIFOLDS 739
It is worth now to make the following considerations.
On the other hand, since g(,) const, e ay ay in tIar the case a Sasaktan antfoId that defines tth an Like usually denote by the eueatue operator.hen, a is known, the eetgona oeat (,) degtned by and ) t given by where R(,U,J,UU) (R(',U)U,U).' llence by (3.27) and (3.28) one gets K(II,Ui) -.Now referring to (2.10) and (3.12) one finds after some calculations ,, Vs A taU a S A VS, + v s A 'a a a (.) , + (no suaation) +aa where we have set (3.32) As is known (see Libermann []] ]), the components of the R'foef./:.ensor are given by b c* a a * a a bc, A a + a* 0).Because of this, we get from (3.31) that bc tb c (3.33) aa a a I follows from (3.33) tha tle componenC of the Rcci tensor are o (see Rosca [16]).In addition, since the 8eaa cu'vauve C is the trace of, the Rieci tensor with respect to g, one finds by (2.7) and (3.3) that M C 4-m (m > 4).Therefore we conclude that tie pseudo-Sasakian manifold under consideration is a space fo (4-m) of hpePboe THEOREI 1.Let (U,,,) be a pseudo-Sasaktan manifold endowed with a c.c. connection and let (resp.fl d/2) be the principal vector field associated with this connection (resp.the fundamental 2-fo on ).One has the following properties: (i) U is divergence free, and U defines an infinitesimal homothety on (li) all the connection forms on are integral relations of invarlance for UU; (Ill) and U) define an U-contact couclrcular pairing, and defines a 3-foliation on (iv) any contact extension f + of U is a relative contact Infinitesimal transformation of q; (v) and U define both an 4nflniteslmal automorphism of all (2q+l)-forms Lq where u is the dual form of U(q<m); q (vi) the Ricci 1-form of M is exterior recurrent, and the Ricci tensor is disjoint; (vii) M is a space-fomn of hyperbolic type (viii) any such submanifold N is defined by a compietely integrable system of differenttai equations whose solution depends on 2m arbitrary constants.co-isooc F0t^o o ?(v,,n,l. We shall consider on M the following three distributions: a) An gnarag distribution D r (i.e.UD'_ D') of dimension 2(m-)+l defined by D= {hi,hi,, i=l m-; i*=i+m}.Cs,(D' D)IS * of dimension defined {h r by D t ,; r* 2m-+1, 2m).
These three distributions have no conunon direction and tley define on a -st'uctu'e of rank 2 (see Sinha []7]).
Accordingly we shall split the principal vector field as follows: U U q) ' t  x to be a oo-soeropgo ogation t that be exterior recurrent (see Lichnerowtcz [la] and Yano and Kon [7]).
tlence one ust rtte d and if ttI(c,R) represent the eZas of g then the recurrence l-form y defines an eleen og l( ,R) (see Ltclnerolcz [6]).In the case under dscusston one gnds (co,pare tth gan nd on [7]) that the necessary aud sufficient condttten for t receive a tropic foliation Fc D Dx Is that the component t of vanishes.In this case the recurrence l-form y of is given by (u-n).
(   One knows (see Lichnerowlcz [18]) that the class of cohomology of w G which is an element of H2+I(Mc;R) is an luvarla,t of tle folIaton.Using the same notation as in section 3 and applying (4.4), we may write w G c(Lu-L,) c(8-Ln) and as is known, one ha, g(ha,hb*) 5al), g( ,iant i.e.UI)xC 1x( ).

( 3 .
28)Making use of (3.5), (3.6), and (3.19), one finds [, U] 4(+2) D D b) An isoropc distribution D x (i e _ orth of dimension defined by D A {h.; r=m-E+l m}.c) A transversal distribution I) According to the considerations of Section I, it follows that M is a CICR c 8ubmanoZd.By definition we have du O.Because of this and (3.1), the exterior differentiation of (4.3) gives dy -2.

(4. 4 )
Equation (4.4) shows that the restrlction M is an exact forum.cOn the other hand, the form of Codb[||on-Vey (see Lichnerowicz[6]) on M we have set c Thus it follows from (3.22) that uwG -CU (Ln).

(4. 7 )
By means of (2.13) and (3.9) one has d(Ln) 2(A)+I(4.8) As Is known, the form R defines the i'sA class oy Cheu of H.
3.2) that [ ta, ta, c, c const.0 which shows that u is an :,ztegeal reZation of invoice for U (see Licl,nerowlcz satisfy (3.1), ?d/2, d O, leads to the identity.Because of this, dl I, that is E s a eZosed system.It follows from this that the system E defining the pseudo-Sasakian antfold endowed enneeton is eompleeZ ;nt'able and ts solution depends on 2 entnts (the number of equations in (3.4)).PSEUDO-SASAKIAN MANIFOLDS 737 From (3.4) and (3.3) we also obtain cu [ (t *=-t (3 5) a a % a and ()