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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6
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1answer
54 views

What is the boundary of a tubular neighbourhood of the projective plane embedded in $\mathbb{R}^4$?

There are various ways to embed the projective plane into $\mathbb{R}^4$ very nicely, see e.g. Wikipedia. Suppose now that I take such an embedded projective plane $P \subset \mathbb{R}^4$ and fix a ...
2
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1answer
64 views

Is this a manifold with boundary?

I am new to differential geometry/manifolds. I have two questions: 1) Are two disks tangentially touching each other a manifold with boundary. ie: is this set a manifold with boundary $$\{(x,y):(x-1)...
1
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0answers
68 views

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), ...
3
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1answer
344 views

The boundary of a manifold is a closed subset.

We want to show that the boundary $\partial M$ of an $n-$manifold M is a closed subset of the manifold. We show that its complement $M\setminus\partial M$ is open in $M$. Indeed, each point $x \in M\...
10
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1answer
77 views

Euler characteristic of a manifold is odd

This was a past exam question: Let $M$ be a compact connected orientable topological $n$-manifold with boundary $\partial M$ so that $H_*(\partial M;\mathbb{Q}) \cong H_*(S^{n-1};\mathbb{Q})$. If $n \...
2
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2answers
25 views

Are there metrics of nonnegative Gaussian curvature on these surfaces?

Let $\Sigma$ be a compact surface of genus $g \geq 1$ and having $r \geq 1$ boundary components. Are there metrics of nonnegative Gaussian curvature on $\Sigma$? If $\Sigma$ were closed, then the ...
0
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0answers
23 views

What is the orientation of the normalized boundary $\partial(M\times N)$ of product manifold?

Assume $M$ and $N$ are two oriented smooth manifold with or without boundaries. Then $M\times N$ is an oriented manifold with corners. Inspired by the theory of cobordism or differential forms, the ...
0
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0answers
35 views

Counting closed orbits on a compact surface topologically equivalent to a sphere

Previously I asked a question about a surface formed by gluing together pieces of spheres, or, contracting overlayed copies of spheres, or configuring $4$ twice pointed spheres inside a region of ...
1
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2answers
54 views

Is it possible to construct a 3D equivalent of Gabriel's Horn in a higher dimensional space?

Gabriel's Horn has the interesting property that it is an infinite surface area bound within a finite volume. I was wondering if there was an extension of this to 3D space in a higher dimensional ...
6
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1answer
1k views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves $\...
1
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1answer
65 views

Do homeomorphic manifolds with boundary have homeomorphic interiors?

Let $M, N$ be manifolds with boundary and $f: M \rightarrow N$ be a homeomorphism. I want to show that $\text{Int}M$ is homeomorphic to $\text{Int}N$. I think I have most of the proof but it relies on ...
5
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1answer
53 views

If the interior of a manifold with boundary is smooth, is the whole manifold smooth?

Let $M$ be a topological manifold with boundary. Let Int$M$, its interior, be a smooth manifold. Is it a known result that $M$ itself will be a smooth manifold with boundary? Can we extend a smooth ...
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0answers
13 views

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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0answers
27 views

Tangent vectors at boundary points

It is well known that we can define the tangent space of a manifold $M$ at a point $p\in M$ as the set of speeds, at time $0$, of curves $\alpha : (-\varepsilon, \varepsilon) \to M$ such that $\alpha(...
1
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1answer
30 views

Does there exist a compact smooth embedding submanifold $N\subset M$ with or without boundary such that $N\supset A$?

Let $M$ be a smooth manifold with or without boundary and $A$ a compact subset of $M$, does there exist a compact smooth embedding submanifold $N\subset M$ with or without boundary such that $N\...
1
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1answer
35 views

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 ...
0
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1answer
19 views

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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0answers
32 views

Why do Guillemin and Pollack avoid the term “submanifold with boundary”?

On p. 60 of the book "Differential Topology" by named authors they state: Theorem. Let $f$ be a smooth map of a manifold $X$ with boundary onto a boundaryless manifold $Y$, and suppose that both $f:...
1
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1answer
85 views

Manifolds with corners

Here is what I'm experiencing. A text introduces manifolds. Then, manifolds with boundary is introduced. After all, manifolds with corners is introduced. First of all, I checked whether Implicit ...
2
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1answer
31 views

Does Proposition 5.18 also hold if $N$ is changed to a smooth manifold with boundary?

Does Proposition 5.18 also hold if $N$ is changed to a smooth manifold with boundary?
2
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0answers
21 views

Extending a triangulation from a manifold boundary to the interior

Let $M$ be a (second-countable topological) compact connected manifold-with-boundary. Suppose $\partial M$ has a triangulation. Does there exist a triangulation of $M$ which extends the triangulation ...
0
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1answer
16 views

What's the meaning of projection with kernel $\Bbb Rv$?

In Lemma 6.13, it says "For any $v\in\Bbb R^N\setminus \Bbb R^{N-1}$, let $\pi_v:\Bbb R^N\to \Bbb R^{N-1}$ be the projection with kernel $\Bbb Rv$". What's the meaning of projection with kernel $\Bbb ...
1
vote
2answers
63 views

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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0answers
17 views

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 ...
4
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0answers
129 views

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 ...
1
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1answer
21 views

Deformation retraction onto an open subset of manifold boundary

I'd like to prove the following result, perhaps with additional assumption if needed -- I don't know whether the claim holds. Let $M$ be a compact connected manifold with boundary $\partial M$. Let $U'...
0
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0answers
14 views

In- and out-boundaries of the unit interval

I am trying to understand orientation of one-point manifolds in order to figure out in- and out-boundaries of the unit interval. So far I have understood that an orientation of a zero-manifold is an ...
0
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0answers
11 views

What's the meaning of “critical value of $F|_{\overline{V_a}}$” and “critical value of $F_c$”?

In Lee's book "Introduction to Smooth Manifolds", "critical value" is defined for smooth maps between smooth manifolds with or without boundary. But in the Proof of Sard's Theorem, the author doesn't ...
3
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1answer
77 views

Smooth map between smooth manifold and boundary of manifold

This is my first question here so I'm asking for your understanding. I also apologize in advance for my English. I'm beginner on smooth manifolds topics and I can't solve the following problem. Let ...
2
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0answers
94 views

Boundary of the image is the image of the boundary.

Suppose that $f:O\to \mathbb R^n$ is a smooth change of variables on the open subset O of $\mathbb R^n$. Let D be an open subset of $\mathbb R^n$ such that $D\cup \partial D$ is contained in O. Show ...
2
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0answers
35 views

Gauss-Bonnet theorem for vector bundles on manifolds with boundary

I hope this question is not a duplicity, but I really failed to find a good reference for it. I am wondering whether there is a generalization of the Gauss-Bonnet theorem to real vector bundles on a ...
0
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0answers
15 views

Do we have $f(M)\subset \operatorname{Int}N$?

Let $M$ be a connected smooth manifold and $N$ a smooth manifold with boundary. If $f: M\to N$ is a smooth map of constant rank and there exists $p\in M$ such that $f(p)\in \operatorname{Int}N$, do we ...
1
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1answer
25 views

Can $dF_p:T_pM\to T_{F(p)}N$ be surjective?

Suppose $M$ is a smooth manifold and $N$ a smooth manifold with boundary, $F:M\to N$ a smooth map, if $F(p)\in \partial N$, can $dF_p:T_pM\to T_{F(p)}N$ be surjective?
2
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0answers
32 views

Are integral curves on a connected manifold always path-connected?

I am working out some math involving integral curves of a gradient field on a smooth, connected manifold with boundary. Let's assume the function that provides the gradient field is smooth or even ...
7
votes
1answer
500 views

The definition of smooth maps given in Introduction to Smooth manifolds by John M. Lee

I'm currently reading Introduction to Smooth Manifolds by John M. lee. I'm trying to understand his definition of smooth maps $F:A\subseteq M\to N$ given in page 45. Let's start from scratch to have ...
0
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0answers
29 views

What is an orbifold with corners

Can one have a formal definition of orbifold with corners? note that it is not parallel to the definition of manifold with corners, as a manifold with boundary is already an orbifold(not with boundry)....
1
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0answers
38 views

Embedding a Riemannian manifold with boundary in a closed manifold

Let $M$ be a complete Riemannian manifold of finite volume, with sectional curvatures bounded by some $K>0$ in absolute value. Let $M_{\geq R}$ be the set of points in $M$ with injectivity radius $\...
5
votes
2answers
121 views

Classification of surfaces

The Classification Theorem for surfaces says that a compact connected surface $M$ is homeomorphic to $$S^2\# (\#_{g}T^2)\# (\#_{b} D^2)\# (\#_{c} \mathbb{R}P^2),$$ so $g$ is the genus of the surface, $...
2
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1answer
674 views

Showing that $\bar{\mathbb{B}}^n$ is a manifold with boundary (Lee ITM Probelm 3-4)

"Show that every closed ball in $\mathbb{R}^n$ is an $n$-dimensional manifold with boundary, as is the complement of every open ball. Assuming the theorem on the invariance of the boundary, show that ...
0
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0answers
45 views

Uniqueness of integral curves starting on boundary?

I'm only aware of ODE uniqueness on open subsets of $R^n$, which is useful when we want to prove uniqueness of integral curves on a smooth manifold. Suppose we have an interal curve $f:[0,\delta]\to ...
1
vote
1answer
54 views

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 ...
5
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0answers
99 views

Is a manifold-with-boundary with given interior and non-empty boundary essentially unique?

Let $M$ be a compact connected manifold-with-boundary such that $\circ M \neq \emptyset$, where $\circ M$ is the boundary of $M$. Let $N$ be a compact connected manifold-with-boundary such that $\circ ...
2
votes
1answer
98 views

Manifolds with Boundary and Maximal Atlas

I was reading Tu's An Introduction to Manifolds and learned about the notion of manifolds with boundary. But there was a point which was not clear to me. Here are the definitions(I will use the word ...
3
votes
1answer
78 views

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 ...
0
votes
1answer
20 views

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' ...
0
votes
1answer
30 views

Question about manifolds with boundary

Prove that if $f:X\to Y$ is a diffeomorphism of manifolds with boundary, then $f$ maps $\partial X$ to $\partial Y$ diffeomorphically. Answer: Let $U\subset H^k$ be an open subset and let $\phi:U\...
0
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0answers
38 views

geodesic balls in Riemannian manifolds with bounded geometry

Let $(M,g)$ be an open (:=complete, non-compact) Riemannian manifold with bounded geometry, in the sense that in some atlas of charts of radius $r_0>0$, the metric and all its derivatives are ...
2
votes
2answers
98 views

Searching for a connected neighborhood of fixed radius for every point of a submanifold

Let $\Gamma=\gamma(\partial D)\subset \mathbb R^3$ be an image of $C^1$-curve, where $D$ is an open ball in $\mathbb R^2$. My question is whether it is possible to find a fixed $\epsilon >0$ such ...
4
votes
0answers
111 views

Spivak Calculus on Manifolds - Tangent space on a boundary point of a manifold

I am an undergraduate student who is studying Spivak's calculus on manifolds. I have several questions in the pages 119 and 120 of the book, which are about the tangent space at a boundary point of a ...
0
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0answers
47 views

Spivak manifolds - definition of $dw$ for a p-form $w$ on a manifold $M$

Spivak says the definition of $dw$ for a k-form $w$ does not make sense on a manifold because $D_j(w_{i_1, \dots , i_p})$ has no meaning. Does it have no meaning because the function w_{i_1, \dots , ...