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%%Page: 78 4
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1831 y(B) y(B) y(B) y(@)1362
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y Fp(.) y(.) y(.) y(.)1905
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Fi(1) y(C) y(C) y(C)
y(C) y(C) y(C) y(C)
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b(the)i(sense)h(of)e(Goze-Haraguc)m()i(it)d(is)h(a)g(nilp)s(oten)m
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3058 y(suc)m(h)34 b(that)49 b([)p Fo(H)8 b(p)p Fp(\()p
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81 6 bop 660 -142 a Fu(A.)30 b(Aw)m(ane:)42 b(Some)30
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a(2)p Fo(q)1790
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2164 y Fp(b)s(e)31 b(the)g(op)s(en)g(ball)45 b Fo(B)5
b Fp(\()p Fo(Q) y Fh(0) y Fo(;)17 b(")p
Fp(\))47 b(with)30 b(cen)m(ter)i(at)e(the)i(p)s(oin)m(t)46
b Fo(Q) y Fh(0) y Fp(and)0 2284 y(with)32
b(radius)49 b Fo(")g Fp(\(for)32 b(the)h(norm)48 b Fn(jj)p
Fp(\()p Fo(u;)17 b(v)t(;)g(w)s Fp(\))p Fn(jj)41 b Fp(=)j
Fo(max)p Fp(\()p Fn(j)p Fo(u)p Fn(j)p Fo(;)17 b Fn(j)p
Fo(v)t Fn(j)p Fo(;)g Fn(j)p Fo(w)s Fn(j)p Fp(\)\).)195
2405 y(Let)61 b Fo(g)51 b Fn(2)d Fo(G)60 b Fp(suc)m(h)46
b(that)61 b Fo(g)t(U)10 b Fn(\\)p Fo(U)75 b Fn(6)p Fp(=)64
b Fn(;)c Fp(and)45 b(let)60 b Fo(Q)k Fp(=)g(\()p Fo(x;)17
b(y)t(;)g(z)t Fp(\))64 b Fn(2)g Fo(U)72 b Fp(suc)m(h)45
b(that)61 b Fo(g)t Fp(\()p Fo(Q)p Fp(\))i(=)0 2525 y(\()p
Fo(x)93 2489 y Fm(0)117 2525 y Fo(;)17 b(y)213 2489 y
Fm(0)235 2525 y Fo(;)g(z)328 2489 y Fm(0)352 2525 y Fp(\))44
b Fn(2)g Fo(U)28 b Fp(.)43 b(Then)34 b(w)m(e)f(ha)m(v)m(e)51
b Fn(j)p Fo(x)22 b Fn(\000)h Fo(x) y Fm(0)1632
2525 y Fn(j)k Fo(<)h Fp(2)p Fo(")17 b Fp(,)48 b Fn(j)p
Fo(y)25 b Fn(\000)d Fo(y) y Fm(0) y
Fn(j)27 b Fo(<)h Fp(2)p Fo(")48 b Fp(and)h Fn(j)p Fo(z)27
b Fn(\000)c Fo(z) y Fm(0) y Fn(j)k
Fo(<)h Fp(2)p Fo(")p Fp(.)195 2645 y(By)j(\()p Fo(a)p
Fp(\),)f(\()p Fo(b)p Fp(\),)h(\()p Fo(c)p Fp(\))f(w)m(e)h(ha)m(v)m(e)64
b Fn(j)p Fo(z)21 b Fn(\000)c Fo(z) y Fm(0)1510
2645 y Fn(j)44 b Fp(=)g Fn(j)p Fp(\() y Fi(P)1856
2605 y Fg(n) y(i)p Fh(=1) y Fo(\015)2042
2660 y Fg(i) y Fp(\))p Fo(q) y Fh(2)2190
2645 y Fn(j)27 b Fo(<)h Fp(2)p Fo(")44 b(<)g Fp(1)33
b(,)d(then)47 b Fn(j) y Fi(P) y Fg(n)3115
2670 y(i)p Fh(=1) y Fo(\015) y Fg(i)3329
2645 y Fn(j)f Fp(is)30 b(an)f(in)m(teger)0 2766 y(less)c(than)42
b(1,)26 b(then)742 2699 y Fi(P)830 2726 y Fg(n)830 2790
y(i)p Fh(=1)965 2766 y Fo(\015) y Fg(i)1088
2766 y Fp(=)44 b(0)d(and)h Fo(z)32 b Fp(=)c Fo(z)1727
2730 y Fm(0) y Fp(.)41 b(By)26 b(the)f(inequalities)39
b Fn(j)p Fo(x)6 b Fn(\000)g Fo(x) y Fm(0)2914
2766 y Fn(j)28 b Fo(<)f Fp(2)p Fo(")41 b Fp(and)h Fn(j)p
Fo(y)10 b Fn(\000)c Fo(y) y Fm(0) y
Fn(j)28 b Fo(<)f Fp(2)p Fo(")0 2886 y Fp(w)m(e)34 b(deduce)g(that)65
b Fn(j)p Fo(s)787 2850 y Fh(0)787 2911 y(2)826 2886 y
Fp(\()p Fo(x)919 2850 y Fm(0)965 2886 y Fn(\000)23 b
Fo(x)p Fp(\))39 b Fn(\000)g Fo(s) y Fh(0)1359
2911 y(1) y Fp(\()p Fo(y) y Fm(0)1533
2886 y Fn(\000)23 b Fo(y)t Fp(\))p Fn(j)54 b(\024)i Fp(2)p
Fo(")p Fp(\()p Fn(j)p Fo(s) y Fh(0)
y(1) y Fn(j)38 b Fp(+)h Fn(j)p Fo(s)
y Fh(0) y(2) y Fn(j)p Fp(\))17 b(.)0
3044 y(And)39 b(also)f(b)m(y)56 b(\()p Fo(a)p Fp(\))17
b(,)57 b(\()p Fo(b)p Fp(\))17 b(,)57 b(\()p Fo(c)p Fp(\))e(w)m(e)40
b(ha)m(v)m(e)78 b Fn(j) y Fi(\022)
y(P) y Fg(n) y(i)p Fh(=1)
y Fo(\013) y Fg(i) y Fp(+)64 b Fo(m)2343
2978 y Fi(P) y Fg(n) y(i)p Fh(=1)2566
3044 y Fo(\014) y Fg(i) y Fi(\023)2710
3044 y Fo(q) y Fh(0) y Fn(j)76 b Fo()c Fp(0)k(suc)m(h)37
b(that)52 b Fo(K)k Fn(\032)49 b Fo(B)5 b Fp(\()p Fo(Q)2483
3995 y Fh(1) y Fo(;)17 b(r)s Fp(\))g(,)35 b(where)53
b Fo(B)5 b Fp(\()p Fo(Q) y Fh(1) y
Fo(;)17 b(r)s Fp(\))51 b(is)35 b(the)h(op)s(en)0 4101
y(ball)31 b(with)h(cen)m(ter)i(at)e(the)h(p)s(oin)m(t)48
b Fo(Q) y Fh(1) y Fp(and)33 b(with)f(radius)49
b Fo(r)s Fp(.)195 4221 y(The)34 b(follo)m(wing)29 b(sets)237
4415 y Fo(E)309 4430 y Fh(1)404 4415 y Fp(=)185 b Fn(f)p
Fo(\015)50 b Fn(2)44 b Fj(Z)31 b Fo(=)i Fn(j)p Fo(\015)5
b(q) y Fh(2) y Fn(j)44 b Fo(<)28 b(r)s
Fn(g)237 4572 y Fo(E)309 4587 y Fh(2)404 4572 y Fp(=)185
b Fn(f)17 b Fp(\()p Fo(\013)39 b Fp(+)g Fo(m\014)6 b
Fp(\))p Fo(q) y Fh(0) y Fo(=)49 b Fn(j)p
Fo(\013)23 b Fp(+)38 b Fo(m\014)6 b Fn(j)28 b Fo(<)f
Fp(2)p Fo(r)s Fp(\()p Fn(j)p Fo(s) y Fh(0)2124
4597 y(1) y Fn(j)21 b Fp(+)h Fn(j)p Fo(s)2384
4536 y Fh(0) y(2) y Fn(j)p Fp(\))55
b Fo(;)72 b(\013)18 b(;)33 b(\014)50 b Fn(2)28 b Fj(Z)m
Fn(g)0 4678 y Fp(and)237 4762 y Fo(E)309 4778 y Fh(\()p
Fg(j;)p Fh(3\))507 4762 y Fp(=)82 b Fn(f)17 b Fp(\()p
Fo(\013)39 b Fp(+)g Fo(m\014)6 b Fp(\))p Fo(z)
y Fh(1) y Fo(s) y Fh(0) y
Fg(j) y Fp(+)38 b Fo(\015)5 b(t) y
Fh(2) y Fg(j) y Fp(+)38 b Fo(u)50 b(=)f
Fn(j)p Fp(\()p Fo(\013)23 b Fp(+)f Fo(m\014)6 b Fp(\))p
Fo(z) y Fh(1) y Fo(s) y Fh(0)2544
4787 y Fg(j) y Fp(+)22 b Fo(\015)5 b(t)2795
4726 y Fh(2) y Fg(j) y Fp(+)22 b Fo(u)p
Fn(j)k Fo()f Fp(0)p
Fo(:)p eop
%%Page: 82 8
82 7 bop -184 -142 a Fp(82)659 b Fu(A.)31 b(Aw)m(ane:)41
b(Some)31 b(a\016ne)f(prop)s(eries)e(of)j(the)f Fk(k)s
Fu(-symplectic)g(manifolds)-184 98 y Fp(Let)926 254 y
Fo(")972 213 y Fm(0) y Fp(=)71 b(min)
y Fi(\022) y Fp(1)p
a Fo(q) y Fh(0) y Fo(;) y Fn(j)p
Fo(D) y Fh(2) y Fn(j)p
a Fp(4)p Fo(q) y Fh(0) y Fp(\()p
Fn(j)p Fo(s) y Fh(0) y(1) y Fn(j)39
b Fp(+)f Fn(j)p Fo(s) y Fh(0) y(2)2237
322 y Fn(j)p Fp(\)) y Fo(;) y(\013)p
a Fp(2) y Fi(\023)2600
254 y Fo(;)-184 490 y Fp(and)32 b(let)49 b Fo(U)229 505
y Fh(1)318 490 y Fp(\(resp.)61 b Fo(U)683 505 y Fh(2)739
490 y Fp(\))32 b(b)s(e)h(the)g(op)s(en)g(ball)e(with)h(cen)m(ter)50
b Fo(Q) y Fh(1) y Fp(\(resp.)61 b Fo(Q)2606
505 y Fh(2) y Fp(\))33 b(and)g(with)f(radius)49
b Fo(") y Fm(0) y Fp(.)-184 612 y(W)-8
b(e)33 b(ha)m(v)m(e) y Fo(G:U) y Fh(1)1683
738 y Fn(\\)p Fo(U) y Fh(2) y Fp(=)44
b Fn(;)p Fo(:)-184 918 y Fd(Case)34 b(2:)167 b Fo(S)368
882 y Fh(0)451 918 y Fp(=)44 b(0.)-184 1038 y(W)-8 b(e)33
b(ma)m(y)f(pro)m(v)m(e)i(the)f(assertion)f(for)g(this)h(case)g(exactly)
g(as)g(for)f(the)h(\014rst)g(case.)-184 1161 y(Observ)m(e)h(that,)f(in)
e(this)i(case,)g(the)g(group)49 b(\000) y Fh(
1185 y(2) y Fp(is)32 b(the)h(free)g(ab)s(elian)e(group)h(on)h
(three)g(generators.)-184 1336 y(\(ii\))c(The)j(fact)f(that)h(the)f(v)m
(ectors)50 b(\()p Fo(T) y Fh(0) y Fo(;)17
b Fp(0\))g(,)47 b(\()p Fo(T) y Fh(1)
y Fo(;)17 b Fp(0\))48 b(and)g(\()p Fo(T) y Fh(2)2141
1336 y Fo(;)17 b(q) y Fh(2) y Fp(\))48
b(are)31 b(linearly)e(indep)s(enden)m(t)k(p)s(ermits)-184
1456 y(us)g(to)f(see)i(that)e(the)h(quotien)m(t)50 b
Fo(M) y Fh( 1481 y(2) y Fp(is)32
b(a)g(compact)h(manifold.)11 1578 y(The)k(orien)m(tabilit)m(y)c(of)i
(this)h(manifold)d(follo)m(ws)h(from)h(the)h(fact)g(that)f(eac)m(h)i
(elemen)m(t)f(of)f(the)h(group)-184 1699 y Fo(H)8 b(p)p
Fp(\(2)p Fo(;)17 b Fp(1)p Fo(;)g Fj(R)9 b Fp(\))49 b(has)33
b(p)s(ositiv)m(e)f(determinan)m(t.)-184 1874 y(\(iii\))24
b(By)k([2)o(])g(and)f([3],)h(t)m(w)m(o)g(compact)f(connected)h
(complete)f(lo)s(cally)d(a\016ne)k(manifolds)41 b Fo(M)3145
1889 y Fh(1) y Fp(and)i Fo(M) y Fh(2)3607
1874 y Fp(are)-184 1994 y(homeomorphic)35 b(if)h(and)h(only)g(if)f(the)
i(fundamen)m(tal)e(groups)54 b Fo(\031) y Fh(1)2249
1994 y Fp(\()p Fo(M) y Fh(1) y Fp(\))f(and)h
Fo(\031) y Fh(1) y Fp(\()p Fo(M)2950
2009 y Fh(2) y Fp(\))g(are)37 b()-184
2115 y(this)h(implies)e(that)55 b Fo(M)687 2079 y Fh(
2139 y(2)923 2115 y Fp(is)38 b(homeomorphic)f(to)h(the)h(torus)55
b Fo(T) y Fh(3) y Fp(if)37 b(and)i(only)f(if)f(the)i
(fundamen)m(tal)-184 2235 y(group)49 b Fo(\031)164 2250
y Fh(1)204 2235 y Fp(\()p Fo(M)346 2199 y Fh(
2260 y(2)527 2235 y Fp(\))55 b(=)g(\000)812 2199 y Fh(
2260 y(2) y Fp(of)49 b Fo(M)60 b Fp(is)32 b(an)g(ab)s(elian)f
(group)h(if)g(and)g(only)g(if)48 b Fo(S) y Fh(0)2921
2235 y Fp(=)c(0.)-184 2516 y Fr(References)-87 2739 y
Fp([1])49 b(Auslander,)33 b(A.:)44 b Fd(Examples)33 b(of)i(lo)-5
b(c)g(al)5 b(ly)34 b(a\016ne)h(sp)-5 b(ac)g(es.)31 b
Fp(Ann.)i(of)f(Math.)h Fr(64)g Fp(\(1964\),)e(255{259.)-87
2887 y([2])49 b(Auslander,)37 b(A.:)50 b Fd(The)38 b(structur)-5
b(e)39 b(of)e(c)-5 b(omplete)37 b(lo)-5 b(c)g(al)5 b(ly)38
b(a\016ne)f(manifolds)p Fp(.)e(T)-8 b(op)s(ology)34 b(3)i(suppl.1)65
3008 y(\(1964\),)c(131{139.)-87 3156 y([3])49 b(Auslander,)e(L.;)j
(Markus,)e(M.:)67 b Fd(Holonomy)45 b(of)g(\015at)g(a\016nely)g(c)-5
b(onne)g(cte)g(d)44 b(manifolds)p Fp(.)f(Ann.)h(of)65
3277 y(Math.)33 b(Princeton)g Fr(62)f Fp(\(1955\),)g(139{151.)-87
3426 y([4])49 b(Aw)m(ane,)55 b(A.:)78 b Fd(Sur)50 b(une)h(g)n(\023)-47
b(en)n(\023)g(er)-5 b(alisation)47 b(des)j(structur)-5
b(es)51 b(symple)-5 b(ctiques)p Fp(.)49 b(Th)m(\022)-46
b(ese)51 b(Strasb)s(ourg)65 3546 y(\(1984\).)-87 3695
y([5])e(Aw)m(ane,)34 b(A.:)60 b Fo(k)s Fd(-symple)-5
b(ctic)34 b(structur)-5 b(es)p Fp(.)34 b(J.)f(Math.)g(Ph)m(ys.)h
Fr(33)f Fp(\(1992\),)e(.)-87 3843 y([6])49 b(Aw)m(ane,)33
b(A.)f Fd(G-esp)-5 b(ac)g(es)49 b Fo(k)s Fd(-symple)-5
b(ctiques)33 b(homo)-5 b(g)n(\022)-47 b(enes)p Fp(.)29
b(J.)j(Geom.)e(Ph)m(ys.)j Fr(13)f Fp(\(1994\),)e(139{157.)-87
3992 y([7])49 b(Blumen)m(thal,)29 b(R.)g(A.:)42 b Fd(F)-7
b(oliate)i(d)31 b(manifolds)f(with)i(\015at)g(b)-5 b(asic)32
b(c)-5 b(onne)g(ction)p Fp(.)28 b(J.)i(Di\013eren)m(tial)d(Geom-)65
4112 y(etry)33 b Fr(16)g Fp(\(1981\),)e(401{406.)-87
4261 y([8])49 b(Bott,R.:)42 b Fd(L)-5 b(e)g(ctur)g(es)32
b(on)g(char)-5 b(acteristic)32 b(classes)f(and)g(foliations)p
Fp(.)e(Lecture)i(Notes)f(in)f(Math.)h Fr(279)p Fp(,)65
4381 y(Springer-V)-8 b(erlag,)31 b(New-Y)-8 b(ork)33
b(\(1972\),)f(1{80.)-87 4530 y([9])49 b(Dazord,)34 b(P)-8
b(.)33 b Fd(Sur)j(la)g(g)n(\023)-47 b(eom)n(\023)g(etrie)33
b(des)i(sous-\014br)n(\023)-47 b(es)34 b(et)i(des)g(feuil)5
b(letages)35 b(lagr)-5 b(angiens)p Fp(.)32 b(Ann.)i(Ecole)65
4651 y(Normale)d(Sup.)i Fr(14)g Fp(\(1981\),)e(P)m(aris,)i(465{480.)
-135 4799 y([10])48 b(Goze,)33 b(M.:)44 b Fd(Syst)n(\022)-47
b(emes)33 b(de)i(Pfa\013)p Fp(.)d(Rend.)h(Sem.)f(F)-8
b(ac.)33 b(Sc.)g(Univ.)f(Cagliari.)e Fr(60)j Fp(\(1990\),)e(167{187.)
-135 4948 y([11])48 b(Goze,)40 b(M.;)i(Haraguc)m(hi,)d(Y.:)55
b Fd(Sur)41 b(les)e(r-syst)n(\022)-47 b(emes)39 b(de)h(c)-5
b(ontact)p Fp(.)38 b(C.)h(R.)f(Acad.)h(Sci.)f(P)m(aris,)i
Fr(294)65 5068 y Fp(\(1982\),)32 b(95{97.)-135 5217 y([12])48
b(de)32 b(Leon-Mendez-Salgado,)g(M.:)43 b Fd(R)-5 b(e)g(gular)34
b(p-almost)f(c)-5 b(otangent)34 b(structur)-5 b(es)p
Fp(.)32 b(J.)g(Korean)g(Math.)65 5337 y(So)s(c.)h Fr(25)f
Fp(\(1988\),)g(273{287.)p eop
%%Page: 83 9
83 8 bop 660 -142 a Fu(A.)30 b(Aw)m(ane:)42 b(Some)30
b(a\016ne)h(prop)s(eries)d(of)j(the)f Fk(k)s Fu(-symplectic)g
(manifolds)658 b Fp(83)49 98 y([13])48 b(Molino,)31 b(P)-8
b(.:)44 b Fd(G)n(\023)-47 b(eom)n(\023)g(etrie)33 b(de)h(Polarisation)p
Fp(.)e(T)-8 b(ra)m(v)j(aux)33 b(en)g(cours)h(Hermann)e(\(1984\),)g
(37{53)f(.)49 244 y([14])48 b(Molino,)54 b(P)-8 b(.:)79
b Fd(G)n(\023)-47 b(eom)n(\023)g(etrie)49 b(glob)-5 b(ale)50
b(des)h(feuil)5 b(letages)51 b(riemanniens)p Fp(.)d(Pro)s(c.)j(Konink.)
f(Nederl.)249 364 y(Ak)-5 b(ad.)33 b(W)-8 b(etensc)m(h.)35
b Fr(85)e Fp(\(1982\),)e(45{76.)49 509 y([15])48 b(Puta,)j(M.:)73
b Fd(Some)47 b(R)-5 b(emarks)47 b(on)h(the)65 b Fo(k)s
Fd(-symple)-5 b(ctic)47 b(manifolds)p Fp(.)e(T)-8 b(ensors.)49
b(N.S.)e Fr(47)g Fp(\(1988\),)249 630 y(109{115.)49 775
y([16])h(Sari,)e(T.:)67 b Fd(Sur)45 b(les)g(vari)n(\023)-47
b(et)n(\023)g(es)42 b(de)j(c)-5 b(ontact)45 b(lo)-5 b(c)g(alement)44
b(a\016nes)p Fp(.)g(C.)g(R.)g(Acad.)g(Sci.)f(P)m(aris)h
Fr(292)249 895 y Fp(\(1981\),)32 b(809{812.)49 1041 y([17])48
b(W)-8 b(olf,)32 b(J.)g(A.:)44 b Fd(Sp)-5 b(ac)g(e)34
b(of)h(c)-5 b(onstant)34 b(curvatur)-5 b(e)p Fp(.)33
b(Univ)m(ersit)m(y)h(of)e(California,)d(0
y(Receiv)m(ed)34 b(June)f(7,)g(1996;)e(revised)j(v)m(ersion)f(No)m(v)m
(em)m(b)s(er)g(1,)g(1996)p eop
userdict /end-hook known{end-hook}if