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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0answers
42 views

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
111 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
116 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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1answer
372 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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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 ...
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2answers
252 views

Geometrically, what is the stereographic projection of a closed $n$-ball?

To show $\overline{B^n}$ is a $n$-manifold with boundary, apparently there is a trick to use stereographic projection after subtracting out the radius connecting $0$ to the north pole. I'm familiar ...
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1answer
43 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 ...
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2answers
228 views

Is every flat manifold with boundary locally isometric to the Euclidean half-space?

Let $M$ be a smooth manifold with boundary, endowed with a smooth Riemannian metric $g$. Suppose $g$ is flat, and let $p \in \partial M$. Is there an open neighbourhood of $p$ which is isometric to ...
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2answers
76 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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71 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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1answer
93 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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1answer
43 views

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
31 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
110 views

Product of manifolds with or without boundary

Let $M_1, M_2, ... M_k$ be smooth manifolds with or without boundary such that at most one of these has nonempty boundary, say $M_a$ has nonempty boundary where $a$ is in $\{1,2, ..., k\}$. Then it ...
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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 ...
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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 $...
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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 ...
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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 ...
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1answer
145 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\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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,\...
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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 ...
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1answer
203 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,...
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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 ...
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1answer
1k views

Inverse Function Theorem for Manifolds with Boundary as the Domain

In Lee's Smooth Manifolds, it is written that the inverse function theorem can fail for manifolds with boundary. As hint, it is given the inclusion of half space into euclidean space $\iota:\mathbb{H}^...
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1answer
163 views

Let $M$ be a smooth manifold and let $N$ be a manifold with boundary. If $dF_{p}$ is an isomorphism, then $F\left(p\right)\in\mbox{int}M $.

Let $M$ be a smooth manifold and let $N$ be a manifold with boundary, both with the same dimension $n$. If $dF_{p}$ is an isomorphism, then $F\left(p\right)\in\mbox{int}N $. I am trying to prove ...
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2answers
773 views

Take a regular coordinate ball and you get a manifold with boundary.

Suppose $M$ is a (topological) manifold of dimension $ n \geq 1$ and $B$, is a regular coordinate ball in $M$. Show that $M\backslash B$ is an $n$-manifold with boundary and whose boundary is ...
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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 ...
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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 ...
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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 ? ...
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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' ...
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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 $\partial\Omega$. ...
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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 $\...
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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.
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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 $...
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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 $F$ ...
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3answers
695 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)\;|\; ...
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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 $...
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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 ...
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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 ...
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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\...
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1answer
110 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 ...
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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 $...
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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 $...