Manifolds and differential geometry. Online supplement

216 Pages·2009·1.179 MB·English

by  Lee J.M.

Supplement for Manifolds and Differential
Geometry
Jeffrey M. Lee
Department of Mathematics and Statistics, Texas Tech Uni-
versity, Lubbock, Texas, 79409
Currentaddress: DepartmentofMathematicsandStatistics,TexasTech
University, Lubbock, Texas, 79409
E-mail address: [email protected]

Contents
Chapter 0. Background Material 1
§0.1. Sets and Maps 1
§0.2. Linear Algebra (under construction) 7
§0.3. General Topology (under construction) 11
§0.4. Shrinking Lemma 12
§0.5. Calculus on Euclidean coordinate spaces (underconstruction) 13
§0.6. Topological Vector Spaces 16
§0.7. Differential Calculus on Banach Spaces 21
§0.8. Naive Functional Calculus. 43
§0.9. Attaching Spaces and Quotient Topology 51
§0.10. Problem Set 53
Chapter 1. Chapter 1 Supplement 57
§1.1. Comments and Errata 57
§1.2. Rough Ideas I 57
§1.3. Pseudo-Groups and Models Spaces 64
§1.4. Sard’s Theorem 69
Chapter 2. Chapter 2 Supplement 73
§2.1. Comments and Errata 73
§2.2. Time Dependent Vector Fields 73
Chapter 5. Chapter 5 Supplement 77
§5.1. Comments and Errata 77
v
vi Contents
§5.2. Matrix Commutator Bracket 77
§5.3. Spinors and rotation 79
§5.4. Lie Algebras 81
§5.5. Geometry of figures in Euclidean space 94
Chapter 6. Chapter 6 Supplement 99
§6.1. Comments and Errata 99
§6.2. sections of the Mobius band 99
§6.3. Etale space of a sheaf 100
§6.4. Discussion on G bundle structure 101
Chapter 8. Chapter 8 Supplement 111
§8.1. Comments and Errata 111
§8.2. 2-Forms over a surface in Euclidean Space Examples. 111
§8.3. Pseudo-Forms 113
Chapter 11. Chapter 11 Supplement 117
§11.1. Comments and Errata 117
§11.2. Singular Distributions 117
Chapter 12. Chapter 12 Supplement 123
§12.1. Comments and Errata 123
§12.2. More on Actions 123
§12.3. Connections on a Principal bundle. 125
§12.4. Horizontal Lifting 130
Chapter 13. Chapter 13 Supplement 133
§13.1. Alternate proof of test case 133
Chapter 14. Complex Manifolds 135
§14.1. Some complex linear algebra 135
§14.2. Complex structure 138
§14.3. Complex Tangent Structures 141
§14.4. The holomorphic tangent map. 142
§14.5. Dual spaces 143
§14.6. The holomorphic inverse and implicit functions theorems. 144
Chapter 15. Symplectic Geometry 147
§15.1. Symplectic Linear Algebra 147
§15.2. Canonical Form (Linear case) 149
Contents vii
§15.3. Symplectic manifolds 149
§15.4. Complex Structure and K¨ahler Manifolds 151
§15.5. Symplectic musical isomorphisms 154
§15.6. Darboux’s Theorem 155
§15.7. Poisson Brackets and Hamiltonian vector fields 156
§15.8. Configuration space and Phase space 159
§15.9. Transfer of symplectic structure to the Tangent bundle 161
§15.10. Coadjoint Orbits 163
§15.11. The Rigid Body 164
§15.12. The momentum map and Hamiltonian actions 166
Chapter 16. Poisson Geometry 169
§16.1. Poisson Manifolds 169
Chapter 17. Geometries 173
§17.1. Group Actions, Symmetry and Invariance 179
§17.2. Some Klein Geometries 183
Bibliography 205
Index 209

Chapter 0
Background Material
0.1. Sets and Maps
Anyreaderwhoalreadyknowsthemeaningofsuchwordsasdomain,codomain,
surjective, injective, Cartesian product, and equivalence relation, shouldjust
skip this section to avoid boredom.
According to G. Cantor, one of the founders of set theory, a set is a col-
lection of objects, real or abstract, taken as a whole or group. The objects
must be clear and distinct to the intellect. If one considers the set of all
ducks on a specific person’s farm then this seems clear enough as long as the
notion of a duck is well defined. But is it? When one thinks about the set
of all ducks that ever lived, then things get fuzzy (think about evolutionary
theory). Clearly there are philosophical issues with how we define and think
about various objects, things, types, kinds, ideas and collections. In math-
ematics, we deal with seemingly more precise, albeit abstract, notions and
objects such as integers, letters in some alphabet or a set of such things (or
a set of sets of such things!). We will not enter at all into questions about
the clarity or ontological status of such mathematical objects nor shall we
discuss “proper classes”. Suffice it to say that there are monsters in those
seas.
Each object or individual in a set is called an element or a member
of the set. The number 2 is an element of the set of all integers-we say it
belongs to the set. Sets are equal if they have the same members. We
often use curly brackets to specify a set. The letter a is a member of the
set {a,B,x,6}. We use the symbol ∈ to denote membership; a ∈ A means
“a is a member of the set A”. We read it in various ways such as “a is a
element of A” or “a in A”. Thus, x ∈ {a,B,x,6} and if we denote the set of
1
2 0. Background Material
natural numbers by N then 42 ∈ N and 3 ∈ N etc. We include in our naive
theory of sets a unique set called the empty set which has no members at
all. It is denoted by ∅ or by {}. Besides just listing the members of a set we
make use of notation such as {x ∈ N : x is prime} which means the set of
all natural numbers with the property of being a prime numbers. Another
example; {a : a ∈ N and a < 7}. This is the same set as {1,2,3,4,5,6}.
If every member of a set A is also a member of a set B then we say
that A is a subset of B and write this as A ⊂ B or B ⊃ A. Thus if
a ∈ A =⇒ b ∈ A. (The symbol “=⇒” means “implies”). The empty set is a
subset of any set we may always write ∅ ⊂ A for any A. Also, A ⊂ A for
any set A. It is easy to see that if A ⊂ B and B ⊂ A then A and B have
the same members–they are the same set differently named. In this case we
write A = B. If A ⊂ B but A (cid:54)= B then we say that A is a proper subset
of B and write A (cid:36) B.
TheunionoftwosetsAandBisdefinedtobethesetofallelementsthat
are members of either A or B (or both). The union is denoted A∪B. Thus
x ∈ A∪Bifandonlyifx ∈ Aorx ∈ B. Noticethathereby“x ∈ Aorx ∈ B”
wemeantoallowthepossibilitythatxmightbeamemberofbothAandB.
The intersection of A and B the set of elements belonging to both A and
B. WedenotethisbyA∩B. Forexample, {6,π,−1,42,{−1}}∩N = {6,42}
(Implicitly we assume that no two elements of a set given as a list are equal
unlesstheyaredenotedbythesamesymbol. Forexample,in{6,a,B,x,42},
context implies that a (cid:54)= 42.) If A∩B = ∅ then A and B have no elements
in common and we call them disjoint.
There are the following algebraic laws: If A, B, and C are sets then
(1) A∩B = B∩A and A∪B = B∪A (Commutativity),
(2) (A∪B)∪C = A∪(B∪C) and (A∩B)∩C = A∩(B∩C) (Asso-
ciativity),
(3) A∪(B∩C) = (A∪B)∩(A∪C) and A∩(B∪C) = (A∩B)∪
(A∩C) (Distributivity). It follows that we also have
(4) A∪(A∩B) = A and A∩(A∪B) = A
We can take A∪B∪C to mean (A∪B)∪C. In fact, if A ,··· ,A are sets
1 n
than we can inductively make sense out of the union A ∪A ∪···∪A and
1 2 n
intersection A ∩A ∩···∩A .
1 2 n
If A and B are sets then A\B denotes that set of all elements of A that
are not also elements of B. The notation A−B is also used and is called
the difference. When considering subsets of a fix set X determined by the
context of the discussion we can denote X\A as Ac and refer to Ac as the
complement of A (in X). For example, if A is a subset of the set of real
0.1. Sets and Maps 3
numbers (denoted R) then Ac = R\A. It is easy to see that if A ⊂ B then
Bc ⊂ Ac. We have de Morgan’s laws:
(1) (A∪B)c = Ac∩Bc
(2) (A∩B)c = Ac∪Bc
A family of sets is just a set whose members are sets. For example,
for a fixed set X we have the family P(X) of all subsets of X. The family
P(X) is called the power set of X. Note that ∅ ∈ P(X). If F is some
family of sets then we can consider the union or intersection of all the sets
in F.
(cid:83)
A := {x : x ∈ A for some A ∈ F}
A∈F
(cid:84)
A := {x : x ∈ A for every A ∈ F}
A∈F
It is often convenient to consider indexed families of sets. For example,
if A ,A , and A are sets then we have a family {A ,A ,A } indexed by
1 2 3 1 2 3
the index set {1,2,3}. The family of sets may have an infinite number of
members and we can use an infinite index set. If I is an index set (possibly
uncountably infinite) then a family F indexed by I has as members sets
which are each denoted A for some i ∈ I and some fixed letter A. For
i
notation we use
F = {A : i ∈ I} or {A } .
i i i∈I
We can then write unions and intersections as ∪ A and ∩ A . De Mor-
i∈I i i∈I i
gan’s laws become
(cid:161)(cid:83) (cid:162) (cid:84)
A c = Ac,
i∈I i i∈I i
(cid:161)(cid:84) (cid:162) (cid:83)
A c = Ac.
i∈I i i∈I i
A partition of a set X is a family of subsets of X, say {A } such that
(cid:83) i i∈I
A ∩A = ∅ when ever i (cid:54)= j and such that X = A .
i j i∈I i
An ordered pair with first element a and second element b is denoted
(a,b). To make this notion more precise one could take (a,b) to be the set
{a,{a,b}}. Since the notion of an ordered pair is so intuitive it is seldom
necessarytoresorttothinkingexplicitlyof{a,{a,b}}. GiventwosetsAand
B, the Cartesian product of A and B is the set of order pairs (a,b) where
a ∈ A and b ∈ B. We denote the Cartesian product of A and B by A× B.
We can also consider ordered n-tuples such as (a ,...,a ). For a list of sets
1 n
A ,A ,...,A we define the n-fold Cartesian product:
1 2 k
A ×A ×···×A := {(a ,...,a ) : a ∈ A }.
1 2 n 1 n i i
If A = A = A = ··· = A then we denote A ×A ×···×A by An. For
1 2 n 1 2 n
any set X the diagonal subset ∆ of X ×X is defined by ∆ := {(a,b) ∈
X X
X ×X : a = b} = {(a,a) : a ∈ X}.
4 0. Background Material
Example 0.1. IfRdenotesthesetofrealnumbersthen,foragivenpositive
integer n, Rn denotes the set of n-tuples of real numbers. This set has a lot
of structure as we shall see.
One of the most important notions in all of mathematics is the notion
of a “relation”. A relation from a set A to a set B is simply a subset of
A × B. A relation from A to A is just called a relation on A. If R is a
relation from A to B then we often write aRb instead of (a,b) ∈ R. We
single out two important types of relations:
Definition 0.2. An equivalence relation on a set X is a relation on X,
usual denoted by the symbol ∼, such that (i) x ∼ x for all x ∈ X, (ii) x ∼ y
if and only if y ∼ x, (iii) if x ∼ y and y ∼ z then x ∼ z. For each a ∈ X the
set of all x such thatx ∼ a is called the equivalence class ofa (often denoted
by [a]). The set of all equivalence classes form a partition of X. The set of
equivalence classes if often denoted by X/∼.
Conversely, it is easy to see that if {A } is a partition of X then we
i i∈I
my defined a corresponding equivalence relation by declaring x ∼ y if and
only if x and y belong to the same A .
i
For example, ordinary equality is an equivalence relation on the set of
natural numbers. Let Z denote the set of integers. Then equality modulo
a fixed integer p defines and equivalence relation on Z where n ∼ m iff1
n−m = kp for some k ∈ Z. In this case the set of equivalence classes is
denoted Z or Z/pZ.
p
Definition 0.3. A partial ordering on a set X (assumed nonempty) is
a relation denoted by, say (cid:185), that satisfies (i) x (cid:185) x for all x ∈ X, (ii) if
x (cid:185) y and y (cid:185) x then x = y, and (iii) if x (cid:185) y and y (cid:185) z then x (cid:185) z. We
say that X is partially ordered by (cid:185). If a partial ordering also has the
property that for every x,y ∈ X we have either x (cid:185) y or y (cid:185) x then we call
the relation a total ordering or (linear ordering). In this case, we say that
X is totally ordered by (cid:185).
Example 0.4. The set of real numbers is totally ordered by the familiar
notion of less than; ≤.
Example 0.5. The power set P(X) is partially ordered by set inclusion ⊂
(also denoted ⊆).
If X is partially ordered by (cid:185) then an element x is called a maximal
element if x (cid:185) y implies x = y. A minimal element is defined similarly.
1
“iff”means”ifandonlyif”.