# Questions tagged [manifolds-with-boundary]

For questions about manifolds with boundaries, as well as manifolds with corners, and other such generalisations of the notion of a manifold.

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### Is $TN$ a smooth embedded submanifold with or without boundary in $TM$?

Let $M$ be a smooth manifold with or without boundary, and $N$ a smooth embedded submanifold with or without boundary in $M$, then the inclusion map $N\to M$ induces $TN\to TM$, is $TN$ a smooth ...
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### If $M$ is a smooth manifold with boundary, $f \in C^\infty(M)$, $b$ a regular value of $f$, then is $f^{-1}((-\infty,b])$ a regular domain in $M$?

The above is taken from John Lee's Introduction to Smooth Manifolds (p. 121). In Proposition 5.47, one supposes that $M$ is a smooth manifold. Does Proposition 5.47 also hold if $M$ is changed to a ...
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### How to conclude each $E_i$ is a compact regular domain by Proposition 5.47?

In Proposition 5.47, $M$ is a smooth manifold, but in Theorem 6.15, $M$ is a smooth manifold with or without boundary, how to conclude each $E_i$ is a compact regular domain by Proposition 5.47?
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### Is any map $f:M\to N$ smooth?

Let $M$ be a 0-dimensional smooth manifold and $N$ a smooth manifold with or without boundary, is any map $f:M\to N$ smooth?
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### Is a compact connected manifold-with-boundary a CW complex?

Suppose $M$ is a compact connected manifold-with-boundary with non-empty boundary. What can be said on whether $M$ can be given a CW complex structure? A similar problem has been discussed for ...
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### Special Riemannian metric on the product

Let $M^2$ be an orientable surface with boundary endowed with a Riemannian metric $g$. We know that if we put in the manifold $M \times \mathbb{R}$ the product metric $\overline{g} = g + dt^2$, then ...
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### How can we apply generalized Stokes' theorem to a non-oriented manifold with boundary?

I do not really know much about the boundary of non-oriented manifold. A boundary of oriented manifold, if it exists, has a sign. If you reverse the orientation, the boundary picks up an extra ...
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### Spin structure and bordism

I have some questions about bordism and spin structures on manifolds. If you have any examples or references I would appreciate it. Is there a 3-manifold $M$, orientable, which does not support 3 ...
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### Is there Method to visualize the object $Disc \times Disc$?

For me its clear how to build up the object $S^1 \times S^1$ , who is our old friend torus, but the product of 'interior'of these objetcs doesn't look clear to me how to build up I tried 'forget' ...
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### $\mathbb S^2$ or $\mathbb RP^2$ on boundary of a 3-manifold

Let $M$ be a 3-manifold with boundary $\partial M$. Suppose $\partial M$ contains a sphere or a projective plane, which is contractable in $M$. Show that $M$ is also contractable. The above statement ...
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### Integration of forms over manifolds

Spivak start the section 'Stokes theorem on manifolds' of his book 'Calculus on manifolds' with the following definition: If $\omega$ is a $p$-form on a $k$-dimensional manifold with boundary $M$ ...
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### Question on the definition of tangent space at boundary points

I'm going thought Spivak's 'Calculus On Manifolds' and I am bit confused with certain things in the definition of outward normal unit vector. For the notation. With $v_{p}$ Spivak denotes the tangent ...
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### Manifold Optimization in Human understandable language

I am coming across a project that involves with "manifold optimization". I don't know what that is. After Googling for a bit, my understanding is that we use a low dimensional surface to approximate ...
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### Several questions in Lee's <Introduction of smooth manifolds>, chapter 1,2

(1), In the proof of Lemma 1.10: Every topological manifold has a countable basis of precompact coordinate balls, what is the function of $B_{r^{'}}(x)$? Why do we need it? Proof of Lemma 1.10 (2), ...
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### Deduce the curvature of the manifold by probing it with geodesics

Suppose in order to probe the curvature of a manifold in $\Bbb R^2$, you decide to shoot off geodesics from the four corners of the square. Consider a set of geodesics, $K,$ each symmetrical about ...
### Why does any connected closed $m$-manifold that can be embedded in $E^{m+1}$ bounds a compact connected $(m+1)$-manifold?
I am reading Sheila Carter and S.A. Robertson's paper Relations Between a Manifold and its Focal Set. In this paper, they use the following facts: Any closed $m$-manifold $M$ that can be embedded ...
### Existence of a special homeomorphism on $\mathbb{T}^2$.
Let $A, B$ be closed topological subspaces of $\mathbb{T}^2$. Suppose that $A$ and $B$ are homeomorphic as topological spaces. My Question: Is it possible to construct a homeomorphism \$h: \mathbb{...