Supplement for Manifolds and Differential Geometry Jeffrey M. Lee

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18 0. Background Material GL(n, F) Dual Spaces, dual basis dual map contragedient multilinear maps components, change of basis tensor product Inner product spaces, Indefinite Scalar Product spaces O(V ), O(n), U(n) Normed vector spaces, continuity Canonical Forms 0.3. General Topology (under construction) metric metric space normed vector space Euclidean space topology (System of Open sets) Neighborhood system open set neighborhood closed set closure aherent point interior point discrete topology metric topology bases and subbases continuous map (and compositions etc.) open map homeomorphism weaker topology stronger topology subspace relatively open relatively closed relative topology
0.4. Shrinking Lemma 19 product space product topology accumulation points countability axioms separation axioms coverings locally finite point finite compactness (finite intersection property) image of a compact space relatively compact paracompactness 0.4. Shrinking Lemma Theorem 0.16. Let X be a normal topological space and suppose that {U α } α∈A is a point finite open cover of X. Then there exists an open cover {V α } α∈A of X such that V α ⊂ U α for all α ∈ A. second proof. We use Zorn’s lemma (see [Dug]). Let T denote the topology and let Φ be the family of all functions ϕ : A → T such that i) either ϕ(α) = U α or ϕ(α) = V α for some open V α with V α ⊂ U α ii) {ϕ(α)} is an open cover of X. We put a partial ordering on the family Φ by interpreting ϕ 1 ≺ ϕ 2 to mean that ϕ 1 (α) = ϕ 2 (α) whenever ϕ 1 (α) = V α (with V α ⊂ U α ). Now let Ψ be a totally order subset of Φ and set ψ ∗ (α) = ⋂ ψ(α) ψ∈Ψ Now since Ψ is totally ordered we have either ψ 1 ≺ ψ 2 or ψ 2 ≺ ψ 1 . We now which to show that for a fixed α the set {ψ(α) : ψ ∈ Ψ} has no more than two members. In particular, ψ ∗ (α) is open for all α. Suppose that ψ 1 (α), ψ 2 (α) and ψ 3 (α) are distinct and suppose that ψ 1 ≺ ψ 2 ≺ ψ 3 without loss. If ψ 1 (α) = V α with V α ⊂ U α then we must have ψ 2 (α) = ψ 3 (α) = V α by definition of the ordering but this contradicts the assumption that ψ 1 (α), ψ 2 (α) and ψ 3 (α) are distinct. Thus ψ 1 (α) = U α . Now if ψ 2 (α) = V α for with V α ⊂ U α then ψ 3 (α) = V α also a contradiction. The only possibility left is that we must have both ψ 1 (α) = U α and ψ 2 (α) = U α again a contradiction. Now the fact that {ψ(α) : ψ ∈ Ψ} has no more than two members for every α means that ψ ∗ satisfies condition (i) above. We next show that ψ ∗ also satisfies condition (ii) so that ψ ∗ ∈ Φ. Let p ∈ X be arbitrary. We have
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102 5. Chapter 5 Supplement We have
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104 5. Chapter 5 Supplement and so
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Chapter 6 Chapter 6 Supplement 6.1.
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6.4. Discussion on G bundle structu
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Chapter 8 Chapter 8 Supplement 8.1.
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8.3. Pseudo-Forms 121 Example 8.2.
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8.3. Pseudo-Forms 123 and if ω = o
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Chapter 11 Chapter 11 Supplement 11
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11.2. Singular Distributions 127 di
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132 12. Chapter 12 Supplement given
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136 12. Chapter 12 Supplement Conve
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138 12. Chapter 12 Supplement for a
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140 12. Chapter 12 Supplement same
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142 13. Chapter 13 Supplement Given
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144 14. Complex Manifolds v ∈ V a
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146 14. Complex Manifolds Now if f
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148 14. Complex Manifolds Thus we h
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150 14. Complex Manifolds Thought o
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152 14. Complex Manifolds and J ∗
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154 14. Complex Manifolds functions
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156 15. Symplectic Geometry On the
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158 15. Symplectic Geometry Definit
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160 15. Symplectic Geometry Theorem
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162 15. Symplectic Geometry (2) A J
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164 15. Symplectic Geometry In the
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166 15. Symplectic Geometry Theorem
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168 15. Symplectic Geometry Lemma 1
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170 15. Symplectic Geometry 15.9.0.
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172 15. Symplectic Geometry We will
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174 15. Symplectic Geometry with re
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176 15. Symplectic Geometry Definit
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178 16. Poisson Geometry Definition
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Chapter 17 Geometries The art of do
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17. Geometries 183 We use the term
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17. Geometries 185 Thus we can form
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17.1. Group Actions, Symmetry and I
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17.2. Some Klein Geometries 191 com
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17.2. Some Klein Geometries 193 The
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17.2. Some Klein Geometries 195 geo
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17.2. Some Klein Geometries 197 fun
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17.2. Some Klein Geometries 199 The
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17.2. Some Klein Geometries 201 Bas
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17.2. Some Klein Geometries 203 Bas
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17.2. Some Klein Geometries 205 Rem
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17.2. Some Klein Geometries 207 fro
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I 17.2. Some Klein Geometries 209 x
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17.2. Some Klein Geometries 211 bet
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Bibliography [Arm] M. A. Armstrong,
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Bibliography 215 [L1] S. Lang, Fund
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Index admissible chart, 75 almost c
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Supplement for Manifolds and Differential Geometry Jeffrey M. Lee

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