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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1answer
34 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 ...
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0answers
11 views
Tangent Vectors : $\gamma\sim \gamma|_{[0,\epsilon)}$?
If $p\in \operatorname{Int} M$, given a smooth curve $\gamma: (-\epsilon,\epsilon)\to M$ such that $\gamma(0)=p$, then $\gamma|_{[0,\epsilon)}:[0,\epsilon)\to M$ is also a smooth curve such that $\...
3
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1answer
56 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 ...
4
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0answers
85 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 ...
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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' ...
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1answer
26 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\...
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0answers
22 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 ...
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0answers
35 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 , ...
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0answers
76 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 ...
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1answer
28 views
Show that the intersection of two objects is a manifold with boundary
The question is to find $a$ for which the intersection of the solid hyperboloid $x^2+y^2-z^2\leq a$ with $x^2+y^2+z^2 = 1$ is a manifold with boundary.
My attempt: Let $I$ be the intersection. ...
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1answer
20 views
A question about diffeomorphism between manifolds with boundary
I don't understand a couple of things about the following proof:
Statement: Suppose $f:X\rightarrow Y$ is a diffeomorphism of manifolds with boundary. Show that $\partial f:\partial X\rightarrow \...
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2answers
92 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 ...
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0answers
92 views
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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1answer
38 views
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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0answers
23 views
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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0answers
35 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), ...
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0answers
45 views
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 ...
1
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1answer
54 views
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 ...
0
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1answer
60 views
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{...
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0answers
30 views
What is the ring and does $\phi^*$ preserves module operations
I have posted this problem recently and I think that as it was too long, not many people took interest in it. So I am reposting it also I am giving you an earlier link :I had a problem in manifold ...
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2answers
57 views
Retract of noncompact surface to its boundary?
Suppose $M$ is a connected, noncompact 2-manifold, and its boundary $\partial M$ is a circle. What's the simplest way to show there is a retraction $r: M\rightarrow \partial M$?
Here are some ...
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0answers
28 views
The orientation induced on the boundary of a manifold.
I just learned about the notion of orientability of a manifold which is difficult and abstract for me. If we consider all basis of a vector space, the matrix that transforms one basis in another basis ...
2
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0answers
31 views
Homotopy type of smooth manifold with boundary
It seems very likely to me that a $n$-dimensional smooth manifold with boundary has the homotopy type of a $(n-1)$-dimensional CW complex. Is that true? Does the manifold need to be compact? What ...
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0answers
18 views
Are the two definitions of that a map $F:A\to N$ is smooth on $A$ equivalent?
In the above pictures, we can see two definitions of that a map $F:A\to N$ is smooth on $A$, are they equivalent?
3
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1answer
67 views
Is this conjecture about the boundary of a surface correct?
I came up with an intuitive conjecture about boundaries of surfaces based on the idea that at a boundary point we can wrap a string across the edge, and the two halves of the string (on opposite sides ...
5
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1answer
70 views
$W$ spin implies $\partial W$ spin
Let $M$ be a compact orientable manifold with the first two Stieffel-Whitney numbers equal to zero (this is my definition of SPIN manifold). Let $B$ be the boundary of $M$; I want to show that $B$ is ...
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1answer
33 views
A difficulty in understanding a part of a paragraph in Guillemin & Pollack p.60
I do not understand the highlighted part of the paragraph given below:
Could anyone explain it for me please?
2
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1answer
77 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 ...
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1answer
23 views
What's the meaning of being expressible as a convergent power series in a neighborhood of each point?
The following pictures are from Lee's "Introduction to Smooth Manifolds".
What's the meaning of being expressible as a convergent power series in a neighborhood of each point? However, I only know ...
3
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0answers
187 views
Constant Rank Theorem for Manifolds with Boundary
I'm trying to answer problem 4-3 from Lee's Introduction to Smooth Manifolds, 2nd edition. The problem says:
Formulate and prove a version of the rank theorem for a map of constant rank whose ...
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1answer
37 views
if $f: X \rightarrow Y$ is a diffeomorphism of manifolds with boundary, then $\partial f$ maps $\partial X$ diffeomorphically onto $\partial Y$.
Prove that if $f: X \rightarrow Y$ is a diffeomorphism of manifolds with boundary, then $\partial f$ maps $\partial X$ diffeomorphically onto $\partial Y$.
Could anyone give me a hint for the proof ...
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0answers
143 views
The boundary is disjoint from the interior in 2d manifold
I'm self-studying on Lee's Introduction to Topological Manifolds, and I'm stuck on a problem (8-6). I'm missing something really basic, I fear.
I must prove that the set of the boundary points of a 2-...
3
votes
3answers
113 views
No where vanishing exact $1$-form on compact manifold.
I found several answers on the following question :
Does there exists a no where vanishing exact $1$-form on a compact manifold without boundary?
All answer says that certainly not. But I cannot ...
1
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1answer
71 views
Orientable manifold $M$ ,then $\partial M$ is orientable
Let $M$ a topological manifold of dimension $n$ with boundary $\partial M$.
We define $M$ to be orientable if $M- \partial M$ is orientable. Here when I say orientable, I mean there is a locally ...
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0answers
29 views
preserves or reverse orientation of sphere surface
Let $\varphi: (0, \infty) \times (0, \pi) \times (0, 2 \pi) \to \mathbb{R}^3 \setminus \{(x,y,z) \in \mathbb{R}^3| y=0, x \geq 0 \}$
$$(p,\phi,\theta) \mapsto (p \sin \phi \cos \theta, p \sin \phi \...
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0answers
62 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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1answer
48 views
The riemannian metric of a neighborhood of the boundary of a compact manifold
Let $M$ be a compact riemannian manifold with boundary $\partial M$. We have that $\partial M$ is also compact and I was able to show that there is some $a>0$ such that the map
$F:[0,a]\times \...
3
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1answer
51 views
Existence of boundary cylindrical neighborhood for a compact manifold
Let $M$ be a compact riemannian manifold with boundary $\partial M\neq \varnothing$. I would like to show that there is some neighborhood $U$ of $\partial M$ which is diffeomorphic to $[0,a)\times \...
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0answers
32 views
Reference request Eigenspace decomposition Hodge Laplacian on forms on manifolds with boundary
I know that on a connected, compact, oriented Riemannian manifold without boundary the Hodge Laplacian $\Delta_k=(d+\delta)^2$ (acting on $k$-forms) admits an orthogonal eigenspace decomomposition of $...
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1answer
39 views
Prove there exists a outward unit normal field on the boundary this manifold
Let $M$ be a compact subset of $\mathbb{R}^3 $ with the standard orientation $\mu =[e_1,e_2,e_3] $ and let $S = \partial{M}$ is its smooth boundary with the induced orientation from $M$. Prove there ...
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1answer
59 views
Manifold with boundary and topological boundary
Why aren't the topological boundary of a manifold with boundary and the boundary of the manifold the same sets?
We defined the boundary of a manifold as follows:
$\partial M:= \lbrace p \in M : \...
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0answers
44 views
Lipschitz and / or piecewise smooth boundary
Let $\Omega_3 = \overline{Q_1 \cup Q_2}$ with $Q_1 = (-1, 3) \times (0,2) \times (-1, 0)$ and $Q_2 = (0,2) \times (-1, 3) \times (0,1)$
Here is what I've done until now:
$\Omega_3$ can be imagines as ...
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0answers
38 views
Image of smooth map lies in interior [duplicate]
Problem 4.2, pg 98, John Lee's smooth manifold: Suppose $M$ is a smooth manifold (without boundary), $N$ is a smooth manifold with boundary, $F:M \rightarrow N$ is smooth. Show that if $p \in M$ is a ...
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0answers
19 views
Understanding the definition of boundary points of a manifold
Let $M$ be a topological manifold. We call it a $n$-manifold with boundary if for each $x\in M$, there is a chart $(U,\phi)$ at $x$ such that $\phi$ is a homeomorphism from $U$ to an open subset of $\...
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0answers
51 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
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1answer
37 views
When both $U$ and $W$ are open in $\mathbb{H}^k$ and $\mathbb{H}^l$, respectively, then why $U\times W$ cannot be open in $\mathbb{H}^{k+l}$
In the book of Analysis on Manifolds by Munkres, at page 202 question 5, it is asked that
Show that if $M$ is a k-manifold without boundary in $\mathbb{R}^m$,
and if $N$ is an l-manifold in $\...
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0answers
79 views
Boundaries of manifolds and algebraic topology
I would like to know how to tackle questions of the following type:
Show that $\mathbb{CP}^{2n}$ is not the boundary of any manifold.
Another such question would be:
Let $\iota: S^1 \to S^3$ be ...
2
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1answer
41 views
Why the boundary of $M_1\# M_2$ is incompressible if the boundaries of $M_i$ are incompressible?
This may be a stupid question, because I am completely new to $3$-manifolds and get stuck (and honestly, a bit confused) in the following problem.
Let $M_i$ ($i=1,2$) be $3$-manifolds with boundary ...
2
votes
1answer
144 views
When is a minimal geodesic a shortest path?
Let $S$ be a surface. What conditions can we place on $S$ so that for any two points $a$, $b$ on $S$, the minimal geodesic from $a$ to $b$ is the shortest path on $S$ from $a$ to $b$ ?
Why I am ...
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1answer
44 views
Existence of a unique “outer” normal vector
A compact set $K\subseteq\mathbb R^3$ is said to have a smooth boundary, if for all $p\in\partial K$ there is an open neighborhood $U$ of $p$ and a continuously differentiable function $\psi:U\to\...
