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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3
votes
1answer
55 views

2 -manifolds that can't be decomposed into two equal parts

The real projective plane has the property that if you divide into two different 2-manifolds, they will not be homeomorphic (i.e. one will be orientable, and the other non-orientable). The sphere ...
3
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0answers
93 views

Lemma 3 of the Milnor's Topology from the differentiable viewpoint.

Lemma 3 in Milnor's "Topology from the differentiable Viewpoint" (p.12) is stated as below; Let $M$ be a manifold without boundary and let $g:M \to \mathbb{R}$ have 0 as regular value. The set of $...
4
votes
2answers
172 views

Is the closed hemisphere diffeomorphic to the closed disk?

This might be silly but it puzzles me. Let $M$ be the closed upper hemisphere of $\mathbb{S}^2$. This is a manifold with boundary. Is it diffeomorphic to a closed disk in $\mathbb{R}^2$? These ...
0
votes
1answer
61 views

compute integral of function of distance to the boundary only

How does one compute the integral $ \int_{\Omega_{\epsilon}} f( d(x))\ dx = ? $ where $d(x)$ is the distance to the boundary and $\Omega_\epsilon := \{ x\in \Omega: d(x)<\epsilon\}$, supposing ...
16
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1answer
747 views

Can you hear the pins fall from bowling game scores?

Let $\mathbb T=\{1,\dotsc,10\}$ represent the ten pins in a standard game of bowling. Given two sets of pins $T\subseteq S\subseteq \mathbb T$, let's write $p_{S\to T}$ to represent the conditional ...
1
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0answers
14 views

Possible to describe random 3D surfaces (geograhical height over limited area) by formula?

Coming from a geographic computer sciences background and working with 3D terrain (so please forgive if my terminology is inappropriate), I was always wondering if it is possible to describe the 3D ...
0
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1answer
144 views

Pullbacks and differential forms, require deep explanation + algebra rules

Can somebody help me understand this. Let $\omega$ be a closed two-form on $\mathbb{R}^3$ and $\eta$ a one-form such that $\omega=d\eta$. $M$ is an orientable manifold with boundary $\partial M$. $i:M\...
1
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0answers
43 views

Manifolds with boundary and foliations

Is there a theory of foliations by manifolds with boundary? Particularly, Is there a generalization of the Frobenius theorem and the Stefan-Sussmann theorem in which the leaves are manifolds with ...
1
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0answers
39 views

A question about the proof of Extension Lemma for Smooth functions

I am trying to understand the proof, but am stuck at the last line. Why does the first equality of the picture below holds? The codomain of $f$ is $R^k$, not some the set of nonnegative numbers. Is ...
0
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1answer
116 views

Giving a counterexample for the extension lemma of smooth functions

I am supposed to give a counterexample showing the conclusion is false when $A$ is not closed. I tried to find one when $M$ is Euclidean space but kept failing... Could anyone please show me a ...
0
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1answer
158 views

Showing that a diffeomorphism preserves the boundary

I am supposed to use the fact presented below to show that the diffeomorphism $F$ in the theorem 2.18 preserves the boundary. I found a way to prove it but it does not use the fact below. I am curious ...
0
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2answers
31 views

Integration of one-form

I am trying to compute $$\int_C i^*\eta$$ $\eta=(x^2+y^2)dz$ and $C=\{(x,y,-1): x^2+y^2=1\}$ and $i$ is the inclusion map This is what I did $$\int_{-1}^{-1}(x^2+y^2) dz=0 $$ Is this correct?
2
votes
1answer
121 views

“Defining a smooth structure on a topological manifold with boundary”

This is a proposition for topological manifolds. However, I am wondering if this prop holds when $M$ is a topological manifold "with boundary." Thus, can the expression like "a smooth atlas on a ...
0
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0answers
115 views

Regular surfaces with boundary and $C^1$ domains

I would like to ask about the equivalence between these two definitions for a $C^1$ domain. In the book Vector Analysis Versus Vector Calculus, we have: Definition 8.2.1: Let $\mathbb{H}^k=\{(t_1,\...
2
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1answer
95 views

Showing that a neighborhood is not homeomorphic to the open subset of $R^n$

This is an exercise from Boothby that I am stuck at. It is trivial that the boundary of $H^n$ is a manifold of dimnension $n-1$ because it is homeomorphic to $R^{n-1}$. However I cannot show that no ...
0
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1answer
202 views

Show that $\mathbb{S}^d$ is homeomorphic to the the boundary of the cube $\partial I^{d+1}$.

How to show that the sphere $\mathbb{S}^d$ and the boundary of the cube $\partial I^{d+1}$ are equivalent topological manifolds? One way is to use the homeomorphism $\varphi: \mathbb S^2 \to I^3 : (x,...
0
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0answers
118 views

Show that if $dF_p$ is nonsingular, then $F(p)\in \text{Int}N$ : Lee's Smooth Manifolds

Suppose $M$ is a smooth manifold without boundary and $N$ is a smooth manifold with boundary and let $F : M \rightarrow N$ is a smooth map. Show that if $p \in M$ is a point such that $dF_p$ is ...
1
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2answers
97 views

What are the possible boundaries of connected compact manifolds?

I'm trying to understand what types of manifolds can occur as the boundary of a compact manifold $M$. When $M$'s dimension $d$ is 1, the boundary is always either empty or two points (corresponding to ...
2
votes
0answers
110 views

Stokes' theorem proof without FTC

To present time I have not found a proof of Stokes' Theorem for manifolds that does not involve the Fundamental Theorem of Calculus in some way. Is it possible to prove the general theorem without ...
2
votes
1answer
43 views

The set that we add to compactify is itself compact.

Let $X$ be a topological space. Let $X_c$ a compactification of $X$. I want to know if $X_c\setminus X$ is always compact, or maybe more simply whether it is always closed in $X_c$. I think it is true ...
1
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0answers
94 views

Are there topological manifolds with boundary that are not compact?

Following this question Are there compact manifolds without boundary? I'm asking if there is any example of a manifold with boundary that is not compact, Is the interval $X=[0,1)$ a counterexample ? ...
1
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0answers
151 views

integrating a 2-form over the sphere

I need to integrate the following form over the 2-sphere: $$w=\frac{xdy\wedge dz-ydx\wedge dz+zdx\wedge dy}{(x^2+y^2+z^2)^\frac{3}{2}}$$ now a direct calculation shows that $dw=0$, thus from stokes' ...
5
votes
1answer
54 views

Why do we sometimes call the boundary of the body by $\partial$?

I very often see the symbol "$\partial$" is used to define the boundary of a body in three-dimensional space. For example, if the body is called $\Omega$ the boundaries is labeled as $\partial\Omega$. ...
2
votes
2answers
76 views

Suppose $M$ is a connected manifold and $B \subseteq M$ is a regular coordinate ball, is $M \setminus B$ connected?

Suppose $M$ is a connected manifold of dimension $n > 1$ and $B \subseteq M$ is a regular coordinate ball, is $M \setminus B$ connected? I think the answer to this is definitely yes, but I'm ...
2
votes
1answer
139 views

Suppose $M$ is a manifold of dim $n \geq 1$ and $B \subseteq M$ is a regular co-ordinate ball. Show $M \setminus B$ is a $n$-manifold with boundary

Suppose $M$ is a manifold of dimension $n \geq 1$ and $B \subseteq M$ is a regular co-ordinate ball. Show that $M \setminus B$ is a $n$-manifold with boundary, whose boundary is homeomorphic to $\...
1
vote
1answer
45 views

If the boundary $\partial M$of a manifold with boundary is connected,is $M$ connected?

Given a manifold with non-empty boundary $\partial M$, I am trying to see if the boundary is connected, will the manifold itself $M$be connected? I can not come up with any counter-example.
1
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0answers
192 views

Embedded extension of embedded submanifolds with boundary

Let $M$ be a smooth manifold (without boundary) and $S \subseteq M$ be a smooth embedded submanifold with non-empty boundary $\partial S$. Does there exist a smooth, embedded extension $\tilde S$ of $...
4
votes
1answer
79 views

2-manifold with involution without fixed points is a boundary of a 3-manifold.

I am preparing for geometry/topology test and I can not handle the following problem from previous years: Let $F$ is closed orientable 2-manifold with involution without fixed points. Prove that $F$ ...
2
votes
1answer
83 views

Why is the boundary of an oriented manifold with its (opposite oriented) copy the empty set?

Excuse the very basic question: I'm following Milnor's Lectures on Characteristic Classes. He defines a relation on the collection of compact, smooth, oriented manifolds of dimension $n$ by letting $...
0
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0answers
85 views

Volume form on a disk and induced orientation 1-form

I'm having trouble with this question: Choose a volume form $\zeta$ on the disk $D(r) = \{(x,y,0)\in \mathbb{R}^3:x^2+y^2\leq r^2\}$ and describe what the induced orientation 1-form looks like in the ...
4
votes
3answers
692 views

Confusion about definition of manifold with boundary

Consider this definition: A space $M$ is a manifold with boundary if each point $x\in M$ has a neighborhood $U_x$ that is homeomorphic to $\mathbb R^n$ or to $\mathbb R^n_+=\{(x_1,\cdots,x_n)\;|\; ...
3
votes
0answers
315 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\...
0
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1answer
93 views

A crucial step in showing that the boundary of a $n-$manifold is a $(n-1)-$manifold

Consider $M$ an $n-$manifold with boundary $\partial M$. In showing that $\partial M$ is an $(n-1)-$manifold a crucial step is not clear to me: Let $x\in\partial M$, then there exists an open ...
2
votes
1answer
109 views

Definition of connected sum

My professor has defined the connected sum of two oriented closed connected triangulated manifolds $M_{1}$ and $M_{2}$ as $N_{1} \cup_{f} N_{2}$ where: $N_{i}$ is the n-manifold with boundary ...
1
vote
1answer
39 views

Understanding a conclusion through “elementary considerations about bilinear forms”

I am having a hard time to understand how "elementary considerations about bilinear forms" can imply the following result: Let $E$ be a function space, 1 the constant function $x\mapsto 1$ and let $...
0
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1answer
45 views

$M\setminus\partial M$ open and proof of dimension

Let $M$ be an $n$-dimensional manifold with boundary. Show that: $M\setminus\partial M$ is an open subset of $M$ and it is an $n$-dimensional manifold, and that $\partial M$ is a closed subset of $M$...
0
votes
1answer
42 views

Coordinate chart of 1-dimensional manifold with boundary

Let $M$ be a 1-dimensional manifold with boundary, $x_0\in M$, and $\varphi:U\rightarrow V$ be a local coordinate chart such that $x_0\in U$. Take $x_0$ such that $\varphi(x_0)=0\in \partial\mathbb{H}$...
3
votes
2answers
159 views

$\mathbb S^2$ 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 ...
1
vote
0answers
53 views

Calculus of variations when target space has boundary

Let $M,N$ be smooth oriented Riemannian manifolds; $M$ closed and $N$ with non-empty boundary. Let $f:M \to N$ be smooth and suppose the image of $f$ intersects $\partial N$. Let $$W:=\{V \in \Gamma(...
7
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1answer
473 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 ...
1
vote
0answers
28 views

Immersive limit of embeddings is injective on the interior?

$\newcommand{\pl}{\partial}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be smooth compact, connected, oriented manifolds of the same dimension. $\M$ can have a boundary....
2
votes
1answer
182 views

Volume of a manifold

Throughout this post, I am presuming $M$ to be an $2$-dimensional manifold that is parametrized by one chart $\varphi$, and I presume $\omega$ be a $2$-form on $M$. Apparently, there is no natural ...
0
votes
1answer
83 views

How to show reflection of $\mathbb{S}^n$ and the identity are not homotopic?

Let $N=\mathbb{S}^n$, $M \subseteq \mathbb{S}^n$ be a hemisphere, including the equator (I am considering $M$ with boundary). Let $f_1:M \to N$ be the inclusion map, and let $f_2:M \to N$ be the ...
2
votes
1answer
258 views

Do trivial homotopy groups imply existence of boundary preserving homotopies?

Let $N$ be a smooth $d$-dimensional connected orientable manifold $N$ which have the following property: For every smooth $d$-dimensional manifold $M$ with non-empty boundary, and for every smooth ...
2
votes
1answer
130 views

Is uniform limit of embeddings injective?

$\newcommand{\pl}{\partial}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be smooth compact, connected, oriented manifolds of the same dimension with non-empty boundaries....
0
votes
0answers
124 views

Boundary of a submanifold

Let be $M$ a smooth manifold with boundary and $N \subset M$ a submanifold with boundary, such $N$ is a closed subset of $M$ in the topological sense. Denote $\partial N$ be the boundary of $N$ as a ...
29
votes
0answers
835 views

Why are functions with vanishing normal derivative dense in smooth functions?

Question Let $M$ be a compact Riemannian manifold with piecewise smooth boundary. Why are smooth functions with vanishing normal derivative dense in $C^\infty(M)$ in the $H^1$ norm? Here I define $...
2
votes
1answer
486 views

Proving diffeomorphism invariance of boundary

I'm trying to follow Lee's book Introduction to Smooth Manifold in details. In chapter 2, Exercise 2.19 asks the reader to prove that if $F:M \to N$ is a smooth diffeomorphism between manifolds with ...
3
votes
1answer
62 views

Are two compact $n$-dim submanifolds with boundary of $\mathbb{R}^n$ with identical boundaries coincide?

Let $U,V$ be the interiors of codimension $0$ compact embedded submanifolds with boundary of $\mathbb{R}^n$. Suppose that $\partial U=\partial V$. Is it true that $U=V$? (In other words, I am asking ...
2
votes
1answer
36 views

Obstructions to existence of a boundary respecting family of primitive exterior derivatives

Let $M$ be an oriented, compact smooth $d$-dimensional manifold with boundary, and let $\omega_t$ be a smooth family of $d$-forms that agree on some open neighbourhood of $\partial M$. (i.e $\...