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### Is $[0,1]$ an *oriented* manifold with boundary? (and Stokes theorem)

The definitions I am using are a manifold with boundary is something locally homeomorphic to $(0,1] \times \mathbb{R}^n$ or $\mathbb{R}^n$. an oriented manifold is one where the transition functions ...
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### Pushfoward of smooth vector field is smooth?

My books are Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (Volume 2) and An ...
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### Does the Riemannian metric induced by a diffeomorphism $F$ exist for a reason other than the existence of vector field pushforwards?

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (...
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### $[a,b]$ as a smooth manifold with boundary has global coordinates?

Consider a compact interval $[a,b]$. If $[a,b]$ had global coordinates, then there would be an homeomorphism $f:[a,b]\to U$ where $U$ is an open subset of $\mathbb{R}$ or an open subset of $[0,\infty)$...
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### Is a smooth immersion $c: [a,b] \to M$ injective if its image is a 1-manifold with non-empty boundary?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. In this question, it is confirmed that the example is an error. ...
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### Boundary of a one-dimensional manifold - choosing an oriented atlas

I am trying to understand how to assign an orientation to the boundary of one-dimensional manifolds using Loring Tu's book on manifolds. This is what I got so far: Let $M$ be an oriented manifold. If ...
I'm confused by Munkres' definition of the path product using the positive linear map. He defines the positive linear map $p: [a,b] \rightarrow [c,d]$ to be the unique map of the form \$x \mapsto mx + ...