PRINCIPAL TOROIDAL BUNDLES OVER CAUCHY-RIEMANN PRODUCTS

The main result we obtain is that given t N M a TS-subbundle of the generalized Hopf fibration t H2+’cP over a Cauchy-Riemann product M _ cP, i.e. N _ H is a diffeomorphism on fibres and oj= ot, if s is even and N is a closed submanifold tangent to the structure vectors of the canonical 5Zstructure on H then N is a Cauchy-Riemann submanifold whose Chen class is non-vanishing.

Let r H2*/-cP* be the generalized Hopf fibration, as given by D.E.BLAIR, [8]. Leaving definitions momentarily aside we may formulate the following: THEOREM A i) Let N be a framed C.R. submanifold of an -manifold M2n+'. Then the f-anti-invariant distribution .1. of N is completely integrable.
ii) Any framed C.R. submanifold of H2.+, (carrying the standard -structure) is either a C.R. submanifold (s even) or a contact C.R. submanifold (s odd).
The converse holds. iv) Any f-invariant submanifoM of H 2.+' having a parallel second fundamental form is totally-geodesic.
It is known that compact regular contact manifolds are S tprincipal fibre bundles over sympleetie manifolds, el. W.M. BOOTHBY & H.C.WANG, [9].
Eversince this (today classical) paper has been published, several "Boothby-Wang type theorems have been established, cf. e.g.A.MORIMOTO, [10], for the case of normal almost contact manifolds, $.TANNO, [11], for contact manifolds in the non-compact case; more recently, we may cite a result of I.VAISMAN, [12], asserting that compact generalized Hopf manifolds with a regular Lee field may be fibred over Sasakian manifolds, etc. There exists today a large literature, cf. K.YANO & M.KON, [7], concerned with the study of the geometry (of the second fundamental form) of a C.R. submanifold of a Kaehlerian ambient space. In particular, following the method of Riemannian fibre bundles (such as introduced by H.B.LAWSON, [13], towards studying submanifolds of complex space-forms, and developed successively by Y.MAEDA, [14], M.OKUMURA, [15] [19]. Assume that f has complemented frames, i.e. there exist the differential 1-  [20]. Let f0 be a compatible Riemaniann metric on M+s, i.e. one satisfying: Compatible metrics always exist, of. D.E.BLAIR, [4]. Such has often been called a metrical f-structure with complemented frames. Let f0(X, _fY) be its fundamental 2-form. Throughout we assume M 2n+' to be an manifold, cf. the terminology in [4], i.e. the given f-structure is normal, its fundamental 2-form is closed and there exist s smooth real-valued functions , E C(M'-+'), a s, such that: We shall need, cf. [4], [21], the following result. Let M2"+', n > 1, be a connected manifold carrying the structure g, 2, r/, ), 1 a "< s. Then x are real constants, are Killing vector fields (with respect to f0 and the following relations hold: Ker(f,) and therefore the horizontal distribution of the Riemannian submersion is precisely . . Trace(h)) be the mean curvature m+| vector of (resp. j). As an application of our (2.9) (2.12) one may derive: provided that {',.. < a : s} consists of mutually orthogonal unit vector fields. In particular, if N i tandem to cack trctr vector l; 14) h(X,.) + DI x 2 at nor ( X). 2) Generally, if N is a submanifold of the aZmanifold M 2"+' and N is normal to some , with x 0 then tangent spaces at points of N are f-anti-invariant, i.e. _fx(Tx(N)) _ Tx(N ) x E N. Indeed, by (2.4) and the Weingarten formula from WI= 0 and _f X is normal to N. We denote by c # the complex projective space with constant holomorphic sectional curvature 1 (with Fubini Study metric) and complex dimension n, and by S 2"+1 the (2n+l)-dimensional unit sphere carrying the standard Sasaldan structure. Let 7t S 2"+cl # be the Hopf fibration and set Ha'+' {(Pl P,) ( The characteristic 1-form of H2+', s even, is defined by" . , 1 a s, X s (l X) I (3.7) where ' P2. As the sum nx + llx x E N, is direct one obtaines s, ax x" The Riemannian sectional curvature of (M2n+', $ restricted to f-sections is refered to as the f-sectional curvature of the 9Zmanifold. (Cf. also [21], p. 183). At this point we may establish i) of the,or. A. Let X, V be respectively a tangent vector field on N and a cross-section in T(N) "1" -> N. We set P X tan(f X), F X nor(f V) and f V nor( V). The following identities hold as direct consequences of definitions: p2 + tF =-I + q, (R) ', FP + fF --0, Pt + tf--0, .L By (4.3) and the above lemma we conclude P[X, Y] 0, i.e. D is involutivc.
Let us prove now ii) in theor. A. We analyse for instance the case s even. Let N a framed C.R. submanifold of H2+s. Let F P 0, and one applies theor.3.1 of [7], p.87. The case s odd being similar is left as an exercise to the reader To prove the converse of ii) in theor.A we need to characterize framed C.R. submanifolds as follows. Let N be a framed C.R. submanifold of an .9manifold M2+. Then (4.1) leads to P ' P, F P 0, f F 0, etc. One obtaines the following statement Let N be a submanifold of the manifold M 2+' such that N is tangent to the structure vectors ,. Then N is a flamed C.R. smanifold of M 2n+s if and only if F P 0. We have proved the necessity already. Viceversa, let us put by definition / p2 + q (R) , L I-Z Since F P 0, the projections < l. make N into a framed C.R. submanifold, Q.E.D. Now the converse of ii) in theor. A is easily seen to hold, i.e. both C.R. submanifolds of (H2+', af, fg), s even, and contact C.R. submanifolds of (H2n+s, , 0' qo' fg), s odd, are framed C.R. . 2 p2 q (R) . . (4.6) Now the notion of framed C.R. submanifold appears to be essentially on old concept. For not only N becomes a C.R. submanifold of the Hermitian manifold H2a+', if for instance s is even, but its holomorphic and totally-real distributions are precisely , .L. Indeed, by (4.6) one has = K Q.E.D.
2) Due to (3.4) there is a certain similarity between manifolds and locally conformal Kaehler manifolds, cf. P.LIBERMANN, [26]. See also [12]. For instance, we may use the ideas in [2] (of. also theor. 3.4 of [7], p.89) to give an other proof of the integrability of the f-anti-invariant distribution of a framed C.R. submanifold. Indeed, let N be a framed C.R. submanifold of H2+', seven. Let X E , Z,W E .1.. By (3.4) To establish iii) let N be an f-invariant submanifold of H2+'. As a consequence of (2.5), for any tangent vector fields X, Y on N one has:

5.-THE CHEN CLASS OF A CAUCHY-RIEMANN SUBMANIFOLD.
Let M be a C.R. submanifold of el:. Let 7r N -M be a TL fibration, as in theor. B. Assume s is even. Then N is a C.R.submanifold of H 2n+' and its totally-real distribution is integrable. We shall need the following: LEMMA The holomorphic distribution of N is minimal. Proof.
Note that we may not use lemma 4. in [17]   Proof.
Let also V tan (ot r V). Using the Gauss and Weingarten formulae of N in H 2"+s on= has: (_D_ x oq)Y (D x aY-W al.Y Xth(X, Y) + + (D X .l.)y + h(X, 9 Y)-f h(X, Y) (5.5) Let us us= (5.1) to substitute in (5.5); a comparisson between the normal components in (5.5)  But .l. is minimal, so the right hand member of (5.7) is zero. Finally, one may follow the ideas in [17], (p.170) to show that since N is integrable and N minimal the (2p+s)-form A is coclosed. As N is compact, A is harmonic. Thus c(N) [A] 0, and our the.or. B is completely proved.