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Questions tagged [manifolds-with-boundary]

Manifolds are typically defined to be without boundaries (every point has a neighbourhood homeomorphic to an Euclidean open disc). Use this tag for the manifolds with boundaries, as well as manifolds with corners, and other such generalisations of the notion of a manifold.

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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 ...
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1answer
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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 ...
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1answer
58 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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29 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
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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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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 ...
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26 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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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?
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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 ...
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1answer
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$W$ 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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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?
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1answer
51 views

Manifolds with Boundary and Maximal Atlas

I was reading Tu's Introduction to Smooth 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 ...
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1answer
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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 ...
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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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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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133 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-...
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3answers
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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 ...
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1answer
52 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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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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52 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 ...
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1answer
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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 \...
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1answer
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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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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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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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55 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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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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37 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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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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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 ...
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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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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 ...
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1answer
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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 ...
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1answer
128 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
40 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\...
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Proof of that if $M$ is a $n$-manifold oriented, then $\partial M$ is a $(n-1)$-manifold oriented

I'm trying understand the result described on the title of the topic by the book "Differential forms and Applications" by do Carmo. The proof given by author can be found below $\textbf{Proposition ...
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1answer
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How is the “surface measure” on a manifold defined?

Let $k,n\in\mathbb N$ with $k\le n$ $M$ be a $k$-dimensional $C^1$-submanifold of $\mathbb R^n$ $\Omega\subseteq\mathbb R^k$ be open, $\phi:\Omega\to M$ be a global chart of $M$ and $$g_\phi(x):=\det\...
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Independence of definition of boundary point of a manifold

I'm using the book Differential Forms and Applications by Do Carmo in order to understand the theorem of Stokes on compact manifolds and I'm stuck in the following lemma: My doubt is why $(f_1^{-1} \...
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Extending functions on boundary into M as a harmonic function

I am trying to show that if $\varphi \in C^{\infty}(\partial M)$ then there is $\psi \in C^{\infty}(M)$ with $\psi|_{\partial M}=\varphi$ e $\Delta \psi=0$, where $M$ is a compact riemannian manifold ...
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Construction of a measure wich represents the surface area of a subset of a piecewise differentiable two-dimensional manifold with boundary

Let $\mathcal M\subseteq\mathbb R^3$ be a union of piecewise differentiable two-dimensional manifolds with boundaries. I want to construct a measure $A$ on $\mathcal M$ such that $A(D)$ is the surface ...
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1answer
57 views

A calculus of Jacobian on a surface

This question might be quite easy for someone used to this kind of object. Let us consider an open bounded domain $\Omega$ with $C^1$ boundary, $\Omega \subset \mathbb{R}^n$, and a vector $v \in \...
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Dirichlet principle on compact manifolds with boundary

How is it the Dirichlet's principle on compact manifolds with boundary $\partial M= \mbox{manifold boundary}$?. I've just found the Dirichlet's principle on domains $U \subset \mathbb{R}^{n}$, i.e, $U$...
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Given a family of 2D curves, find a 3D manifold whose geodesics project to the plane curves

Below is an image of the family of 2D curves for reference. The image is arbitrary. Still trying to formulate a concise question. I know that the geodesics for flat Euclidean Space are straight lines....
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205 views

Geodesic curvature change under conformal metrics

Suppose that $\sigma_0$ is a fixed metric on a compact riemannian 2-manifold $M$ with boundary $\partial M$. Let $\sigma=\rho \sigma_{0}$, where $\rho=e^{2\varphi}$ with $\varphi \in C^{\infty}(M)$, ...
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How to write Impermeable Boundary Condition for a scalar field in a circle

How can Impermeable Boundary Condition for field $f(r,\theta)$ in a circle be written? $R$ is radius of the circle and $f(r>R,\theta)=0$. $r$ and $\theta$ are polar coordinates.
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Is the geodesic diameter on simply connected domains with boundary always realized by points on the boundary?

Assuming standard euclidean metric, the geodesic diameter, of simply connected polygons in the plane is realized by a shortest connection between two vertex points. This result is e.g. referenced in "...
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1answer
84 views

Lemma in Milnor's Differential Topology

I'm reading the proof of a lemma in Milnor's Topology from the Differentiable Viewpoint, specifically Lemma 4 of Chapter 2. I am caught up on a detail. Essentially, it amounts to the following: Let $...
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1answer
197 views

Show that the closed $n$-ball $B^n(a)$ is a manifold

Show that the closed $n$-ball $B^n(a)$ is a manifold. I know how to show that $S^{n-1}(a)=\partial B^n(a)$ is an $n-1$ manifold without boundary. We consider the function $f(x)=a^2-\Vert x\Vert ^2$. ...
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Open subsets of the connected sum $M_1\# M_2$ [duplicate]

I'm trying to solve a problem in John Lee's ITM (Problem 4-19), but seems that i need helps now. Here's the problem : Let $M_1 \# M_2$ be a connected sum of $n$-manifolds $M_1$ and $M_2$. Show ...
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Show that ∂(M×N)=M × ∂N.

Let M smooth manifolds (without boundary) and N is a smooth manifold with boundary. Could someone help me to show that $∂(M × N) = M × ∂N$? I saw a suggestion here on the site how to do it, but I'm ...