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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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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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1answer
62 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 ...
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1answer
60 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
29 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?
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1answer
42 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
20 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 ...
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108 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
28 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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119 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
69 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 ...
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1answer
47 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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21 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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43 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
38 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 \...
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1answer
45 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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28 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
22 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
45 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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24 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
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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0answers
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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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30 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 ...
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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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67 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 ...
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1answer
38 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 ...
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1answer
113 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
37 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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1answer
52 views

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
69 views

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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35 views

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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0answers
42 views

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
54 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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0answers
56 views

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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208 views

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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1answer
176 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
80 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
145 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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1answer
101 views

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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2answers
72 views

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 ...
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0answers
41 views

About a regular value of a manifold with boundary

I don't understand the proof of the following theorem. Thorem: Let $f:X→N$ be a smooth mapping, where $X$ is $m$ dimensional smooth manifold with boundary and $N$ is $n$ dimensional smooth manifold. ...
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209 views

Homeomorphism between $[0,\infty)^n$ with Upper Half-Space $\mathbb{H}^n$ [duplicate]

I've been working through an exercise in Lee's Smooth Manifold. The author ask us to show that the space $\bar{\mathbb{R}}_+^n := [0,\infty)^n$ is homeomorphic to upper half-space $\mathbb{H}^n := \...
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0answers
21 views

Orientability of manifolds with boundary

Is it true that the exterior of any knot is an orientable manifold? It is intuitive to me that an oriented manifold naturally inherits an orientation to its codimension $0$ submanifold, at least when ...
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1answer
70 views

Definition of the preimage orientation

Guillemin and Pollack give quite confusing (at least for me) definition of the preimage orientation (see below). I don't understand the part starting from the last display. Namely: How exactly does ...
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44 views

Does any 2-manifolds besides the sphere, real projective plane, and disk have a finite fundamental group?

The sphere, real projective plane, and disk have finite fundamental groups. Do any other 2-manifolds have finite fundamental groups?
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1answer
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Constructing boundary charts for the n-dim closed balls

This is an exercise from John Lee's book. I am having extreme difficulty with constructing explicit boundary charts for $M$. I have no idea at all how to explicitly express the boundary charts for $M$....
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1answer
24 views

Boundary of Seifert Fibered Space is a $T^2$ or $K^2$

I'm getting my feet wet with Seifert Fibered Spaces in Hatcher's 3-manifold papers. Elsewhere, it is said that this follows easily from the definition. I am not seeing it. I think we would need to ...
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1answer
31 views

Defining an open submanifold with boundary - John Lee book's proposition and exercise

Above is from p.13 of John Lee's Introduction to Smooth Manifolds. I am curious if this proposition also holds when $M$ is a topological manifold with boundary, thus making it into a smooth manifold ...