-
Title: Smooth Structures
-
Series: Manifolds
-
YouTube-Title: Manifolds 12 | Smooth Structures
-
Bright video: https://youtu.be/VzldIIxogyA
-
Dark video: https://youtu.be/JmHRe1k2jCQ
-
Ad-free video: Watch Vimeo video
-
Quiz: Test your knowledge
-
Dark-PDF: Download PDF version of the dark video
-
Print-PDF: Download printable PDF version
-
Thumbnail (bright): Download PNG
-
Thumbnail (dark): Download PNG
-
Subtitle on GitHub: mf12_sub_eng.srt missing
-
Timestamps (n/a)
-
Subtitle in English (n/a)
-
Quiz Content
Q1: Which of the following maps can be a transition map for some manifold?
A1: $ \omega: \mathbb{R}^n \rightarrow \mathbb{R}^{n+1}, ~ x \mapsto x$
A2: $ \omega: [0,1] \rightarrow \mathbb{R}^{2}, ~ x \mapsto \binom{x}{1}$
A3: $ \omega: \mathbb{R}^n \rightarrow \mathbb{R}^{n}, ~ x \mapsto x$
A4: $ \omega: \mathbb{R}^2 \rightarrow \mathbb{R}^{2}, ~ x \mapsto 0$
Q2: Let $\omega: \mathbb{R}^n \rightarrow \mathbb{R}^{n}$ be a $C^k$-diffeomorphism for $k\geq1$. What is not correct in general?
A1: $\omega$ is a $C^{k-1}$-diffeomorphism.
A2: $\omega$ is a homeomorphism.
A3: $\omega$ is invertible.
A4: $\omega$ is continuous.
A5: $\omega$ is $(k+1)$-times continuously differentiable.
Q3: A pair $(M,\mathcal{A})$ is called a $C^k$-smooth manifold if
A1: $M$ is topological manifold and $\mathcal{A}$ is an atlas.
A2: $M$ is topological manifold and $\mathcal{A}$ is a maximal $C^k$-atlas.
A3: $M$ is topological manifold and $\mathcal{A}$ is a $C^k$-atlas.
-
Last update: 2024-10
