Differential-Geometry of Manifolds Fall 2017 Manifolds, guaranteed and verified 100% Pass

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Manifolds
Def. Let 𝑈 and 𝑉 be open sets in ℝ𝑛 . A differentiable function, ℎ: 𝑈 → 𝑉 with
a differentiable inverse ℎ−1 : 𝑉 → 𝑈, is called a diffeomorphism
(“differentiable” will mean 𝐶 ∞ from here on).

Def. A subset, 𝑀 ⊆ ℝ𝑛 , is called a differentiable manifold (or just a
manifold) of dimension 𝑘 if for each point 𝑥 ∈ 𝑀 there is an open
set 𝑊 ⊆ ℝ𝑛 , an open set 𝑈 ⊆ ℝ𝑘 , and a diffeomorphism:
ℎ: 𝑊 ∩ 𝑀 → 𝑈.
ℎ is called a system of coordinates on 𝑊 ∩ 𝑀.
ℎ−1 : 𝑈 → 𝑊 ∩ 𝑀 is called a parameterization of 𝑊 ∩ 𝑀.

𝑀 𝑊∩𝑀
𝑥

ℎ


ℎ−1 𝑈




The set {ℎ𝛼 , 𝑊𝛼 } of coordinate functions and sets 𝑊𝛼 that cover 𝑀 is called an
atlas.



Ex. A point in ℝ𝑛 is a zero dimensional manifold.
An open set in ℝ𝑛 is an 𝑛-dimensional manifold.

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Notice that if (ℎ1 , 𝑊1 ) and (ℎ2 , 𝑊2 ) are two coordinate systems on
𝑊1 , 𝑊2 ⊆ 𝑀, where ℎ1 : 𝑊1 → 𝑈1 and ℎ2 : 𝑊2 → 𝑈2 , then:

ℎ12 = ℎ2 ℎ1−1 : ℎ1 (𝑊1 ∩ 𝑊2 ) → ℎ2 (𝑊1 ∩ 𝑊2 )

is a differentiable map of an open set in ℝ𝑘 into an open set in
ℝ𝑘 , and is called a transition function between the coordinate
systems (ℎ1 , 𝑊1 ) and (ℎ2 , 𝑊2 ).


𝑊1 𝑊1 ∩ 𝑊2 𝑊2



𝑀




ℎ2−1
ℎ1
ℎ2
ℎ1−1



𝑈1 𝑈2
ℎ2 ℎ1−1
ℎ1 (𝑊1 ∩ 𝑊2 ) ℎ2 (𝑊1 ∩ 𝑊2 )




Def. An atlas (ℎ𝛼 , 𝑊𝛼 ) is called smooth if all of the transition functions are smooth.