Topics in differential geometry

357 Pages·2004·2.079 MB·English

by  Michor.

Topics in
Difierential Geometry
Peter W. Michor
Institut fu˜r Mathematik der Universita˜t Wien, Strudlhofgasse 4, A-1090 Wien,
Austria.
Erwin Schro˜dinger Institut fu˜r Mathematische Physik, Boltzmanngasse 9, A-1090
Wien, Austria.
[email protected]
These notes are from a lecture course
Difierentialgeometrie und Lie Gruppen
which has been held at the University of Vienna during the academic year 1990/91,
again in 1994/95, in WS 1997, in a four term series in 1999/2000 and 2001/02, and
parts in WS 2003 It is not yet complete and will be enlarged considerably during
the course.
Typeset by AMS-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
Draft from September 15, 2004 Peter W. Michor,
iii
TABLE OF CONTENTS
0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
CHAPTER I Manifolds and Vector Fields . . . . . . . . . . . . . . . 3
1. Difierentiable 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 Difierential Forms and De Rham Cohomology . . . . . 67
6. Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 67
7. Difierential 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 flelds . . . . . . . . . . . . . . . . . . . . . . . . . . 189
CHAPTER V Bundles and Connections . . . . . . . . . . . . . . . 205
19. Derivations on the Algebra of Difierential Forms
and the Fro˜licher-Nijenhuis Bracket . . . . . . . . . . . . . . . . . . 205
20. Fiber Bundles and Connections . . . . . . . . . . . . . . . . . . 213
21. Principal Fiber Bundles and G-Bundles . . . . . . . . . . . . . . 222
22. Principal and Induced Connections . . . . . . . . . . . . . . . . 238
23. Characteristic classes . . . . . . . . . . . . . . . . . . . . . . 256
24. Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
CHAPTER VI Symplectic Geometry and Hamiltonian Mechanics . . . 277
25. Symplectic Geometry and Classical Mechanics . . . . . . . . . . . 277
26. Completely integrable Hamiltonian systems . . . . . . . . . . . . 298
27. Extensions of Lie algebras and Lie groups . . . . . . . . . . . . . 303
28. Poisson manifolds . . . . . . . . . . . . . . . . . . . . . . . . 310
29. Hamiltonian group actions and momentum mappings . . . . . . . . 319
30. Lie Poisson groups . . . . . . . . . . . . . . . . . . . . . . . 340
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 347
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Draft from September 15, 2004 Peter W. Michor,
iv
Draft from September 15, 2004 Peter W. Michor,
1
0. Introduction
In this lecture notes I try to give an introduction to the fundamentals of difierential
geometry (manifolds, (cid:176)ows, Lie groups, difierential forms, bundles and connections)
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
Frobenius theorem for distributions of non constant rank, and we give a quick proof
in two pages of the Campbell - Baker - Hausdorfi 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
on vector bundles I treat the Lie derivative for natural vector bundles, i.e. functors
which associate vector bundles to manifolds and vector bundle homomorphisms
to local difieomorphisms. 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 a careful treatment to this situation. It follows a standard presentation of
difierential forms and a thorough treatment of the Fro˜licher-Nijenhuis bracket via
the study of all graded derivations of the algebra of difierential forms. This bracket
is a natural extension of the Lie bracket from vector flelds to tangent bundle valued
difierential 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 flrst
on general flber bundles (without structure group), with curvature, parallel trans-
port and Bianchi identity, and only then add G-equivariance as a further property
for principal flber bundles. I think, that in this way the underlying geometric ideas
are more easily understood by the novice than in the traditional approach, where
too much structure at the same time is rather confusing.
We begin our treatment of connections in the general setting of flber bundles (with-
out structure group). A connection on a flber 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-
tion is not deflned along the whole of the curve in the base in general - if this is the
case, the connection is called complete. We show that every flber bundle admits
complete connections. For complete connections we treat holonomy groups and the
holonomy Lie algebra, a subalgebra of the Lie algebra of all vector flelds on the
standard flber.
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
usual structure equations can be derived from the basic properties of the Fro˜licher-
Nijenhuis bracket. Induced connections are investigated thoroughly - we describe
tools to recognize induced connections among general ones.
IftheholonomyLiealgebraofaconnectiononaflberbundlewithcompactstandard
flber turns out to be flnite dimensional, we are able to show, that in fact the flber
Draft from September 15, 2004 Peter W. 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 ofiers some didactical
advantages besides presenting new results: the geometric content of a connection
is treated flrst, 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].
Draft from September 15, 2004 Peter W. Michor,
3
CHAPTER I
Manifolds and Vector Fields
1. Difierentiable 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
2
u : U u(U) Rn, where U is an open neighborhood of x in M and u(U) is an
! (cid:181)
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 preflx pure.
A family (U ;u ) of charts on M such that the U form a cover of M is called
fi fi fi A fi
an atlas. The mapp2ings u := u u 1 : u (U ) u (U ) are called the chart
fifl fi– ¡fl fl fifl ! fi fifl
changings for the atlas (U ), where U := U U .
fi fifl fi fl
\
Anatlas(U ;u ) foramanifoldM issaidtobeaCk-atlas, ifallchartchangings
fi fi fi A
2
u : u (U ) u (U ) are difierentiable of class Ck. Two Ck-atlases are called
fifl fl fifl fi fifl
!
Ck-equivalent, if their union is again a Ck-atlas for M. An equivalence class of Ck-
atlases is called a Ck-structure on M. From difierential topology we know that if M
has a C1-structure, then it also has a C1-equivalent C -structure and even a C1-
1
equivalent C!-structure, where C! is shorthand for real analytic, see [Hirsch, 1976].
By a Ck-manifold M we mean a topological manifold together with a Ck-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 difierentiable structures.
For example, every 4-dimensional manifold is smooth ofi 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 flnitely many, see [Milnor, 1956]. But the most
surprising result is that on R4 there are uncountably many pairwise inequivalent
(exotic) difierentiable structures. This follows from the results of [Donaldson, 1983]
and [Freedman, 1982], see [Gompf, 1983] for an overview.
Note that for a Hausdorfi C -manifold in a more general sense the following prop-
1
erties are equivalent:
(1) It is paracompact.
Draft from September 15, 2004 Peter W. Michor,
4 Chapter I. Manifolds and Vector Fields 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
1
synonymously for C , it will be Hausdorfi, separable, flnite dimensional, to state
1
it precisely.
Note flnally that any manifold M admits a flnite atlas consisting of dimM +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-
dard inner product x;y = xiyi. The n-sphere Sn is then the subset x Rn+1 :
h i f 2
x;x = 1 . Since f(x) = x;x , f : Rn+1 R, satisfles df(x)y = 2 x;y , it is of
h i g hP i ! h i
rank 1 ofi 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
2
U+ := Sn nfag; u+ : U+ ! fag?; u+(x) = x1¡hxx;;aaia;
¡h i
U := Sn a ; u : U a ?; u (x) = x¡hx;aia:
¡ nf¡ g ¡ ¡ ! f g ¡ 1+hx;ai
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) = jyyj22¡+11a+ y 22+1y for y 2 fag? nf0g
j j j j
and (u u 1)(y) = y . The latter equation can directly be seen from the drawing
¡– ¡+ jyj2
using ‘Strahlensatz’.
1.3. Smooth mappings. A mapping f : M N between manifolds is said to be
!
Ck if for each x M and one (equivalently: any) chart (V;v) on N with f(x) V
2 2
there is a chart (U;u) on M with x U, f(U) V, and v f u 1 is Ck. We will
¡
2 (cid:181) – –
denote by Ck(M;N) the space of all Ck-mappings from M to N.
Draft from September 15, 2004 Peter W. Michor,
1.5 1. Difierentiable Manifolds 5
ACk-mappingf : M N iscalledaCk-difieomorphismiff 1 : N M existsand
¡
! !
is also Ck. Two manifolds are called difieomorphic if there exists a difieomorphism
between them. From difierential topology (see [Hirsch, 1976]) we know that if there
is a C1-difieomorphism between M and N, then there is also a C -difieomorphism.
1
There are manifolds which are homeomorphic but not difieomorphic: on R4 there
are uncountably many pairwise non-difieomorphic difierentiable structures; on ev-
ery other Rn the difierentiable structure is unique. There are flnitely many difierent
difierentiable structures on the spheres Sn for n 7.
‚
A mapping f : M N between manifolds of the same dimension is called a local
!
difieomorphism, if each x M has an open neighborhood U such that f U : U
2 j !
f(U) N is a difieomorphism. Note that a local difieomorphism 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
1
sections which will appear later. C (M) is a real commutative algebra.
1
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 . Thezero setoff isthesetwheref vanishes,
f 2 6 g
Z(f) = x M : f(x) = 0 .
f 2 g
1.5. Theorem. Any (separable, metrizable, smooth) manifold admits smooth par-
titions of unity: Let (U ) be an open cover of M.
fi fi A
2
Then there is a family (’ ) of smooth functions on M, such that:
fi fi A
2
(1) ’ (x) 0 for all x M and all fi A.
fi
‚ 2 2
(2) supp(’ ) U for all fi A.
fi fi
‰ 2
(3) (supp(’ )) is a locally flnite family (so each x M has an open neigh-
fi fi A
2 2
borhood which meets only flnitely many supp(’ )).
fi
(4) ’ = 1 (locally this is a flnite sum).
fi fi
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
subset S in M. Choose a metric on M. For each U and each x U there is an
2 U 2
y S and n N such that the ball B (y) with respect to that metric with center
1=n
2 2
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.
2 U
This is then a countable subcover of .
U
Now let (U ) be the given cover. Let us flx flrst fi and x U . We choose a
fi fi A fi
2 2
chart (U;u) centered at x (i. e. u(x) = 0) and " > 0 such that "Dn u(U U ),
fi
‰ \
where Dn = y Rn : y 1 is the closed unit ball. Let
f 2 j j • g
e 1=t for t > 0;
¡
h(t) :=
0 for t 0;
‰ •
Draft from September 15, 2004 Peter W. Michor,
6 Chapter I. Manifolds and Vector Fields 1.7
a smooth function on R. Then
h("2 u(z) 2) for z U;
f (z) := ¡j j 2
fi;x
0 for z = U
‰ 2
is a non negative smooth function on M with support in U which is positive at x.
fi
We choose such a function f for each fi and x U . The interiors of the
fi;x fi
2
supports of these smooth functions form an open cover of M which reflnes (U ), so
fi
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 f 2 n i n • g
and denote by W the closure. Then (W ) is an open cover. We claim that (W )
n n n n n
is locally flnite: Let x M. Then there is a smallest n such that x W . Let
n
2 2
V := y M : f (y) > 1f (x) . If y V W then we have f (y) > 1f (x) and
f 2 n 2 n g 2 \ k n 2 n
f (y) 1 for i < k, which is possible for flnitely many k only.
i • k
Consider the non negative smooth function g (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 flnite sum, and everywhere positive, thus (gn=g)n N is a
P 2
smooth partition of unity on M. Since supp(g ) = W is contained in some U
n n fi(n)
we may put ’ = gn to get the required partition of unity which is
fi n:fi(n)=fi g
f g
subordinated to (U ) . ⁄
fi fi A
P 2
1.6. Germs. Let M and N be manifolds and x M. We consider all smooth
2
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 j j
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).
x1
Note that for a germs at x of a smooth mapping only the value at x is deflned. 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
x1
constructionworksalsoforothertypesoffunctionslikerealanalyticorholomorphic
ones, if M has a real analytic or complex structure.
Using smooth partitions of unity ((1.4)) it is easily seen that each germ of a smooth
function has a representative which is deflned on the whole of M. For germs of real
analytic or holomorphic functions this is not true. So C (M;R) is the quotient of
x1
the algebra C (M) by the ideal of all smooth functions f : M R which vanish
1
!
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
2
is simply a pair (a;X) with X Rn, also denoted by X . It induces a derivation
a
2
Draft from September 15, 2004 Peter W. Michor,