E. Bompiani ( E d.)
Problemi di geometria differenziale in grande Lectures given at the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Sestriere (Torino), Italy, July 31-August 8, 1958
C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy
[email protected] ISBN 978-3-642-10894-5 e-ISBN: 978-3-642-10895-2 DOI:10.1007/978-3-642-10895-2 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2011 st Reprint of the 1 ed. C.I.M.E., Florence, 1958. With kind permission of C.I.M.E.
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CENTRO INTERNATIONALE MATEMATICO ESTIVO (C.I.M.E)
Reprint of the 1st ed.- Sestriere, Italy, July 31-August 8, 1958
PROBLEMI DI GEOMETRIA DIFFERENZIALE IN GRANDE
C.B. Allendoerfer:
Global Differential Geometry of imbedded manifolds ........................................................................... 1
P. Libermann:
Pseudo-groupes infinitesmaux……................................... 39
C.B. Allendoerfer: Global Differential Geometry of imbedded manifolds
1
- 2 -
Lemma (Morrey) . With eaoh point P of Xn are associated n fu notions 4>
i
(i
= 1, . .. , nJ wioh are
0'" over Xn and have linearly in-
dipendent gradients at P . This lemma is an important result in its own. right . Then 4>i (p) have independent gradients in N(P). Oover Xn with
N(P i )
i
= 1. . . q
, This gives
4>i~(i
= 1 . . . n,
~
= 1 . . . q) .
Take the-
se as ooordinates in Enq . This is an imbedding whioh i s 0'" and 10cally one-to one . Hence it induoes a 0'" Riemann metric. The res ult now follows from the above theorem of Boohner.
'3. ISOIIETRIO IMBEDDING . When Xn has a Riemann. metric, we may f urther require that the given metrio coinoide with that induced by the imbedding, 1. e. that the imbedding be isometric . The results are Janet (1926)
If Xn is
0"',
it oan be looally imbedded with
preservation of the metrio in En(n+1)/2 . Nash-Kuiper
(195~
- Annals of Mathematics)
If Xn is 0 1 and
dompaot, and if it oan be differentiably imbedded in EN (N ~ lit1), then it has a 0 1 isometrio imbedding in EN , This result is effioieni regarding dimension, but i9 true only for 0 1; the case of the torus in E3 shews it · •• be false $tr O? Nash (1956 - Annals . of Mathematios) . I f Xn is O(h) (3 ~ h ~ (0)
a~d 1. oo.paot , it has an isometrio O(h) imbedding in an Euclidean spaoe of dimension (n/2) · (3n+11) . When Xnis non-compact, t he dimension required is 3n 3/2 + 7n 2
~ 11n/2 . The cases of 02 and
0'" are open .
4. RIGID IWBEDDING . If an isometric imbedding is unique to wi thin motion 1n the euolidean spaoe, it is said to be "rigid" , Suff i cie nt
2
- 3 -
cunditions for rigid imbedding will be given later in this series of lectures ,
5. NOTATIONS FOR IMBEDDED MANIFCLDS Local coordinates in En +N Lo~al
coordinates in Xn
AIsc: p,/IJ,
7"
Let Xn be imbedded in En+N
r = 1 ... n+N)
ya (a,
~,
xi (i,
j, k
= 1. . . n)
= n+1. .. n+N.
The imbedding is given locally by the functions
Then (1 )
These are a base for the tangent vectors to Xn , and so any tangent veator .is a linear combination of the dxi . It will be convenient to choose an orthonormal base for the tangent vectors, e,?- , such that 1
a In this notation a
= 8lJ ..
e~ ei = 0
•
Since 4>i are linearly independent :
= bO'
b O'
Mereover
ij
ji
since
The b~. are the coefficients of the second fund~ment~L form. lJ (b)
Define
Curv~ture .
n~ J
= -wiO' A wU:J = dw~J
+ wik
t\
w~ J
Then
n~J = -n~1 These are the curvature forms . They satisfay the further identities : Ricci Bianchi (c) The
w~ are uniquely determined by 4>i, and hence also the n~ J
J
Since 4>i give the metric, w~ and n~ are called "intrinsic". ~y oo ntrest wi and O'
wP
O'
are not intrinsic .
Theorem : There exist unique
w]
which are B~ew-symmetric and sati-
sfy
(d) For a hypersurface there is only one second fundamental form
5
- 6 b ij · Because of (6) the expressions bijb kl - bilb kj are uniquely determined by n~, and hence are intrinsic. J
Theorem: If rank (b ij ) )0 3, the b ij are uniquely determined to within sign; 1. e. the imbedding is rigid (locally). Proof : Put b ij into the diagonal fQrm
Then assume A1 j 0; A2 j 0; A3 j O. Also AiAj are known. So AfA~A~ is known;since A A j 0, A is known to within sign. From A1 A1. all 2 3 1 other Ai are determined . This theorem is due to Beeg (1895). A generalisation for Xnc: En +N (N > 1) is included in my thesis (Amer,Jour.Math , 1939)
6
CHAPTER II THE THEOREM OF GAUSS-BONNET 1, ELEMENTARY CASES . (a) . The simplest form of the G, B. theorem is the familiar formula for the angle sum for a plane triangle . For later purposes ccnconsider the following proof
Draw the auter
,,
normals to the edge. at the vertices, and thus form the ouhr
a 1 , a 2,
~nlLes
a 3 , From an interior point draw· normals
to the edges and form angles ~1' ~2' ~3 ' Clearly ai
= ~i
; ~fli
= 2~
(b) , I f the triangle has sides whibh are differentiable curves, the
f~rmula
lao1 +!. 1
f
becomes
k ds
Ci
= 2~
If C is a simple closed curve, this
Jc
k ds
= 2~
~"
et ~
The prlof is not elementary and is due to H,Hopf , (0), If the triangle,is on a sphere, with great circle boundaires, we have this familiar formula
where Yi are the interior angles: Yi
=
~-ai '
Henoe
(d) If the edges are differentiables ourves, the formula becomes
7
- 8 -
J:°i
La. i + !
K ds +.!!!!
g
r
2
= 277
)
where Kg is the geodesio ourvature . (e) , Finally for an arbritary triangle on a surface (1 )
where KT is the gaussian ourvature, dA the element of area, and R the interior of the triangle. I oall this (1) the
G~u33-Bonnet t~.or'.
for
~
;oLiedron
(really a triangle here!) •
2. THE G.B . THEOREV FOR A COVrACT SURFACE . 2
Let X be a oompaot differentiable manifold of 2 dimension , There exists a differentiable triangulation . Applr (1) to eaoh triang1e and sum. Then V
= no.
of vertioes
=0
E
= no .
of edges
= ffITdA
F = no. of
!a. i = 277(E - V)
!IK~ds !ffKTdA !277
fa~ee
= 277. F
Then 277(E-V) +
II! T dA = 2n . F
or
where
( 7)
X.
is the Euler-Poinoare oharacteristio . Or better
II
K dA =,i2 T 2
X)
where 0 2 is the surface area of a unit 2-aphere .
8
- 9 3. THE INDEX THEOREM FOR A UNIT VECTOR FIELD . On X2 (c.e~pact) consider a unit vector field, u, lIhich is continous and differentiable except possibly a finite number of points P. Such vectcr fields can alllays be defined . We now discuss the in-
dex of the singularity
u at a singular point P.
~f
Consider a neighbourhoed of P, say R, which is bounded by a differentiable curve C e 1 , e 2 , At a point,
On R choose a differentiable frame : en C we
Q,
can write
where u 1 u 1 + u 2u 2
=1
.
Let ¢ be the positive angle from e 1
~9
u- Then
It is evident that
fcd¢
= 2w . I
where I is some integer.
In order to put our eepression for d¢ in a form invariant wit h respeot to the oh.loe of the frame, lie define
At once lie have : dtjJ
= Du
+
",1 2
Hence 2w I
= f c d¢ = Ic Du +
HOllever
and so 2w I =
Ic Du
+ 9
II,R 012
f c ",12
•
- 10 -
A direot oaloulation shows that
Thus from the/theorem of Stokes, I does not depend on C, as long as C enoloses a single singularity P. Hence we oall I the index
of the
siniu~ar.ity
of u at P
Let us now consider all of
X, Surround eaoh Pi with a Ci' whose interior is R . • Then from 1
Stokes
lie
have
also from 0) ~
(4)
f.
'1
Du
+
~
JJR K
-1 'T'
dA
= 211. n •
Combining these two equations gives us
This shows that II is independent of the veotor field u . Also from (2) II
= X.
We may prooeed in the opposite direotion. By a simple example we show that II
=l
. Then
from (4) - with a modifioation to han-
dle disoontinuities of u on C like outer angles - we prove (1), This is the best proof of (1) . Also direotly from (5) ve (3). This proves all our results .
10
w~
oan deri-
- 11 -
4. THE G, -B . THEOREII FOR A OOIlP.AOT IMBEDDED SURFAOE . Let X be oompaot and be imbedded in E3 . As before we oheese a looal frame suoh that e 1 and e 2 are tangent te X2 and e 3 is the outer normal. Then the Gauss map
(where
a2
is the unit 2- sphere in E~) is defined by the normal ve-
otor field e 3 , Frem a theorem of Kroneoker (see Hadamard, Appendix to vol . 2, Tannery "theorie des Funotions") the degree D ef this map (an integer) is given by the fermula ;
or D
= in II
X
2 KTdA
From a theorem of H , H~pf (llath Annalen 1925, pp.340-367) D
-12
Therefore we have another proof of (2), namely
JJ X2K T dA = 271 X'. . The above assumes, of oourse, that X2 is differentiable (0 0 ) is e~.ugh) . The situation ohanges if X2 is a polyedron with ourved faoes and edges . In the most elementary oase, let X2 be a reotilinear tetrahedron - what should the theorem be? It will olearly be a generalization of the angle-sum theorem fer a triangle . To obtain this, we prooeed for the triangle . We draw outer normals to the three faoes at eaoh vertex . There three normals are the edges of a trihedral angle, whioh oan be measured as a s.lid angle (the t.tal solid angle at a point is ohosen to be 471) . Oall these solid angles a 1 . Then the proof given for a triangle shows
11
- 12 -
that
Since
X= 2
for the BUl'face of a tetrahedron this is c,onsistent
with (2) above. If the polyhedron has curved edges and faces , the formula becomes (6 )
where
~
depends in a way
whi~h
will not be disoussed here on the
outer dihedral angle at each point of the corresponding edge . Formula (6) is important for another reason - it is the polyhedral G. -B , formula in a space of dimension 3 ' It remaina unchanged even when the polyhedron lies in a gener~l
x3-
there 1s no jnte,ra~
0-
ver the interior of the polyhedron . This is markedly diffel'ent from the case of two dimensions , and is characteristic of .dd dimensions of higher dimension .
5. THE GENERAL CASE OF THE G, -B. THEOREY, We have seen that there are three appreaches to the G, -B. theorem fol' a compact manifold : (a) Derive the polyhedral G. B, Theorem for dimension n . Triangulate Xn , and apply the polyhedral formula to each simplex
I
add up
and use various combinatorial relationships. (b) Derive the formula for the index at a singuLarity of a unit vector field on Xn , prove by a simple example as ' above that
X
~I =
and thus establish G.B .. (c) Imbed Xn in some euclidean space , ann use the results of Hopf and Kroneoker regarding the degree of the normal map to prove the theorem. Before discussing these methods of proof, we should first try
12
- 13 -
to guess the form of the result t4 be proved . We proceed on the assumption that X is a hypersurfa c e - We wish to find an appropriate generalization for K . It is m4st reasonable to define K
T
= det , (b, . ), since this is the expression suggested by the IJ
Gaussian map. Remember that sec6nd order minors of (b ij ) are intrinsic . Hence for n even, the Laplace expansion of det(b .. ) giIJ
ves us the following intrinsic expressi6n for ATdV : Ei
It dV T
"
,0
= 2 (s~ ' ;ln n 2 (n /2).
11 A ' "
i2
1\
A
/1
. 01n-l
in
(n eve n ) J
where dV is the volume element of Xn , Since this expression is intrinsic, it makes sense even if xn is not a hypersurface - so we adopt it in general. This leads us to
t~e
conjecture
( 7)
When n 1s odd, we are in a little diffioulty. The det(b ij )
~an
not be expressed intrinsically, and other considerations (which .appear later.) suggest that for n odd we define KT
X!
=0
. Since
0 for an odd dimensional compact manifold, we have (7) tri-
viall·y verified for n odd as well. I shall return to the questiO n of n odd a little later Let us now discuss various methods for proving (7). (a) The polyhedral formula of G. B., for Xn was proved by Allendoerfer-Weil (Trans . Am . Wath . Soc . 1943). The formula for the integrands for the faces of various dimensions, and the methods of combining outer angles of various dimensions when the polyhedra are added together are much too complicated to present there _ Reuders must be referred to the paper mentioned above . (b) On Xn (n even Or odd) define a unit vector field u whic h is continuous except at a finite number of points, p ., To det'.ine 1
13
- 14 -
the index fr u at P, proceed as abu'e . Choose a differentiable (n-l)-sphere S in Xn such that P lies in its interi.r R (a cell) In R choose a c.ntinuous frame e 1 ... e n . Then we wish to find an (n-l)-form depending on u, d1T=
JSn-l 1T
~~
and
KdV
n~
such that :
except at P
7'
n
1T
+ -fRK 7' d V =.Q. .I 2
}
n even,
will then be analogous tn the form Du used before . The expres-
sion for
1T
is very complicated and is
giv~n
by Chern (Annals of
Math 1944, p. 747) . This expression also appears (but not explicitly) in
Allend~erfer-Weil
and is discussed further in Allendoer-
fer (Bull.Am Math , Soc . 194S). For n odd, we have the formulas d1T = 0
JSn- l 1T -- -On •I 2 Proceeding as described for n=2 , we then derive
JXn K7' d V =.2!: 2 o =
LI ,
LI
n even n odd
Thus LI does not depend on u . A special example shows that in each case LI =
t . Hence
(7) is proved.
(c) When Xn can be imbedded as a hyper surface, the result (7) follows as for n=2 from the results of Kronecker and Hopf. The
probl~m there is the case where Xn is imbedded in En +N (N > 1) This problem was solved
>
independently by Allendoerfer (Am , J8urn
of Math . 1940) and Fenchel (J.Lond.n Math.Soc . 1940). We assume n even and N odd - the latter being no restriction.
14
- 15 At each point of Xn construct the SN-1 (unit sphere) in the normal space . !he points on all these SN-1 lie on a "tube", T, which is a hypersurface in En+N. A calculation shows that
where t U are local coordinates in the normal space and dS is the volume element of SN-1 , Therefore
JXn JSN- 1
det(tUb:,".)dSdV IJ
Xn
= -
on+N-1
t
X (tube)'
The integration over·S N- 1 can be carried through explicitly (see H, Weyl - Am . Journ . of Yath. 1939, although the results are actually due to Killing) . The result is
lBut
t (tube) = 2..1 (X n ),
so we arrive again at our result (7)
When this result was published, it seemed toc cover a special class of Xn, namely those which could be imbedded in En +N , From the theorem of Nash it is now clear that this proof is valid in all generality ,.
6. RELATED RESULTS OF CHERN
(Chern-Lashof
Am , Jourri . of Yath .
1957). Consider n even or odd \I 1\
and
The following the.rems are proved (a)
15
compac t COO "•
- 16 J < 3.0 n +N- 1) then Xn is h.meomorphic to an
(b)
n-sphere . (c)
If
J = 2 On+N-1, the X lies in an Bn+1 and is a convex
hypersurface in En+1. The c.nverse is als. true. These theorems generalise results of Wilnor and others for a closed curve in E2; IIKI ds ~ 2~
(a)
7.
(b)
If
IIKI ds < 4~ , the curve is n.t knotted
(c)
If
IIKI ds = 2~ ,Convel plane curve.
RELATED RESULTS OF WILNOR . Let Xn (n odd) b.e a cupact hyp·ersurface in Eo+1, Then Xn is
the b,undary .f a region R. The the.rem .f ij.pf (Wath . Ann -1926) sh.ws that 1
On where
t, (R)
1
Idet(b .. )dV ='t,(R) l.J
is the inner characteristio of R.
Wiln.r (Comm . Wath,Helv . -1956) sh.wed that f(R)
where
til is
.=
t'(X n ) mod 2
the semi-characteristio
x·= ~'-~1+" ' ±~k =~ it.
where
~i
n = 2k+1
[~ • +~ 1 +. .. +~ k 1
are the Betti numbers. Hence mod 2
Wilnor has further results on "immersions". His results have been generalized for n.n-hypersurfaces by Kervaire (Wath.Ann,-1956).
16
- 17 8. FINAL REMARKS . (a) Refer back to the formula : d Du
=
K dA r
This appears to violate Stokes theorem which would have required the left side to be zero . The reason it is not a contradictioniis that Du is singular at a point P in R. Nevertheless dDu is regular everywhere in R. This gives a clue which we shall exploit later to derive integers out of pairs of differential forms. (b) "For those who understand fiber bundles" : The form v above really belongs to the tangent bundle, and is regular there . dv
=-
K dA r
is tho inverse of a form on X with respect to the
projection map p . The pair (v, KrdA) are called "transgressive" .
17
CHAPTER III INTEGRALS OF DIFFERENTIAL FORMS 1. First we recalt s.me elementary n.tions
.n
X.n .
If
dw
If
w
Since
ddw
Poincar~
that da
= w.
=0 = da
I
Let w be an r-form
is "cl.sed"
W
w is "derived".
= 0, a derived f.rm is closed .
Lemma : If dw
= 0,
then locally there exist a such
The maxime regi.n in which a exist will be specified
later The.rem
: If
wn is derived on a c.mpact In I then
'tr suppose wn
=d
a n - 1 , Then by St.kes
But. (}X n is zer. ,
2.
INT~GRAL
I xn wn = O.
JXn wn = f!dxn a n - 1
FORWULAS ON Xn .
The above the.rem permits us t. derive many f.rmulas inv.lving integrals on Xn . We give a simple example . The.rem (Wink.wski). Let X2 be a convex surfa~e in E3 . Then I(H+pK)dA
=0
, where p fa the supp.rt functl.n .f the tangent
plane . Take e 1 and e 2 as tangent vect.rs and e 3 as the .uter n.rmal. Define x t. be the position vector .f a p.int .n X2 . Define a linear form B by the scalar triple produot :
Then
A direct calculati.n sh.ws that dB
= 2(H+pK)dA 19
~
Since de is derived.
Jx de = O.
19 -
Henoe the thetreD.f,lltws.
Ftr the prtblem tf Winkt.ski disoaseed in his leotures by Calabi,a similar methtd (Chern,Amer.Jtur.Wath.1957) shtw~ the unicity d
the stlutitn fir the oaee If 1 2C &3. Sinoe the IOly
80-
nalysis required is the thetrem tf Sttkes, this result requires tnly weak ctnditi.ns tn the differentiability .f the funotitns inv.lved . This result can be integrated as a unioity
t~etrem
f.r a
certain system .f partial differential equati.ns whioh was previ.usly kn.wn .nly f.r the analytloal oase. This system .f equati.ns is imptrtant in hydrtdyn.mics and Sttkee has used this meth.d .f differential getmetry tt derive imptrtant unicity thetrems tf this kind in applfed mathematios.
3. THE THEOREV OF DE RHAW. Let c r be a sm .. th (1. e. Coo, singular) ohain in a ctmpact Then f .
J.or wr
defines a sm.th ctchain,
The ctbtundary .f f, of, is an, r+l c.ohain suoh that
Hence
=J
Theore. : If w is ol.sed, f
or
F.r in this oas~ of: O.
wr is a ctoyole.
Further, if wr : da r - 1, then
and
Hence og : f
Theore.
If w is derived, then f
20
: Jor wr
is a c.b.undary.
- 20 -
AIsl there is the the.rem :see Whitney,
~e.metric
Integrati.n f.r
pr .. t)
n.or ••
If f
= fc
aP
f
Let B(X,R) be the
g
= f /3q , then the cup-pr.duct cg
p
cohlm.lt~y
V g
= Jc
a P" /3q p+q
f
algebra derived fr.m the sml.th f.rms
whlse respective differences are derived . Fir each w define the singular ctchain hw by h,w . c
= Jc w
Then the ab.ve argument may be summarized by the statement : h induces a htmlm.rphism h .n c.htm.l.gy classes h"
B(X,R)'" H(X,Il)
The the.rem .f de Rham states that h' is an algebraic is.morphism .f H(X,R) .nt. tbe singular c.htm.l.gy algebra (cup-product)
H(X,R) .f X. De Rham stated the the.rem s.mewhat differently, Clnsider the h.m.l.gy with integral otefficiente in X, Then fir dimensi.n r there is a set .f r-cycles zl" , zp which f.rm a h.m.l.gy basis f.r the spaoe .f r cycles. Given anr-ftrm wr , the integrals
f
wI' 2i
are called the fundamental peri.ds tf w , De Rham stated (1) If dw
= 0,
and if its fundamental peri.ds are all zer.,
then w is der!ved. (2) Given a set .f fundamental peri.ds th.re exisu a cl.sed f.rm w whioh realises these peri.ds , The the.rem .f De Rham Is the fundatitn .f much subsequent w.rk .n otnnectl.ns between ge.metry and t.ptllgy. It has, however, a fundamental weakness! , i.e . it refers .nly the c.h.mology with real c.efficiente. In .ther w.rds usir.g differential forms
21
- 21 -
it permits us to describe that portion of the homology of X which does not involve torsion . The really
int~resting
geometric homo-
logy, however, is that of homology with integral coefficients involving torsion , It is the object of the next lecture to describe my approach to a generalization of de Rham theorem which handles this situation .
22
CHAPTER IV COHOMOLOGY WITH INTEGRAL COEFFICIENTS Tbis &bapter is based on the paper "On the Cohomology of sm •• th manifolds" Allendoerfer-Eells; Comm.Vath . Helv . 1958 . In our proof of the formula for the index of the singularity of a vector field, we have already seen the basic index, which indeed gave rise t. this the.ry, namely : In order to get an integer in terms of the integrals of differential forms, we can (and indeed must) use singuLar differential forms . We will use a paLr of such forms are regular, d~
=e
,
and
Ic
e - ~c w
(e,~)
such that/where they
= I . We begin with a very
simple example of such a pair. In the plane let ~
Then gular, dw
~
- 1 -"-11
xdy - ydx x 2 + Y2
is regular except at the origin; and where it is re-
= O.
Moreover
Jc w =
o {
i f C does not include 0 •
± 1 if C inoludes 0, the sign depending of the orientation of C.
As an elementary example (to fix the ideas), let us to consider tbe integral coh,mology of a compact X2 . We triangulate X2 into simplices U1 ... u p , and define a 2,dimensional cochain by scribing the values (integers)
ai
= f ,ui
. We wish to show how
can be realized by differential forms. Consider a simplex
U
and neighbourhoods
23
p ~ e
f
- 23 Define c.ordinates x, y in U3 so that 0
€
~"
Then define
ru -. - Trr 1 We need
:0
extend its domain of definition over all X 2 , To do so
¢
we use the standard "bump function"
¢
00
is C
of U3 and
Define
on
U3
on
x-u 3 •
Then w is 0 00 on X - 0 and singular at 0, Also define is in fact 000 on X, since at 0 Then
e Can
dw
= e,
which
be taken to be zero.
Je-IG.I=l ~
~O"
But for any other simplex of the triangulation, p
Je-f w=o p
top
0
In this way define a pair (e, ,w,) for each simplex ~l" 1
1
the preassigned cochain have the integral values a,
1
define
~
= f,~,. 1
Let Then
= ~ia,w, 1 1
The pair (J,~) then realizes the cochain f in the sense that
2, THE GENERAL CASE. This some method olearly works for n-dimensional cochains in Xn , where we choose
24
- 24 -
Adx • n
For ooohains of dimenehn r (1'rl
Henoe smooth in En _ Bn-r
8r
=
f:
w
in
En _ Bn - r
in En- r, i f
I, I>1
Henoe smooth in En _fJa n - r . Then {
0 if
(T
dues not i nterseot Bn - r
± otherwise
From pairs of forms of this kind we oan build a pair (~,~)J representing a given r-ooohain by the methqd explained above.
25
- 25 -
3. SYSTEMATIC DISCUSSION. We now begin a systematio treatment of suoh pairs. Let
(8r,~r-1) be a pair of forms on Xn suoh that (1) 8 r is smooth exoept on a singular set a smooth, locally finite
polyh~dron
e(8) which lies on
of dimension
(2) ~r is smooth exoept on a singular set
e(8) to be a olosed subset of
Define
n-r-1.
e(~) whioh lies on
a smooth, looally finite polyhedron of dimension quire that
~
~ n-r ,
{We re-
e(~)J.
a chain c is admissible for the pair (e,~))if
=J
Define: R [(8,w),oJ
c
8 ..
I
1)0
w
called the "residue".
We then further require that
(3)
R[(8,~),cJ
be an integer for every admissible chain c
Then (8,w) are called a (Z,r) pair (Z standing for the integers).
Remark. Instead of the integers Z , we can consider any integral subdomain of the reals A, in particular Z , or Z . Then 2 p we have (A, r) pairs. The theory to follow still holds, but for simplioity in exposition we consider Inly Z We shall need the following deformation theorem : If 0t (O~t~l) is a smooth deformation If Co whioh is admissible for
(8,~),
then R[(e,~),ooJ = R[(8,~),c1].
Now we wish to construot a oohomology algebra of our pairS
(8, ~) Define
where
26
_ 26 _
V
e(8 2 )
= e(w1 ) V
e(w 2 )
e(8 1 + 8 2 ) = e(8 1 ) e(w 1 + ( 2 ) a-( 8, w) :
( a 8, a CAl )
Uhfortunat~ly
se
fo'!'
these do not form a Z-modu1e, fot the inver-
of (8,w) is not defined if e(w)
f¢
To get around this difficulty we intr.duce the equivalence classes :
(6 1 ,w1 )
~
(8 2 ,w2 )
R[(8 1 ,CAl 1 ),c]
for all
0
if
= R[(8 2 ,w 2 ),c]
J
admissible for both pairs . This is an equivalence rela-
tion : symmetry and reflexivity are trivial. The transitivity requires the above deformation theorem. Denote such an equivalence class by [8,w]. These do form a Z-module. For the differential operator we define . d [8,w]
= [0,8]
. Then
dd
=0 .
To introduce produots we oonsider [8 11\ 8 2, w1 A8 2 ] with singularities e(8 11\ 8 2 )
= e(8 1 ) IJ e(8 2 )
e(w 1 1\8 2 )
= e(w 1 ) V e(8 2 )
The dimensions of these singularities are larger than that allowed by our definition of a(Z,r) pair above . We can deal with this problem either by oonsidering improper integrals for chains which intersect the Singularities or by reducing the dimensions of the singularities by 100al deformations of the forms. We omit tne details.
27
- 27 In summary we now have a cohomology algebra of classes of
[e, wjl
which
'lie
write
H(x, Z).
We can now state the basic theorem :
The:-e is an isomorph.ism : H(X,Z)
~
H(X, Z).
The proof is a slight modification of the modern proof of de Rham's theorem using sheaves(see the book of Hirzebruch). This requires the proof of a Poincar6 lemma : If
d [e,w] = 0 , then locally there exists a ['T1,~] such that
re,w] = d ['T1,~] This can be proved by the standard deformation argument, but the position of the singularities must be watched closely . It is at this point that we need the hypbthesis that the
~ingularities
lie on polyhedra.
3 , REFORMULATION
We can reformulate the theorem in another way ,
Let us recall that for the integral homology of X there are two kinds of fundamental cycles (1)
The ordinary cycles
tfc = 0 •
(2)
The cycles mod p : chains c such thatlO0
= m.c',
where
m
is an integer . These are "cycles mod m". We can find a base of the homology of a given dimension in X of the form C
where c Co) j
(1" ) 1 ..... 1
are ordinary cycles and
C
(1'. )
a.
( T. )
c. J
1
1
1
are cycles mod Ti
Further an integral cochain is uniquely determined if we know its values (integers) on each of these chains. We call the integrals of a pair [e,w] over the chains its fundamental periods, Hence the theorem can be stated :
28
- 28 _
(1) If d[e,w]
= 0,
and if all its fundamental periods are zero,
then (e,w] is derived . (2) Given a set of fundamental peritds there is a closed [e,~l
whose integrals have the presoribed values .
4. THE THEOREM USING A TRIANGULATION, When we triangulate X, the theorem has other interesting aspeots, Let K denote the complex of the triangulation and K' the dual oomplex . Then we choose the singularities of
e and
~ to lie on K·,
Let_Cr(K,Z) be the module of classes tf pairs (9,w] and Cr(K,Z). the module of Simplicial cochains of K with integral coefficients. Theorem . Then the map h
h([e,w])
=J e
c
o
is an isomorphism onto, satisfying dh
=
-14.
sinoe i f the residues of (e,w) are all zero, from the
argum~nt
(ctnstruction) given in
h .~~ 1'+1 = ~
Aist
12
-
Jw
(bc
It is an isomorphism
(e,w] = 0: it is onto above ,
e
Cr +1
and
h(O,eJ.C rt1 =
- ~e 1'+1
Therefore h induces an isomorphism ofOr(X,Z)HHr(K,Z)
= Hr(X,Z) .
Coronrsries : I . Every cthomoltgy olass of nr(K,Z) has a representative
(e,w) with e defined (regular) and closed on I . This
e is
the form given by the de Rham theorem for a coho-
mology class having integral periods on ordinary cycles . The form
w
giv~s
us the opportunity of determining its values on the chai ns
which are cycles mod p .
29
- 29 -
2. Any smooth olosed r-form on X is derivable from a smooth (r-l)-form with singularities lying on an (n-r)-oyole .
5. THE CASE OF A RIEMANN VETRIC . Suppose that our Xn is a smloth (or analytio) olmpaot manifold, oarrying a smooth (or analytio) Riemann struoture . Using a teohnique due to de Rham we oan introduoe for x
"Gre~n's
r
forms", whioh are symmetrio double forms
y, satisfying d g !x,y) x r
= Sy g r +l(x,y),
By an argument similar to that of de Rham (se the lie
gr(x,y)
pape~
for details)
prove :
Theore. : Every oohomology olass of Hr(X,Z) oan be represented by a (Z,r)-pair (B,w) suoh that B is harmonio on X; moreover B is
unique in its oohomology olass This theorem is, then, a generalization of Hodge's theorem . It suggests that the methods of Hodge oan be adapted to use this theorem to obtain information on the torsion of algebraio varieties
30
CHAPTER V PONTRJAGIN CLASSES ON
xn
1. TANGENT CLASSES. We recall that O~ as defined earlier are defined in each coorJ
dinate neighborhood . They are not global forms, but certain combinations of them are global - we call these invariant forms. We have already seen one example of an ' invariant form : i
A 0 i n-1 . n
This is an invariant under proper orthogonal transformations of the underlying frame and hence is well-defined on an orientable manifold. Other examples are 4k
= Tr (0 II
0
~ n
A" J: A U)
2k factors (The trace of a product of an odd number of 0 is identically zero).
64k is a form of degree 4k, having the properties (1) It is invariant under change of frame, and hence is globally defined . (2)
d(6 4k )
=0
.
This is a consequence of Bianchi's identity . Hence 6 4k represents a cohomology class with real ooefficients.
(3) This cohomology class is independent of the metric, and ir.deed of the oonnection (A.Weil) . Hence it depends only on the differential
structur~
of X .
(4) Its values on a set of fundamental cycles of X are integers
31
- 31 -
(Putrjagin) . (5) Considered as forms in the bundle of tangent frames, 6 4k are derived (Weil). If the bundle of frames has a cross-seotion, i. e . Xn is parallelisable, this oohomology class is zero. These olasses are called the tangent Pontrjagin Charaoteristic classes of Xn . Similar olasses oan be defined for any bundle on Xn ,
2 . THE NORMAL CLASSES . We reoall the equations of struoture of XnC
En+N
~
dw~ + w~ /I w~ + wk 1 O' 1 J
A wO'
:;:
i
0
dwC; + wO' 1 j
A wj + wO' A wP = 0 P i i
dwO' + w,:"
wT = 0 A wj + wO'Ii T
P
P
P
J
where nk = i
- wkO' A wO'i auP
We define the • normal ourvature" nO' = dWO' + wO' T P P
A wT = P
- we: II w~ J
This satisfies tae Bianohi identity , In terms of these forms we oan also.4efin' forms invariant under orthogonal transformations in the normal spaoe . Consider first Xn
e
E2n (n even). Then we oan form O'
.. A n n-1 O'n
n= 2 n / 2 (n/2)!
A proof identioal to that for G. B. shows that '
~ J n = LI
xn
32
- 32 -
where I are now the indices of the singularities of a unit normal vector field . This invariant is called the Whitney invariant; it appears to depend upon the imbedding. If ~I
= 0,
there exists a
global field of normals to Xn . For XnC En+N, lie can aleo define the normal Pontrjagin classes by means of the invariant forms 4k ~ N •
These are also closed and hence define cohomology classes . They are invariants of the differential structure of Xn as a result of the theorem (proof by Kobayashi) :
Theor,. :
Preof
since
Since
=
33
CHAPTER VI STIEFEL-WHITNEY CLASSES 1 . INTRODUCTION . This chapter is based upon an unpublished manuscript "A generalization of the Gauss-Bonnet Theorem" by James Eells which is itself an extension of my own paper (Annals of Math . 195Q). The Stiefel-Whitney classes Wr(X) are cohomology X with se) for
coe~ficients l~r: B - E telle que
PIP soi t
l'identite. L'etude de l'existenoe de seotions dans un espaoe fibre oonduit
a
la theorie des obstaoles (voir des exemples dans (1)).
Une struoture fibree subordonnee a
la structure E(B,F,G,H)
est une struoture fibree E(B,F,G',R') ou G' est un sous-groupe de Gi alors H' est un sous-eepaoe de H. On demontre (11) que l' existenoe d'une tel Ie struoture subordonnee (probleme de la reduction du groupe structural dans la terminologie de Steenrod est equivalente
[26V
a l'existenoe d'une seotion dans l'espace fibre
RIG', de base B, dont les fibres sont isomorphes a l'espaoe homogene GIG'. On trouvera des exemples d'espaces fibres dans la suite de oet expose.
4. ESP ACES ETALES . FAISCEAUX. Un espaoe topolo.g ique E' est dit etale (14), (15) dans un espace topologique E s'il existe une applioation p de E' dans E verifiant la propriete : tout x' Eo E' U tel que la restriction de pace
possed.e un voisinage ouvert
voisi~age
soit un homeomorph i-
sme. L'ensemble des inverses des restrictions de p jouissant de cette propriete forme un atlas oomplet de E sur E' compatible ayeo le pseudogroupe r
des applioations identiques des ouverts o de E. Inversement soit un atlas oomplet de E sur E' compatible avec Ie pseudogroupe roi si fi et fj sont deux oartes looa l es de
46
- 7 P.Libermann
buts 0 '
i
et 0 '
j'
dans
0'
i
n0 '
f- 1
les applications
j
~e
t f- 1 j
co!ncident, d'ou une application bien d~termin~e de E' dans E dont la restriction a. 0 '
i
est (f )-1 . Done pour que E' soit i
etaLe dans E, iL faut et iL suffit qu'iL existe un atLas compLet de E sur E' compatibLe avec Le pseudogroupe des appLications identiqueJ 4e! ouverts de E. Etant donnee une application continue f d'un ouvert 0 d'un espace topologique E dans un espace topologique EI, on appelle jet local (ou
~)
en x de f et on note
/'f (ou x € D) la x classe des applications continues d'un ouvert de E dans E' qui coincident avec f dans un vOisinage ouvert de x ; on designera source (resp.but) de jAf Soit
A
Ie point
x(resp.f(x».
x
J (E,E') l'ensemble de tous les jets locaux de E dans
E'; soit f, de source 1 I application jAf : x
looale de E dans
UC E
~
/f x
une application continue dans E'; de
U
dans
JA(E,EI) est une carte
JA(E,EI); l'ensemble de oes oartes looales est
un atlas de E sur JA(E,EI)
compatible avec Ie pseudogroupe des
applicatione identiques des ouverts de E oar
jAf x
= jAx
fl
en-
I
JA(E,E') une topologie
tratne x' = x. Cet atlas transporte sur
A.
engendree par les buts des cartes jAf
tout ouvert de J (E,E')
est Ie but d'une carte looale ou une reunion de tels buts : ~A spaoe J (E,E'), muni de cette topoLogie, est etaLe sur E par l'application a qui a. tout source. L'espace
X ~ JA(E,E') fait correspondre sa
JA(E,E') n'est pas separe en general .
Soit L(E,E') une classe locale d'applications de E dans E, o'est-a.-dire un ensemble d'applications d'ouverts de E dansIE' rifiant la oondition : si U =
Ui U., 1
pour qu'une application con-
tinue de U dans E' appartienne a. L(E,E'), il faut et il suffit que sa restriction a. chaque ouvert de U. appartienne
47
~
1
a
L(E,E').
- 8 -
P.Libermann
L' ensemble
~
des jets locaux correspond ant
a
tous les f £ L(E, L')
est un espaoe etale sur E : o'est un sous-ensemble ouvert de A. J (E, E') j si f E L(E, E') a pour source U, l' application
est une seotion s de
~
tion oontinue de U dans
aU.dessus de U
l
(o'est-~-dire
A. x
x- j f
une applica-
telle que as soit l'application iden-
tique de.U)j inversement s.oit s un.e seotion .de
diff~rentiel1e6
d ' alg~bre
mani~re
exterieure west un t'!
dijjerentie~~e
ext6rieures sur V
n
ext6risure {on d6finit Ie pro-
6vidente) . On d6finit Ie produit int6-
rieur d ' un champ de veateurs
X et d ' une p-forme diff6rentielle
ext6rieure w i a est la (p-l)-forme diff6rentiel1e ext6rieure : p
x ~ i(X )(w ~ )
x
•
x Pour les formes diff6rentiel1ea ext6rieures, on d6finit l ' apdr3p
teur d (ou differ e ntiation ext6rieure); yoir Allendoerfer [1), Lichner.owicz (22), at cet operateur jouit des propri6tes : 1) d augments Ie degre d ' une unite
2) d est ·lineaira 3) d(.w " w ) - dw /I. w p q p q
4) dd
=
0
61 fest une
O~forme
(o'est .. a. .. dire une fonction numerique),
la forme df est la fonction continue diff6rentielle
bll
~
x
~
d f (ou d f e st la x
x
x, de f d6finie dans §,5) , 8i xl) . . " xn
54
SOLt
des
- 15 P.Liberm'lnn
coordonnees locales au voisinage d'un point de V , df peut s en
crire localement : df = 2: 1!.- dxi. (lx i
On peut definir de meme un champ
~oca~
de tenseurs : c'est
un relevement oontinu d'un ouvert U ('lppele source du champ) d'lns I' espace des tenseurs de tYP,e donne. En particulier on designera par transform'ltion infinitesim'lle locale (ou plus brievement t.i.l.) un champ local de vecteurs. On supposera dans l'l suite de cet expose que tous les champs et toutes les applications sont de classe em (beaucoup ~es r'sul~ tats enonces sont valables sous des hYgotheses moins restrictives une application biunivoque de cl'lsse e sera designee par diffeomorPhls Soit fun diffeomorphisme de source
UCV, but f(U)C,V; f n . n
transforme tout champ de vecteurs X de source W en une t.i.l. X' (de source
f(Uf'\ W) : X' = 'fI Xf -1 (ou fI est Ie prolongement de
faux vecteurs defini dans §5); toute forme differentielle west transformee en une forme w'; f(x) ~ w (j1f)-1; de meme f se prox x longe a tout autre champ local de tenseurs de type quelconque. A tout
de vecteurs X (que l'on suppose d'abord defini
ch~mp
globalement sur V ) est associe "op'rateur ddrivde de Lie e(X) n
que l'on peut definir de l'l maniere suivante : La solution du systeme differential defini sur la variete V par Ie champ X n
definit une application: que f(x,o) = x
(x,t)
~
f(x,t) de VnXR dans Vn telle
= X", quel que soit x . Pour t suffiot t =0 samment voisin de 0, chaque application f~ (definie par f t (x) =
= f(x,t)
et
-af (~
) est un
diffeomorphism~
pe de transformations
a
et X detinit un noyau de grou-
un parametre (Ie champ sera dit integra-
'Ole si X d.efinit un groupe, ce qui a lieu notamment pour tout champ si Vn est compacte). Si T est un champ de tenseurs sur Vn , il est transforme par f t en Tt ; par aefinition
55
- 16 P.Libermann
e(X)T
(8 )
e(X)T est un champ de tenseurs de
m~me type que T.
En particulier
on demontre (voir E.Carhn [4] et H.Cartan (5]) que 1 'on a :
e(X)Y = -e(y)X = (X, yl,
(9)
OU [X, y] est Ie crochet des deux champs X et Y. Si au voisinage d'un point X (resp. y) a pour composantes Xi (resp.y i ), [X,Y] a pour composantes
Zi = L(Xi oyi _ yj j
ox j
OX~).
ox J
De mame pour une forme differentielle exterieure w sur V
n
on demontre la relation
8(x)w
( 10)
= i(X)dw
+ di(X)w
(OU i(X) est l'operateur produit interieur). L'operateur e(X) jouit des proprietes suivantes
e(X) d
= e(X)e(y)
e.(Y) e(x)
dx, y] = e(x) iCy)
i (y) e(x)
e[x,Y] (11 )
= de (X)
e(AX)w
= Ae(X)w
+ (e(X)A)w
ou
A est
une fonction numerique.
Pour toute fonction numerique f, on a e(X)f x
= i(X)df
j
en chaque
V, (i(X)dr) n'est autre que X f (voir §5); (i(X)df) ne n x x x depend que du vecteur X alors que [X,Y] et (e(X)w) (ou west x x x une forme de degre ~1) est fonction du jet j!X du relevement €
X de V
n
On
dans T(V ). n
defi~it
de mame l'operateur derivEe de Lie pour les t.i. 1.
X car X definit un noyau de groupe de transformations au voisinage de chaque point de sa source. Si X a pour source
U
c: V
n
,
pour un champ local de tenseurs T, de source U', e(X) T a pour iilource (eventuellement vide) U AU'. SoH X une t. i. 1. sur V , n
56
- 17 P.Libermann
de souroe
/'xx
U!I x . Le jet LocaL
du reHvementX de V dans n T(V ) sera appele ger1lte de t. i. L.; oe germe dHinit un germe de n
groupe
~
un
param~tre.
On definit de mIme un germe de tenseur
de type queloonque. Un ger1lte en x de t.i. L. definit un operateur
deriuee de Lie pour Les ger1ltes en x de tenseurs; par definition : 8(jA.X)/'-T = /(8(X)T) x x x X et Ie ohamp de tenaeurs T sont definis au voisinage de x.
(12) o~
On definit en partioulier A.
=j
x
8(j~X)(j~Y) = / x [X, y)
et
(8(X)w).
Sur l'operateur derivee de Lie en oaloul tensoriel olassique, voir K. Yano [27).
7.
pSEUDOGROUPES INFINITESIMAUX. FAISCEAUX D'ALGEBRES DE LIE
ASSOCIES [21).
¢
DEFINITION 1 :
etant l'ensemble des ouverts d'une topolo-
gie (; sur V , on designera par pseudogroupe infini Us i1llaL, de n
topologie sous~jaoente ~ un ensemble E de t.i.l. verifiant les axiomes suivants : 1) tout Xli: E
U Ii ¢.
a pour souroe
2) si X et X', de souroes U et U' appartiennent a E, alors Ie oroohet [X, X'
1
et 130 t. i.l. pX + /-LX'
(quelles que soient les
oonstantes reelles p et /-L), de souroe Uf'\U', (eventuellement vide), appartiennent 3)
U
si
partienne
a
E
~
= VU i
i
, pour que 130 t. 1. L
E, i1 faut et il suffit que sa restriction
Ui appartienne
a
a
a
ohaque
E.
L'ensemble des noyaux de groupes dant
X, de SOUFce U, ap-
a
un parametre oorrespon-
tous les X€ E engendre par composition et reunion un pseu-
dogroupe P(E). Inversement soit
r
un pseudogroupe de transforma-
57
- 18 -
P.Libermann
tions sur V ; on d'signera par pseudogroupe infinitesimat attach~
ar
n
et l'on notera
t'simal tel que
Q(r)
Ie plus grand pseudogroupe infini_
p(~(r))C r, en prenant pour r et !l(r) la meme
topologie sous-jacente. Soit E un
pseu~ogroupe
J "- (E) l'ensem-
infinit'simal et soit
ble des germes de t.1.1. definis par tous
lesX~
E. L'espace J"-(E)
est 'tal' sur E (cfr.§4); des axiomes pr'c'dents, il r'sulte que J "- (E) des germes de source x est une algebre de Lie;
l'ensemble
x
la loi de composition (X, Y)
~ lx, y)
'tant continue, J"-(E) est un
faisceau d'atgebres de Lie que l'on d'signera par faisceau associe
a E.
Si tout X~ E est une t. i.
let
globale, E est une algebre de Lie
JA.(E) est. Ie faisceau constant V x~ . Nous ne supposerons n
pas cette condition r'alis'e dans 111. suite. Le. pseudogroupe infini-
prolong~
t'simal E sur Vn se
en un
ps.udogroup~
infinit'simal .E(P,q)sur
la vari't' T (p, q) (V ) des tenseurfj de type (p,' q); en effet tou~ X€.E defini\ n
un noyau de groupe de transformations f
t
se prolongeant en transfor-
mations
f(P, q) sur T(P, q) (V) ce qui d'termine une t.1.1. X(p, q) t n sur cette derniere vari't'; X(p,q) sera appel' prolongement de X.
On v'rifie que les prolongements de tous les pseudogroupe infinit'simal
que de E; pour tout et
J~(p,q)(E(P,q))
Xf E
forment un
E(P,q) appel' protongement hoLoedri-
) . . T (P, q) (VIes algebres de LIe J "-(E) . p,q n x sont isomorphes (x est suppos' la projection x()~
de x (p, q) sur V ) . n On peut defini r de meme Ie prolongement de E
~
s . J (V n ' Vn),
espace des s-jets de V dans V n n On d'signera p'>r JS(E) l'ensemble des s-jets des relevements loc'>ux de V
n
d'>ns T(V ) d'termin's par tous les
X ~ E; de I,>
n
d'finition de E, il r'sulte que pour tout
x4 V
n
l'ensemble
JS(E) de t ous l es jets de source x forme un esp'>ce vectoriel et x
58
- 19 P.Libermann
cet espace est de dimension finie
d; en effet JS(E) est un x x sous-espace vectoriel de ds(V ), espace vectoriel des s-jets de x n source x de taus les relevements 10caux de V dans T(V ) et n
n
est de dimension finie: Sl. x 1 , ... ,x n
sont des coord on-
nees locales sur V dans un voisinage U de x, une t.i.l. X n lin n 1 n composantes X (x , ... ,x ), ... , X (x , ... ,x'): le jet jSX x 1 n est defini pOlr les scalaires (x , ... , x ) ,
01
pour
x
oo.t
+ ,",
+ a..
X
)
""vec a +... +a = ~d = O,l, ... ,s). n 1 a. x COXl)o.l ... (ox"') ~ 8i s'> s, dim. JS'(E) > dim. JS(E) et par suite dim./(E) x x x dim.Js(E). x L'espace vectoriel JS(E) n'est pas en general une algebre
~
x
de Lie car le crochet definit une application de JS(E)i JS(E)
x
dans J s - 1 (E) x
j Sx x
JSY x
et
:
x
js-1[X, y] est determine par les jets x . , 1 j y [X,y] a comme composantes L(X jOX ). puisque ox j ox J le jet
2.I:.._
DEFINITION 2. Le pseudogroupe infinitesimal E est dit com-
plet d'ordre q si E est l'ensemble des solutions de Jq(E), q etant le plus petit entier a jouir de cette propriete. Alors E est 1 'en'semble des solutions de
J q ' (E) pour q" >' q.
Au voisinage d'un point de V , Jq(E) est l'ensemble des son
lutions d'un systeme
Lq
d'equations aux derivees partielles d'or-
dre q, lineaire, homogene par rapport aux Xi at. a leurs derivees CD
partieJles, les coefficients etant des fonctions C
des coordon-
nees locales: Ie system.e Iq est compLete11lent integrable c'esta-dire
a
tout jet
a q determine par L q , correspond au mains une
solution de L q . 8i s < q, JS(E) est defini localement par un systeme si
L S obtenu en supprimant dans Iq les equations d'ordre >s;
s > q, JS(E) est defini localement par un systeme L S obtenu
59
-
20 -
P.Libermann
par ddrivations tot ales successives des equations de Iq aux
qUe
les on ajoute dventuellement de nouvelles equations pour Ie ren_ dre completement intdgrable. En chaque point x, dim JS(E) = x dim ~s(V ) - pS ou pS est Ie rang en x du systeme IS considere x n x x comme systeme lineaire; si pS est independant de x, Ie pseudogroux pe infinitesimal est appele pseudolroupe infinit~simaL de Lie : J q ( E ) est a lor sun e sou s - v a r i e t
e
