Linked Questions

2
votes
1answer
110 views

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. ...
1
vote
0answers
78 views

What is the induced orientation on a 1-manifold with boundary that is the image of closed interval under these maps?

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, Prof Jack Lee says that the example is ...
6
votes
1answer
366 views

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 ...
4
votes
2answers
647 views

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 (...
4
votes
3answers
687 views

“A manifold with boundary has dimension at least 1” if it has a dimension and if it has nonempty boundary?

My book is An Introduction to Manifolds by Loring W. Tu. As can be found in the following bullet points Can a topological manifold be non-connected and each component with different dimension? Is $[...
5
votes
1answer
137 views

Does classification of 1-manifolds with boundary give induced orientation of image of closed interval under a smooth immersion?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. I have been trying to wrap my head around this for about 2 hours (...
3
votes
1answer
166 views

$[a,b]$ 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 $[0,\infty)$...
2
votes
0answers
160 views

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 ...
2
votes
0answers
86 views

Munkres positive linear map definition of path product (page 328)

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 + ...
5
votes
2answers
3k views

Composition of injections (proof)

I'm trying to prove that composition of injections is an injection. I want to know if this is a good proof: Composition of injections is an injection. Let $f:S_1\rightarrow S_2$ and $g:S_2\...
19
votes
1answer
2k views

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 ...