Topics in differential geometry (web draft, April 2007)

413 Pages·2.711 MB·English

by  Michor P.W.

Topics in
Differential Geometry
Peter W. Michor
Institut fu¨r Mathematik der Universit¨at Wien, Strudlhofgasse 4, A-1090 Wien,
Austria.
Erwin Schr¨odinger Institut fu¨r Mathematische Physik, Boltzmanngasse 9, A-1090
Wien, Austria.
[e-mail]
These notes are from a lecture course
Differentialgeometrie und Lie Gruppen
whichhasbeenheldattheUniversityofViennaduringtheacademicyear1990/91,
again in 1994/95, in WS1997, in a four term seriesin1999/2000 and 2001/02, and
parts in WS 2003 It is not yet complete and will be enlarged.
TypesetbyAMS-TEX
ii
Keywords:
Corrections and complements to this book will be posted on the internet at the
URL
http://www.mat.univie.ac.at/~michor/dgbook.html
DraftfromApril18,2007 PeterW.Michor,
iii
TABLE OF CONTENTS
CHAPTER I Manifolds and Vector Fields . . . . . . . . . . . . . . . 1
1. Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . 1
2. Submersions and Immersions . . . . . . . . . . . . . . . . . . . 13
3. Vector Fields and Flows . . . . . . . . . . . . . . . . . . . . . 18
CHAPTER II Lie Groups and Group Actions . . . . . . . . . . . . 37
4. Lie Groups I . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5. Lie Groups II. Lie Subgroups and Homogeneous Spaces . . . . . . . . 52
6. Transformation Groups and G-manifolds . . . . . . . . . . . . . . 56
7. Polynomial and smooth invariant theory . . . . . . . . . . . . . . 72
CHAPTER III Differential Forms and De Rham Cohomology . . . . . 85
8. Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 85
9. Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . 97
10. Integration on Manifolds . . . . . . . . . . . . . . . . . . . . . 105
11. De Rham cohomology . . . . . . . . . . . . . . . . . . . . . . 111
12. Cohomology with compact supports and Poincar´e duality . . . . . . . 120
13. De Rham cohomology of compact manifolds . . . . . . . . . . . . 131
14. Lie groups III. Analysis on Lie groups . . . . . . . . . . . . . . . 137
15. Extensions of Lie algebras and Lie groups . . . . . . . . . . . . . 147
CHAPTER IV Bundles and Connections . . . . . . . . . . . . . . 155
16. Derivations on the Algebra of Differential Forms
and the Fro¨licher-Nijenhuis Bracket . . . . . . . . . . . . . . . . . . 155
17. Fiber Bundles and Connections . . . . . . . . . . . . . . . . . . 163
18. Principal Fiber Bundles and G-Bundles . . . . . . . . . . . . . . 172
19. Principal and Induced Connections . . . . . . . . . . . . . . . . 188
20. Characteristic classes . . . . . . . . . . . . . . . . . . . . . . 207
21. Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
CHAPTER V Riemannian Manifolds . . . . . . . . . . . . . . . . 227
22. Pseudo Riemann metrics and the Levi Civita covariant derivative . . . 227
23. Riemann geometry of geodesics . . . . . . . . . . . . . . . . . . 240
24. Parallel transport and curvature . . . . . . . . . . . . . . . . . 248
25. Computing with adapted frames, and examples . . . . . . . . . . . 258
26. Riemann immersions and submersions . . . . . . . . . . . . . . . 273
27. Jacobi fields . . . . . . . . . . . . . . . . . . . . . . . . . . 287
CHAPTER VI Isometric Group Actions and Riemannian G-Manifolds . 303
28. Homogeneous Riemann manifolds and symmetric spaces . . . . . . . 303
29. Riemannian G-manifolds . . . . . . . . . . . . . . . . . . . . . 306
30. Polar actions . . . . . . . . . . . . . . . . . . . . . . . . . . 320
CHAPTER VII Symplectic Geometry and Hamiltonian Mechanics . . . 343
31. Symplectic Geometry and Classical Mechanics . . . . . . . . . . . 343
32. Completely integrable Hamiltonian systems . . . . . . . . . . . . 364
33. Poisson manifolds . . . . . . . . . . . . . . . . . . . . . . . . 369
34. Hamiltonian group actions and momentum mappings . . . . . . . . 379
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 408
DraftfromApril18,2007 PeterW.Michor,
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DraftfromApril18,2007 PeterW.Michor,
1
CHAPTER I
Manifolds and Vector Fields
1. Differentiable Manifolds
1.1. Manifolds. A topological manifold is a separable metrizable space M which
is locally homeomorphic to Rn. So for any x M there is some homeomorphism
∈
u : U u(U) Rn, where U is an open neighborhood of x in M and u(U) is an
→ ⊆
open subset in Rn. The pair (U,u) is called a chart on M.
From algebraic topology it follows that the number n is locally constant on M; if
n is constant, M is sometimes called a pure manifold. We will only consider pure
manifolds and consequently we will omit the prefix pure.
A family (U ,u ) of charts on M such that the U form a cover of M is called
α α α A α
an atlas. The mapp∈ings uαβ :=uα◦u−β1 :uβ(Uαβ)→uα(Uαβ) are called the chart
changings for the atlas (U ), where U :=U U .
α αβ α β
∩
Anatlas(U ,u ) foramanifoldM issaidtobeaCk-atlas,ifallchartchangings
α α α A
u :u (U ) u∈(U ) are differentiable of class Ck. Two Ck-atlases are called
αβ β αβ α αβ
→
Ck-equivalent, iftheirunionisagainaCk-atlasforM. AnequivalenceclassofCk-
atlasesiscalledaCk-structureonM. FromdifferentialtopologyweknowthatifM
has a C1-structure, then it also has a C1-equivalent C -structure and even a C1-
∞
equivalentCω-structure,whereCω isshorthandforrealanalytic,see[Hirsch,1976].
By a Ck-manifold M we mean a topological manifold together with aCk-structure
and a chart on M will be a chart belonging to some atlas of the Ck-structure.
But there are topological manifolds which do not admit differentiable structures.
For example, every 4-dimensional manifold is smooth off some point, but there are
such which are not smooth, see [Quinn, 1982], [Freedman, 1982]. There are also
topological manifolds which admit several inequivalent smooth structures. The
spheres from dimension 7 on have finitely many, see [Milnor, 1956]. But the most
surprising result is that on R4 there are uncountably many pairwise inequivalent
(exotic)differentiablestructures. Thisfollowsfromtheresultsof [Donaldson,1983]
and [Freedman, 1982], see [Gompf, 1983] for an overview.
Note that for a HausdorffC -manifold in a more general sensethe following prop-
∞
erties are equivalent:
(1) It is paracompact.
DraftfromApril18,2007 PeterW.Michor,
2 ChapterI.ManifoldsandVectorFields 1.3
(2) It is metrizable.
(3) It admits a Riemannian metric.
(4) Each connected component is separable.
In this book a manifold will usually mean a C -manifold, and smooth is used
∞
synonymously for C , it will be Hausdorff, separable, finite dimensional, to state
∞
it precisely.
NotefinallythatanymanifoldM admitsafiniteatlasconsistingofdimM+1(not
connected) charts. This is a consequence of topological dimension theory [Nagata,
1965], a proof for manifolds may be found in [Greub-Halperin-Vanstone, Vol. I].
1.2. Example: Spheres. We consider the space Rn+1, equipped with the stan-
dardinnerproduct x,y = xiyi. Then-sphereSn isthenthesubset x Rn+1 :
h i { ∈
x,x = 1 . Since f(x) = x,x , f : Rn+1 R, satisfies df(x)y = 2 x,y , it is of
h i } hP i → h i
rank 1 off 0 and by (1.12) the sphere Sn is a submanifold of Rn+1.
In order to get some feeling for the sphere we will describe an explicit atlas for Sn,
the stereographic atlas. Choose a Sn (‘south pole’). Let
∈
U+ :=Sn\{a}, u+ :U+ →{a}⊥, u+(x)= x1−hxx,,aaia,
−h i
U− :=Sn\{−a}, u− :U− →{a}⊥, u−(x)= x1−+hhxx,,aaiia.
From an obvious drawing in the 2-plane through 0, x, and a it is easily seen that
u is the usual stereographic projection.
+
-a
x
1
0 z=u- (x)
y=u (x)
+
x-<x,a>a
a
We also get
u−+1(y)= |yy|22−+11a+ y22+1y for y ∈{a}⊥\{0}
| | | |
uansidng(u‘−St◦ruah−+l1e)n(sya)t=z’.|yy|2. Thelatterequationcandirectlybeseenfromthedrawing
1.3. Smooth mappings. Amappingf :M N betweenmanifoldsissaidtobe
→
Ck if for each x M and one (equivalently: any) chart (V,v) on N with f(x) V
∈ ∈
there is a chart (U,u) on M with x U, f(U) V, and v f u 1 is Ck. We will
−
∈ ⊆ ◦ ◦
denote by Ck(M,N) the space of all Ck-mappings from M to N.
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1.5 1. DifferentiableManifolds 3
ACk-mappingf :M N iscalledaCk-diffeomorphismiff 1 :N M existsand
−
→ →
is also Ck. Two manifolds are called diffeomorphic if there exists a diffeomorphism
betweenthem. Fromdifferentialtopology(see[Hirsch,1976])weknowthatifthere
isaC1-diffeomorphismbetweenM andN,thenthereisalsoaC -diffeomorphism.
∞
There are manifolds which are homeomorphic but not diffeomorphic: on R4 there
are uncountably many pairwise non-diffeomorphic differentiable structures; on ev-
eryotherRn thedifferentiablestructureisunique. Therearefinitelymanydifferent
differentiable structures on the spheres Sn for n 7.
≥
A mapping f : M N between manifolds of the same dimension is called a local
→
diffeomorphism, if each x M has an open neighborhood U such that f U : U
∈ | →
f(U) N is a diffeomorphism. Note that a local diffeomorphism need not be
⊂
surjective.
1.4. Smooth functions. The set of smooth real valued functions on a manifold
M will be denoted by C (M), in order to distinguish it clearly from spaces of
∞
sections which will appear later. C (M) is a real commutative algebra.
∞
The support of a smooth function f is the closure of the set, where it does not
vanish,supp(f)= x M :f(x)=0 . Thezerosetoff isthesetwheref vanishes,
{ ∈ 6 }
Z(f)= x M :f(x)=0 .
{ ∈ }
1.5. Theorem. Any (separable, metrizable, smooth) manifold admits smooth par-
titions of unity: Let (U ) be an open cover of M.
α α A
∈
Then there is a family (ϕ ) of smooth functions on M, such that:
α α A
∈
(1) ϕ (x) 0 for all x M and all α A.
α
≥ ∈ ∈
(2) supp(ϕ ) U for all α A.
α α
⊂ ∈
(3) (supp(ϕ )) is a locally finite family (so each x M has an open neigh-
α α A
∈ ∈
borhood which meets only finitely many supp(ϕ )).
α
(4) ϕ =1 (locally this is a finite sum).
α α
P
Proof. Any (separable metrizable) manifold is a ‘Lindelo¨f space’, i. e. each open
cover admits a countable subcover. This can be seen as follows:
Let be an open cover of M. Since M is separable there is a countable dense
U
subsetS inM. Choose a metric on M. Foreach U and each x U there isan
∈U ∈
y S andn NsuchthattheballB (y)withrespecttothatmetricwithcenter
1/n
∈ ∈
y and radius 1 contains x and is contained in U. But there are only countably
n
many of these balls; for each of them we choose an open set U containing it.
∈ U
This is then a countable subcover of .
U
Now let (U ) be the given cover. Let us fix first α and x U . We choose a
α α A α
chart (U,u) ce∈ntered at x (i. e. u(x) = 0) and ε > 0 such that∈εDn u(U U ),
α
⊂ ∩
where Dn = y Rn : y 1 is the closed unit ball. Let
{ ∈ | |≤ }
e 1/t for t>0,
−
h(t):=
0 for t 0,
(cid:26) ≤
DraftfromApril18,2007 PeterW.Michor,
4 ChapterI.ManifoldsandVectorFields 1.7
a smooth function on R. Then
h(ε2 u(z)2) for z U,
f (z):= −| | ∈
α,x
0 for z / U
(cid:26) ∈
is a non negative smooth function on M with support in U which is positive at x.
α
We choose such a function f for each α and x U . The interiors of the
α,x α
∈
supportsofthesesmoothfunctions formanopencoverofM whichrefines(U ), so
α
by the argument at the beginning of the proof there is a countable subcover with
corresponding functions f ,f ,.... Let
1 2
W = x M :f (x)>0 and f (x)< 1 for 1 i<n ,
n { ∈ n i n ≤ }
anddenotebyW theclosure. Then(W ) isanopencover. Weclaimthat(W )
n n n n n
is locally finite: Let x M. Then there is a smallest n such that x W . Let
n
∈ ∈
V := y M :f (y)> 1f (x) . If y V W then we have f (y)> 1f (x) and
{ ∈ n 2 n } ∈ ∩ k n 2 n
f (y) 1 for i<k, which is possible for finitely many k only.
i ≤ k
Considerthenonnegativesmoothfunctiong (x)=h(f (x))h(1 f (x))...h(1
n n n− 1 n−
f (x)) for each n. Then obviously supp(g ) = W . So g := g is smooth,
n 1 n n n n
−
since it is locally only a finite sum, and everywhere positive, thPus (gn/g)n∈N is a
smooth partition of unity on M. Since supp(g )=W is contained in some U
n n α(n)
we may put ϕ = gn to get the required partition of unity which is
α n:α(n)=α g
subordinated to (U ){ . (cid:3) }
αPα∈A
1.6. Germs. Let M and N be manifolds and x M. We consider all smooth
∈
mappings f : U N, where U is some open neighborhood of x in M, and we
f f
→
put f g if there is some open neighborhood V of x with f V = g V. This is an
∼x | |
equivalence relation on the set of mappings considered. The equivalence class of a
mapping f is called the germ of f at x, sometimes denoted by germ f. The set of
x
all these germs is denoted by C (M,N).
x∞
Note that for a germs at x of a smooth mapping only the value at x is defined. We
may also consider composition of germs: germ g germ f :=germ (g f).
f(x) ◦ x x ◦
If N = R, we may add and multiply germs of smooth functions, so we get the
real commutative algebra C (M,R) of germs of smooth functions at x. This
x∞
constructionworksalsoforothertypesoffunctionslikerealanalyticorholomorphic
ones, if M has a real analytic or complex structure.
Usingsmoothpartitionsofunity((1.4))itiseasilyseenthateachgermofasmooth
functionhasarepresentativewhichisdefinedonthewholeofM. Forgermsofreal
analytic or holomorphic functions this is not true. So C (M,R) is the quotient of
x∞
the algebra C (M) by the ideal of all smooth functions f : M R which vanish
∞
→
on some neighborhood (depending on f) of x.
1.7. The tangent space of Rn. Let a Rn. A tangent vector with foot point a
∈
is simply a pair (a,X) with X Rn, also denoted by X . It induces a derivation
a
∈
DraftfromApril18,2007 PeterW.Michor,
1.8 1. DifferentiableManifolds 5
X :C (Rn) R by X (f)=df(a)(X ). The value depends only on the germ of
a ∞ a a
→
f ataandwehaveX (f g)=X (f) g(a)+f(a) X (g)(thederivationproperty).
a a a
· · ·
If conversely D :C (Rn) R is linear and satisfies D(f g)=D(f) g(a)+f(a)
∞
→ · · ·
D(g) (a derivation at a), then D is given by the action of a tangent vector with
foot point a. This can be seen as follows. For f C (Rn) we have
∞
∈
1
f(x)=f(a)+ df(a+t(x a))dt
dt −
Z0
n 1
=f(a)+ ∂f (a+t(x a))dt(xi ai)
∂xi − −
i=1Z0
X
n
=f(a)+ h (x)(xi ai).
i
−
i=1
X
D(1)=D(1 1)=2D(1), so D(constant)=0. Thus
·
n
D(f)=D(f(a)+ h (xi ai))
i
−
i=1
X
n n
=0+ D(h )(ai ai)+ h (a)(D(xi) 0)
i i
− −
i=1 i=1
X X
n
= ∂f (a)D(xi),
∂xi
i=1
X
where xi is the i-th coordinate function on Rn. So we have
n n
D(f)= D(xi) ∂ (f), D = D(xi) ∂ .
∂xi|a ∂xi|a
i=1 i=1
X X
Thus D is induced by the tangent vector (a, n D(xi)e ), where (e ) is the stan-
i=1 i i
dard basis of Rn.
P
1.8. The tangent space of a manifold. Let M be a manifold and let x M
∈
and dimM =n. Let T M be the vector space of all derivations at x of C (M,R),
x x∞
the algebra of germs of smooth functions on M at x. (Using (1.5) it may easily be
seen that a derivation of C (M) at x factors to a derivation of C (M,R).)
∞ x∞
So T M consists of all linear mappings X : C (M) R with the property
x x ∞
→
X (f g)=X (f) g(x)+f(x) X (g). The space T M is called the tangent space
x x x x
· · ·
of M at x.
If(U,u)isachartonM withx U,thenu :f f uinducesanisomorphismof
∗
∈ 7→ ◦
algebras C (Rn,R) = C (M,R), and thus also an isomorphism T u : T M
u∞(x) ∼ x∞ x x →
T Rn, given by (T u.X )(f) = X (f u). So T M is an n-dimensional vector
u(x) x x x x
◦
space.
We will use the following notation: u=(u1,... ,un), so ui denotes the i-th coordi-
nate function on U, and
∂ :=(T u) 1( ∂ )=(T u) 1(u(x),e ).
∂ui|x x − ∂xi|u(x) x − i
DraftfromApril18,2007 PeterW.Michor,
6 ChapterI.ManifoldsandVectorFields 1.10
So ∂ T M is the derivation given by
∂ui|x ∈ x
∂(f u 1)
∂ (f)= ◦ − (u(x)).
∂ui|x ∂xi
From (1.7) we have now
n n
T u.X = (T u.X )(xi) ∂ = X (xi u) ∂
x x x x ∂xi|u(x) x ◦ ∂xi|u(x)
i=1 i=1
X X
n
= X (ui) ∂ ,
x ∂xi|u(x)
i=1
X
n
X =(T u) 1.T u.X = X (ui) ∂ .
x x − x x x ∂ui|x
i=1
X
1.9. The tangent bundle. For a manifold M of dimension n we put TM :=
T M,thedisjointunionofalltangentspaces. Thisisafamilyofvectorspaces
x M x
par∈ameterized by M, with projection π :TM M given by π (T M)=x.
F M → M x
For any chart (Uα,uα) of M consider the chart (πM−1(Uα),Tuα) on TM, where
Tuα : πM−1(Uα) → uα(Uα)×Rn is given by Tuα.X = (uα(πM(X)),TπM(X)uα.X).
Then the chart changings look as follows:
Tuβ ◦(Tuα)−1 :Tuα(πM−1(Uαβ))=uα(Uαβ)×Rn →
→uβ(Uαβ)×Rn =Tuβ(πM−1(Uαβ)),
((Tu (Tu ) 1)(y,Y))(f)=((Tu ) 1(y,Y))(f u )
β α − α − β
◦ ◦
=(y,Y)(f u u 1)=d(f u u 1)(y).Y
◦ β ◦ −α ◦ β ◦ −α
=df(u u 1(y)).d(u u 1)(y).Y
β ◦ −α β ◦ −α
=(u u 1(y),d(u u 1)(y).Y)(f).
β ◦ α− β ◦ −α
So the chart changings are smooth. We choose the topology on TM in such a
way that all Tu become homeomorphisms. This is a Hausdorff topology, since X,
α
Y TM maybeseparatedinM ifπ(X)=π(Y),andinonechartifπ(X)=π(Y).
∈ 6
So TM is again a smooth manifold in a canonical way; the triple (TM,π ,M) is
M
called the tangent bundle of M.
1.10. Kinematic definition of the tangent space. Let C (R,M) denote the
0∞
space of germs at 0 of smooth curves R M. We put the following equivalence
→
relation on C (R,M): the germ of c is equivalent to the germ of e if and only if
0∞
c(0) = e(0) and in one (equivalently each) chart (U,u) with c(0) = e(0) U we
∈
have d (u c)(t)= d (u e)(t). The equivalence classes are also called velocity
dt|0 ◦ dt|0 ◦
vectors of curves in M. We have the following mappings
C (R,M)/ C (R,M)
0∞ ∼ u 0∞
α AAAC ev0
Aβ
A
uA u
TM M,
π
M w
DraftfromApril18,2007 PeterW.Michor,