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Title: Examples of Smooth Manifolds
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Series: Manifolds
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YouTube-Title: Manifolds 13 | Examples of Smooth Manifolds
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Bright video: https://youtu.be/wMt5iaaQxjU
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Dark video: https://youtu.be/DvVIHv0IcU0
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: mf13_sub_eng.srt missing
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $U$ be an open subset of $\mathbb{R}^n$. Is this a smooth manifold?
A1: Yes, an atlas is given by one chart and can be extended to a smooth structure.
A2: No, there is no way to define an atlas for $U$.
Q2: Is the map $\omega: [0,1] \rightarrow \mathbb{R}$ given by $$ \omega(x) = \sqrt{1-x}$$ a $C^1$-diffeomorphism?
A1: No, it’s not differentiable at $0$.
A2: No, it’s not differentiable at $1$.
A3: Yes, it’s differentiable everywhere and bijective.
A4: Yes, it’s differentiable everywhere, bijective and the inverse is also differentiable.
Q3: Is the map $\omega: (0,1) \rightarrow \mathbb{R}$ given by $$ \omega(x) = \sqrt{1-x}$$ a $C^1$-diffeomorphism?
A1: No, it’s not differentiable at $\frac{1}{2}$.
A2: No, it’s not differentiable at all points.
A3: Yes, it’s differentiable everywhere and bijective.
A4: Yes, it’s differentiable everywhere, bijective and the inverse is also differentiable.
Q4: Is the sphere $S^n$ with the atlas given in previous videos a $C^\infty$-smooth manifold?
A1: Yes, it’s a $C^\infty$-smooth atlas which can be extended to a maximal one.
A2: No, it’s not a $C^\infty$-smooth atlas!
A3: No, it’s a $C^\infty$-smooth atlas but it cannot be extended to a maximal one.
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Last update: 2024-10
