Chapter 1(Smooth Manifolds)

Topological Manifolds

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Smooth Structure

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Examples of Smooth Manifolds

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Manifolds with Boundary

Definition

Definition. Closed n-dimensional upper half-space Define ${\mathbb{H}}^n$ as ${\mathbb{H}}^n=\left\{\left(x^1,\dots ,x^n\right)\in {\mathbb{R}}^n:x^n\ge 0\right\}$. Then closed n-dimensional upper half-space ${\mathbb{H}}^n\subseteq {\mathbb{R}}^n$ $\text{Int}{\mathbb{H}}^n=\left\{\left(x^1,\dots ,x^n\right)\in {\mathbb{R}}^n:x^n>0\right\}$. $\partial {\mathbb{H}}^n=\left\{\left(x^1,\dots ,x^n\right)\in {\mathbb{R}}^n:x^n=0\right\}$.

We just need to define a chart $\left(U,\phi \right)$ for $\mathbf{M}$ Interior chart if $\phi \left(U\right)$ is an open subset of ${\mathbb{R}}^n$ Boundary Chart if $\phi \left(U\right)$ is an open subset of ${\mathbb{H}}^n$ such that $\phi \left(U\right)\cap \partial {\mathbb{H}}^n\ne \varnothing$

Theorem 1.37. Topological Invariance of the Boundary

If $\mathbf{M}$ is a topological manifold with boundary, then each point of $\mathbf{M}$ is either a boundary point or an interior point, but not both. Thus $\partial \mathbf{M}$ and $\text{Int}\mathbf{M}$ are disjoint sets whose union is $M$.

Differences about boundary in topological of manifold

In the point of view the closed unit ball ${\bar{\mathbb{B}}}^n$, is a manifold with bounday, whose manifold boundary is ${\mathbb{S}}^{n-1}$ as well as view it as a subset of ${\mathbb{R}}^n$ then topological boundary is ${\mathbb{S}}^{n-1}$.

However, if we think of ${\bar{\mathbb{B}}}^n$ as a topological space in its own right, then as a subset of itself, it has empty topological boundary. Also, if we think of it as a subset of ${\mathbb{R}}^{n+1}$, its topological boundary is all of ${\bar{\mathbb{B}}}^n$ is all of itself.

Note that ${\mathbb{H}}^n$ is tiself a manifold with boundary, and its manifold is the same as its topological boundary as a subset of ${\mathbb{R}}^n$.

Every interval in $\mathbb{R}$ is a 1-manifold with boundary, whose manifold boundary consists of its endpoints.

Definition. Closed & Open Manifold

• Closed Manifold

  A compact manifold without boundary

• Open Manifold

  A noncompact manifold without boundary

Proposition 1.38

Let $\mathbf{M}$ be a topological $n$-manifold with boundary

1. $\text{Int}\mathbf{M}$ is an open subset of $\mathbf{M}$ and a topological $n$-manifold without boundary.
2. $\partial \mathbf{M}$ is a closed subset of $\mathbf{M}$ and a topological $\left(n-1\right)$-manifold without boundary.
3. $\mathbf{M}$ is a topological manifold $⟺$ $\partial \mathbf{M}=\varnothing$
4. If $n=0$, then $\partial \mathbf{M}=\varnothing$ and $\mathbf{M}$ is a $0$-manifold.

Smooth Structures on Manifolds with Boundary