Topics in differential geometry

359 Pages·2006·2.347 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.
[email protected]
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 considerably during
the course.
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.ps
DraftfromFebruary21,2006 PeterW.Michor,
iii
TABLE OF CONTENTS
0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
CHAPTER I Manifolds and Vector Fields . . . . . . . . . . . . . . . 3
1. Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . 3
2. Submersions and Immersions . . . . . . . . . . . . . . . . . . . 15
C. Covering spaces and fundamental groups . . . . . . . . . . . . . . 20
3. Vector Fields and Flows . . . . . . . . . . . . . . . . . . . . . 23
CHAPTER II Lie Groups . . . . . . . . . . . . . . . . . . . . . 43
4. Lie Groups I . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5. Lie Groups II. Lie Subgroups and Homogeneous Spaces . . . . . . . . 58
CHAPTER III Differential Forms and De Rham Cohomology . . . . . 67
6. Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 67
7. Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . 79
8. Integration on Manifolds . . . . . . . . . . . . . . . . . . . . . 87
9. De Rham cohomology . . . . . . . . . . . . . . . . . . . . . . 93
10. Cohomology with compact supports and Poincar´e duality . . . . . . . 102
11. De Rham cohomology of compact manifolds . . . . . . . . . . . . 113
12. Lie groups III. Analysis on Lie groups . . . . . . . . . . . . . . . 119
CHAPTER IV Riemannian Geometry . . . . . . . . . . . . . . . 131
13. Pseudo Riemann metrics and the Levi Civita covariant derivative . . . 131
14. Riemann geometry of geodesics . . . . . . . . . . . . . . . . . . 144
15. Parallel transport and curvature . . . . . . . . . . . . . . . . . 152
16. Computing with adapted frames, and examples . . . . . . . . . . . 162
17. Riemann immersions and submersions . . . . . . . . . . . . . . . 175
18. Jacobi fields . . . . . . . . . . . . . . . . . . . . . . . . . . 189
H. Hodge theory . . . . . . . . . . . . . . . . . . . . . . . . . . 204
CHAPTER V Bundles and Connections . . . . . . . . . . . . . . . 207
19. Derivations on the Algebra of Differential Forms
and the Fro¨licher-Nijenhuis Bracket . . . . . . . . . . . . . . . . . . 207
20. Fiber Bundles and Connections . . . . . . . . . . . . . . . . . . 215
21. Principal Fiber Bundles and G-Bundles . . . . . . . . . . . . . . 224
22. Principal and Induced Connections . . . . . . . . . . . . . . . . 240
23. Characteristic classes . . . . . . . . . . . . . . . . . . . . . . 258
24. Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
CHAPTER VI Symplectic Geometry and Hamiltonian Mechanics . . . 279
25. Symplectic Geometry and Classical Mechanics . . . . . . . . . . . 279
26. Completely integrable Hamiltonian systems . . . . . . . . . . . . 300
27. Extensions of Lie algebras and Lie groups . . . . . . . . . . . . . 305
28. Poisson manifolds . . . . . . . . . . . . . . . . . . . . . . . . 312
29. Hamiltonian group actions and momentum mappings . . . . . . . . 322
30. Lie Poisson groups . . . . . . . . . . . . . . . . . . . . . . . 343
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
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0. Introduction
InthislecturenotesItrytogiveanintroductiontothefundamentalsofdifferential
geometry(manifolds,flows,Liegroups,differentialforms,bundlesandconnections)
which stresses naturality and functoriality from the beginning and is as coordinate
free as possible. The material presented in the beginning is standard - but some
parts are not so easily found in text books: we treat initial submanifolds and the
Frobeniustheoremfordistributionsofnonconstantrank,andwegiveaquickproof
in two pages of the Campbell - Baker - Hausdorff formula for Lie groups. We also
prove that closed subgroups of Lie groups are Lie subgroups.
Then the deviation from the standard presentations becomes larger. In the section
onvectorbundlesItreattheLiederivativefornaturalvectorbundles, i.e. functors
which associate vector bundles to manifolds and vector bundle homomorphisms
to local diffeomorphisms. I give a formula for the Lie derivative of the form of a
commutator, but it involves the tangent bundle of the vector bundle involved. So I
also give acarefultreatment tothissituation. Itfollows astandardpresentation of
differential forms and a thorough treatment of the Fro¨licher-Nijenhuis bracket via
thestudyofallgradedderivationsofthealgebraofdifferentialforms. Thisbracket
isanaturalextensionoftheLiebracketfromvectorfieldstotangentbundlevalued
differential forms. I believe that this bracket is one of the basic structures of dif-
ferential geometry, and later I will base nearly all treatment of curvature and the
Bianchi identities on it. This allows me to present the concept of a connection first
on general fiber bundles (without structure group), with curvature, parallel trans-
port and Bianchi identity, and only then add G-equivariance as a further property
forprincipalfiberbundles. Ithink, thatinthiswaytheunderlyinggeometricideas
are more easily understood by the novice than in the traditional approach, where
too much structure at the same time is rather confusing.
Webeginourtreatmentofconnectionsinthegeneralsettingoffiberbundles(with-
out structure group). A connection on a fiber bundle is just a projection onto the
vertical bundle. Curvature and the Bianchi identity is expressed with the help of
the Fro¨licher-Nijenhuis bracket. The parallel transport for such a general connec-
tionisnotdefinedalongthewholeofthecurveinthebaseingeneral-ifthisisthe
case, the connection is called complete. We show that every fiber bundle admits
completeconnections. Forcompleteconnectionswetreatholonomygroupsandthe
holonomy Lie algebra, a subalgebra of the Lie algebra of all vector fields on the
standard fiber.
Then we present principal bundles and associated bundles in detail together with
the most important examples. Finally we investigate principal connections by re-
quiring equivariance under the structure group. It is remarkable how fast the
usualstructureequations canbederivedfromthebasicpropertiesoftheFro¨licher-
Nijenhuis bracket. Induced connections are investigated thoroughly - we describe
tools to recognize induced connections among general ones.
IftheholonomyLiealgebraofaconnectiononafiberbundlewithcompactstandard
fiber turns out to be finite dimensional, we are able to show, that in fact the fiber
DraftfromFebruary21,2006 PeterW.Michor,
2 Introduction
bundle is associated to a principal bundle and the connection is an induced one.
We think that the treatment of connections presented here offers some didactical
advantages besides presenting new results: the geometric content of a connection
is treated first, and the additional requirement of equivariance under a structure
group is seen to be additional and can be dealt with later - so the student is not
required to grasp all the structures at the same time. Besides that it gives new
results and new insights. This treatment is taken from [Michor, 87].
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3
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.
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4 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 5
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,
½ ≤
DraftfromFebruary21,2006 PeterW.Michor,
6 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
½ ∈
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 ){ . ¤ }
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
∈
DraftfromFebruary21,2006 PeterW.Michor,