Linked Questions

0
votes
1answer
41 views

It is not clear to me if the disjoint union of a $d_{1}$-manifold and a $d_{2}$-manifold is still a manifold? [duplicate]

While it is clear that a disjoint union of two $d$-manifolds is a $d$-manifold, it is not clear to me if the disjoint union of a $d_{1}$-manifold and a $d_{2}$-manifold is still a manifold and if yes ...
3
votes
3answers
368 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 $[...
9
votes
1answer
367 views

What is/are the definitions of local diffeomorphism onto image?

In summary: Actually, I think the confusion arises from a distinction between (local diffeomorphism)-onto image and local-(diffeomorphism onto image). See (C1) at the end. Firstly, I believe this is ...
3
votes
2answers
275 views

Is $[0,1) \cup \{2\}$ a manifold with boundary? My issue is the $2$.

This has been asked about here: Understanding topological and manifold boundaries on the real line, and Sharkos said Personally I'd say $M$ wasn't a valid manifold with boundary because the $\{2\}$ ...
5
votes
2answers
131 views

What does it mean for a vector field to be “along” $\partial M$? I think “along” is a generalization of “on”.

My book is An Introduction to Manifolds by Loring W. Tu. The following is an entire subsection (Subsection 22.5) of the section that introduces manifolds with boundary (Section 22, Manifolds with ...
-1
votes
2answers
154 views

Do homeomorphic smooth manifolds, like diffeomorphic ones, have the same dimension? [duplicate]

For smooth manifolds $A$ and $B$ with respective dimensions $a$ and $b$. If $A$ and $B$ are diffeomorphic, then $a=b$. I guess the same is true for homeomorphic topological ($C^0$, I guess) manifolds (...
0
votes
1answer
143 views

Is $x^3-6xy+y^2=-108$ a regular submanifold but not a regular $k$-submanifold?

My book is An Introduction to Manifolds by Loring W. Tu. Let $S = \{x^3-6xy+y^2=-108\}$, and let "submanifold" and "$k$-submanifold" mean, respectively, "regular" and "regular $k$-submanifold". As in ...
0
votes
2answers
203 views

Manifolds of zero dimension and $\mathbb R^0$?

Tu Manifolds Section 5.4 Example 5.13 (Manifolds of dimension zero). In a manifold of dimension zero, every singleton subset is homeomorphic to $\mathbb R^0$ and so is open. Thus, a zero-...
2
votes
0answers
94 views

Does outward-pointing vector field mean each tangent vector at the boundary is outward-pointing?

My book is An Introduction to Manifolds by Loring W. Tu. The following is an entire subsection (Subsection 22.5) of the section that introduces manifolds with boundary (Section 22, Manifolds with ...
1
vote
0answers
66 views

Since not all compact subspaces of $\mathbb R^n$ are manifolds with boundaries, how can we this apply index of a vector field formula?

My book is From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave. I didn't study much of the definitions or theorems in Chapters 1 to 10 that might already be found in An Introduction to ...
1
vote
0answers
43 views

Local expression for a vector field along $\partial M$

My book is An Introduction to Manifolds by Loring W. Tu. The following is an entire subsection (Subsection 22.5) of the section that introduces manifolds with boundary (Section 22, Manifolds with ...