Cattaneo Gasparini, Ida

De Cecco, Giuseppe

English

Università del Salento: ESE - Salento University Publishing

COMPLEMENTARY OISTRIBUTIONS ANO PONTRJAGIN CLASSESby IVA CATTANEO GASPARINI (Roma) and GIUSEPPE VE CECCO (Le~~e) (*)SUMMARY.- ~tom a ne~e6~any ~on~on 60~ the e~6ten~e 06 k(> 2) ~omplemen~ycLùbU.bLlUOn6 on a marU601d we dedLl~e ~onne~Uon6 beAween Ponttjag-i.n mMe,~ o~cLùbU.bLttion6 and 06 ~an6veJtMl bLlndle6.Let M be a riemannian smooth orientab1e manifo1d of dimension n (evenor odd) and suppose that M has k comp1ementary (smooth) distributions ofdecomposed into the direct sum of the subspacesoriented n.-p1anes (i=l, ... ,k); l. e. for every pointlspace T (M) can bepof T (M) where dim Ti = n. (and hence n,+ ... +n k = n).p p lP E M the tangentl kT , ... ,TP PThen one says that M admits an "a1most product" or "a1most mu1tiproduct"structure.In a paper of 1969 ([3J) one of the present authors showed that the vani-shing of certain Pontrjagin c1asses is a necessary condition for the existenceof k comp1ementary distributions on M. After a review of these resu1ts weprove the fo11owingTHEOREM ALe.t ,\I be a ~emanrUan 6mooth o~enta.ble marU601d onFLl~h~o~e let M have k ~ompleme~y cLùbU.bLttion6 E. C TM and let thetbLlndleo.=n-n., ,(, ,(, andF~nalty let p lE)"-bLlndle E. Then ~64~E H (M;lR) denote the ~-th Jteal Pon~jag.{n dM6 06 thePh In) p I E Il ",t f~{. O "h,~ >(~) This work vas carried out in the framewot'k 07 the activities of theGNSAGA (CNR - Ita1y).- 2 -one hMP(O.)=o![ -.(Using Bott's "Vanishing Theorem" one can, in a certain sense invert thepreceding resu1t obtainingTHEOREM BLe.t M be a iUemamùan ~moo.th oiUe.,t.tabR.e marù6o.e.d 06 ml!.Y!<I'<'on .1.E. c nl and tu.(.the bwuUe Q.. = Hl/E ..( (({.=n-n ..( .(andPhlQ.)p lE).( ~ "110 tcl.6 .o and h + ., = I[In order to give a self-contained presentation we reca11 the necessarypreliminaries.1. PreliminariesLet M be a smooth (paracompact) manifo1d and let E be a vector mq-bundle on M. As is well known, the total (rea1) Pontrjagin classis defined byp(E) of Ewhere n is the curvature of an arbitrary connection on E and 4rp (E)EH (M;IR)r- 3 -'*where H (M; ffi) is the de Rham cohomology ring of M.the r-the Pontrjagin class of E.P (E)ri s ca 11 edClearly p (E) = O for 4r > n if n = dim M. Moreover if E is an orienrted bundle of even dimension q, then the class p (E), which is loca llyq/2represented by the closed form (21T)-q det n, equals the square of theEuler class; this latter is strictly connected with the Euler-Poincaré characteristic of the manifold under consideration, if E = TM.If E is the tangent bundle TM, then the classes are also called Pon-trjagin class of M and are often denoted by P (M).rLet n is the curvature form of a connection on the principal fibrebundle of orthonormal frames, then the explicit expression of Pontrjaginclasses is give by(l)( 1. l )whereP (TM)r(1.2) e: 5) .l ••• 1-l 5s! l: o(i •... ,i;j •... j)n.. I\ ...l\n.(i) l 5] 5 J]J2 J5-] Jss is an even integer and o(i l , ... ,i s ; jl , ... ,js) is the generalysedKronecker symbol.For n even, the n-form(1.3) n l-l(-2-)!J l: '. ." n..l ••• l l l] n ] 2 ... "n.l n-l l ncalled Gauss curvature form of M, is a representative of the Euler class ofl )See J.A. Thorpe (6J- 4 -TM and if M is compact and orientable, the Gauss-Bonnet theorem says thatf f~ n(n) = x(M)Mwhere X(M) is the Euler-Poincaré characteristic of M '"and f:M + G (M)nis an orientation of M,n-planes tangent to M.2. A vanishing theorem'"G (M)nbeing the Grassmann bundle of the orientedWe can now prove the following theorem ([3J)(2.1). TheorernLu: M be. a Jvi.e.manrU.an J.>mooth oJvi.e.n.tab.te. marU.60.td 06 cWne.l1J.>~on n w~chadrndJ.> k comp.te.me.n.taJt.y IJ.>moo.tft ) fu.tJvi.bu..ti.o l1J.> 06 oJvi.e.n.te.dP (M)II.Proof.n .- p.tanu-<.nu.t.t 6011. 2Jt>max(n 1, •. ,nk )'"Let E be the principal fibre bundle of the orthonorrnal frarnes (associated'"te the tangent). Its structural group is G = SO(n) (the retation group). Werecall that the Lie algebra ~ of SO(n) can be identified with space ofthe skew-syrnmetric matrices of order n.'"Let us consider the subbundle E of E formed of the frames "adapted"to the distributions, viz. the orthogonal framesvectorsle. l(i=l, ... ,n), so that thele lo.Jo = n +... +n . +1, ••. , n +... +n .j l J-l l Jforrn a basis for Tj . The bundle E can be regarded as having structural group- 5 -A connection w on E is represented by a l-form which takes values inthe Li e a1gebra 71 of G, where 9' i s the di rect product of the Lie alg~bras of SO(n). Hence one obtainsrw .. = O11w.. +w .. =OlJ Jli,j = l •...• nw = O i F jCI.8.l Ji ,j = l, ... , k; et., S . = n + .•• n + l , ... ,n + ... +nl J l j-l l jAnalogous relations hold for the components of the curvature form n.Therefore,if 2r > max(n1 •...• nk).then each term in (1.2) will have a factor Qa.8.l Jwith iFj; the assertion p (TM) = P (M) = O then follows from (1.1).,r r -'Remarks(2.2) lf k = n and therefore n. = llVi (i .e. the manifold is paral-lelizable) then P (M) = O Vr. lt follows that the adapted connectionrvanishes. The manifold is then flato(2.3) For k = 2 (and obviously for k = l) the theorem is not meaningful.and consequently P (M) = Or(2.4) It is worth noticìng that the existence of a distribution ofq-planes ìmplies the existence of a distribution of (n-q)-planes. An argumentanalogous to the one followed above, using the Gauss curvature form, yieldsthe fo 11 owi ngbundle by Q. = TI1/E ..l 1it follows that- 6 -(2.5) TheoremLu M be et omooth e.ompeta o!ÙentaN'.e mo.iU6o.e.d 06 cUmeno.{.on even netnd ouppoo e tha.t .i..t hIV.I o. futtùbu.tio n 06 o!Ùen-ted q- pwneo w.{.th q odd(1 .:'. q < nl. Then the EulVt- Po.{.ne.o.Jté e.1uvu1e.te.Jt.{.oue. o6 M .{.o null..3. Proof of theorems A and B.(3.1) To every d;str;but;on there corresponds a smooth subbundle E.1of the tangent bundle w;th the fibre of dimension n.. Denote the quot;ent1From the Wh;tney dual;ty formula p(TM)=p(Q.)p(E.)1 1(3.2) p (TM) = p (Q.) +P l (Q. )p (E.) +... +Pl (Q . )P l (E. ) +p (E.)r r 1 r- 1 l 1 1 r- 1 r 1where the product between classes ;s the "cuo product" in the r;ngIf 2r> n., then r> n./2 and hence P (E.) = O. If moreoverl 1 r 1;,H (M;IR).(3.3)thenPh(Q.)P (E.) = Ol S lp (TM) = p (Q.)r r lVh,s ~ l, h+s = r2r > ni .Not;ce that, since the Pontrjag;n r;ng may have div;sors of zero, cond;-t;on (3.3) does not ;mply that either Ph(Q.) = O or p (E.) = O. On accountl s 1of our assumpt;ons, from theorem (l. l) one can conclude thatTh;s proves theorem A.(3.4) It ;s well known that if E. c TM ;s ;somorph;c to an integrablel- 7 -subbund1e, then by Bott's theoremP(Q.)=Or lr > 2q.lwhere qi = n-n i · Hence in the assumptions of theorem A, if m = max(n1 ,· .. ,nk)<4q i'then one has for the integers h for which m < h < 4q.lPr(Qi)=0 2r>hwhithout assuming that Qi be integrab1e. This is meaningfu1 if k > 2.(3.5) Converse1y 1et us assume Q. to be integrab1e. Then, under our as-lsumptions, one has at the same timeP (TM) = Orhence an account of (3.2)P(Q.)=or l2r > max(m,4q.)lPh(Q.) P (E.) = Ol S lThis ends the proof of theorem B.RemarkVh, s .:. l , h+s = r.The resu1ts of theorem (2. l) ho1d in the more genera1 situation of analmost mu1tifo1iated riemannian structures on a manifo1d, i.e.are not necessare1y comp1ementary. Infact by increasing the numberwhere E.lof distributions, with a suitab1e choice of the metric ([7J) it is possible goback to the previous situation.- 8 -R E F E R E N C E S[lJ R. Bott, Lec.tuJ;.v, 011 c.haJtac.-feJtùUc. da.M v, al1d 6o.~OIU, Lect. Notesin Math. 279, Springer (1972), 1-94[2] l. Cattaneo Gasparini, C0I1I1e66-i.OI'Ù adaUa-te a U.rla 6.:l:/urt..twta qu.a6-i. ptWd('ite, Ann. Mat. pura e appl. 63 (1963),133-150.[3J l. Cattaneo Gasparini, CaJtv~ e da66~ c.aJt~eJt-i.6uc.he, Rend. Sem. Mat.Un. Torino. vol. 28 (1968-69), 19-30.[4] F.W.Kamber, Ph. Tondeur, Foua-fed Bu.Yldie6 aYld ChaJtaueJt-i.6uc. C.taMv"Lect.Notes in Math. 493, Springer (1975)[5J H.V. Pittie, ChaJtaueJt-i.6-f~c da66e6 06 60~01U, Pitman Pub. (1976)16J J. A. Thorpe, Sec.uoYla..i'. CaJtva-faJtv, aHd ChaJtac-f~uc. c.ea.66e6, Ann. ofMath., 80 (1964), 429-443.[71 I. Vaisman, A.inl06t-mu.U.i.6oUa-te /Uemal1rtWl1 maI'Ù60ùu., An. Sti. Univo"Al.I. Cuza", sect.I,16 (1970), 97-104.

Complementary distributions and Pontrjagin classes

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Complementary distributions and Pontrjagin classes

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