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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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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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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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2answers
75 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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1answer
138 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
43 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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177 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
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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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1answer
75 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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81 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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3answers
615 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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289 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
90 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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1answer
92 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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1answer
37 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 $...
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1answer
44 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 $M$...
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41 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 $\varphi(x_0)=0\in \partial\mathbb{H}$...
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2answers
153 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 ...
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0answers
51 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(...
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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 ...
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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....
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1answer
151 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 ...
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1answer
69 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 ...
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1answer
243 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 ...
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1answer
127 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....
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110 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 ...
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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
450 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 ...
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1answer
58 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 ...
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1answer
35 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 $\...
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1answer
46 views

Can we choose the anti exterior derivative in a way that respect the behaviour on the boundary?

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 $\...
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1answer
185 views

If $M$ is a compact $d$-dimensional manifold with nonempty boundary, then $H^d(M)=0$.

The accepted answer to this question contains the following statement: If $M$ is a compact $d$-dimensional manifold with nonempty boundary, then $H^d(M)=0$. This does not look obvious to me, so I ...
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1answer
79 views

Smoothness of the squared distance to the boundary of an open subset

If $M$ is a Riemannian manifold and $U$ a regular domain (i.e. $U$ is an open subset and $\partial U$ is a smooth submanifold of $M$), is the function $d( \cdot, \partial U)^2 : U \to \Bbb R$ smooth? ...
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1answer
52 views

Are homotopic maps with identical boundary values homotopic through a boundary preserving homotopy?

Let $M,N$ be smooth $d$ dimensional manifolds. Suppose $f_0,f_1:M \to N$ are homotopic, and that $f_0|_{\partial M}=f_1|_{\partial M}$. Is there a homotopy $f_t$ such that $f_0|_{\partial M}=f_t|_{\...
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1answer
163 views

If the symbol $\partial \Omega$ is used to represent the boundary of $\Omega \subset \mathbb{R}^2$, is smooth or differentiable a pre-requisite?

If we can use the symbol $\partial \Omega$ to represent the boundary of $\Omega$, for instance $\Omega$ is in $\mathbb{R}^2$ ($\mathbb{R}^3$), and thus $\partial \Omega$ is a curve (surface), do we ...
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22 views

Finding an orientable surface with a given boundary

I'm going over the problems here, and I'm currently stuck on 6G-3. Intuitively, I try extend the curve $C$ to a short cylinder and try to pull it out where the two loops meet. However, I find this ...
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1answer
124 views

Is the disc minus two holes homotopy equivalent to the disc minus a wedge sum of two holes?

There is an obvious homotopy (given by the usual inverted pant diagram) between the identity map from disc to disc and the map that deforms the separate holes to their wedge sum. But this does not in ...
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1answer
134 views

Prove every compact $1$-manifold with boundary always has an even number of boundary points.

There's a claim that Milnor makes in his book Topology from the Differentiable Viewpoint Every compact $1$-manifold with boundary always has an even number of boundary points. I'm not quite sure ...
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2answers
102 views

If manifolds $M_1$ and $M_2$ are connected, $M_1\# M_2$ is connected.

Show that if $M_1$ and $M_2$ are connected $n$-manifolds and $n>1$, then $M_1 \# M_2$ is connected. $M_1 \# M_2$ is the connected sum of the two manifolds. This is problem 4.18(b) from Lee's ...
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1answer
131 views

Preimage orientation of a manifold

Let $0$ be a regular value of the smooth function $g:\mathbb{R}^n\rightarrow\mathbb{R}^p$ and $M=g^{-1}(0)$. Define the preimage orientaton of $M$ as such that $$[\triangledown g_1,\ldots,\...
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1answer
603 views

Showing that $\bar{\mathbb{B}}^n$ is a manifold with boundary (Lee ITM Probelm 3-4)

"Show that every closed ball in $\mathbb{R}^n$ is an $n$-dimensional manifold with boundary, as is the complement of every open ball. Assuming the theorem on the invariance of the boundary, show that ...
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Intersection of a minimal surface and a closed ball

Let $u:\Omega\subset\mathbb{R}^2\to\mathbb{R}$ such that $u$ satisfies the differential equation of minimal surfaces on the region $\Omega$. Let $p\in(\Omega\times\{0\})\subset\mathbb{R}^3$ and a ...
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1answer
155 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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1answer
72 views

Diffeomorphism between boundary domains

Let $\mathbb R^m_{-}$ be the closed left half space and let $f: U \to V$ be a diffeomorphism between relatively open sets in $\mathbb R^m_{-}$, that is, there exists an extension $\hat f: \hat U \to \...
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0answers
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Homotopy groups of the boundary of a projectively or conformally compactified hyperbolic manifold

Say that we have a projectively or conformally compactified hyperbolic manifold, and we know the homotopy groups (so just the fundamental group) of the hyperbolic manifold. Can we say anything about ...
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134 views

definition of essential annulus contained in the boundary

I read John Hempel's paper "3-manifolds viewed from the curve complex". In the proof of Theorem 3.2, he calls annuli in the boundary of a solid torus essential. (see picture below from his article). ...
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1answer
924 views

Closed vs. compact surface

Wikipedia defines a surface to be a two-dimensional manifold, and a closed surface to be a surface that is compact and without boundary. Am I correct that this definition of "closed surface" is not ...
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1answer
363 views

How to prove the existence of partitions of unity for smooth manifolds with boundary?

I have tried looking through various books and websites and could not find a proof of the existence of partitions of unity for smooth manifolds with boundary. I would like a proof or a reference to ...
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1answer
295 views

Regular values for maps between manifolds with boundary

Let $h:X \to Y$ be a smooth map between manifolds with boundaries. How does one characterize a regular value for $h$ ? I am sifting through Milnor's Topology from the Differentiable Viewpoint and ...
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2answers
210 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 ...