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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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 $M$...
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1answer
257 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
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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....
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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(...
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1answer
484 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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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
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1answer
180 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
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 ...
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123 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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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 ...
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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 $\...
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1answer
47 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
200 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 ...
2
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1answer
82 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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2answers
749 views

Gauss-Green Theorem from generalized Stoke's Theorem.

I am trying to deduce the next identity (Green-Gauss theorem) $$\int_\Omega \dfrac{\partial u}{\partial x_i} dx = \int_{\partial \Omega} uv_i dS$$ from the generalized Stoke's theorem for manifolds. ...
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1answer
53 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
168 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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23 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
74 views

Does the intrinsic and extrinsic distances coincide on the interior?

Let $M$ be a Riemannian manifold with boundary. Consider the interior of $M$ (which we denote by $M^\circ$). $M^\circ$ is an open submanifold of $M$. Let $d_M$ be the induced distance function (...
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1answer
313 views

Isometric immersions between manifolds with boundary are locally distance preserving?

Let $M$ be a compact, connected, oriented smooth Riemannian manifold with non-empty boudary. Let $f:M \to M$ be a smooth orientation preserving isometric immersion. Is it true that $f$ is locally ...
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1answer
137 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
141 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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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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89 views

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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144 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
75 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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35 views

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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1answer
976 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
374 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
858 views

Euler Characteristic of a boundary of a Manifold

I need some guidance in understanding a specific passage of the following result taken from [tom Dieck Algebraic Topology, page 456] Proposition 18.6.2. Let $B$ be a compact $(n+1)$-manifold with ...
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1answer
302 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
128 views

What is the boundary of two manifolds with boundary?

I know that if $M$ is a manifold without boundary and $N$ a manifold with boudary, then $\partial(M\times N)=M\times \partial N$, but, if I have the product of two manifolds with boudary, it's known ...
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0answers
133 views

Manifold with boundary and boundary points?

I was curious that since the boundary of a manifold with boundary is boundary-less, $\left(\partial (\partial M)=\emptyset\right)$, then whether the following example of a disc with an open interval ...
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3answers
258 views

Is there a (what is the) intrinsic definition of boundary?

It is asked to show that the closed disk $\overline{D}^2=\{(x,y)\in \Bbb{R}^2:x^2+y^2\leq 1\}$ (with the topology induced from $\Bbb{R}^2$) is not a regular surface. It seems obvious that we have a ...
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1answer
47 views

Why are those two manifolds with boundary diffeomorphic to $D^2\times S^1$?

I have this problem and I don't know why I can't finish it: Let $S=\{x\in \mathbb{R}^4\mid \vert\vert x \vert\vert=2\}$ the sphere of dimension $3$ and radius $2$. Let $T_+$ (resp. $T_-$) the set ...
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1answer
174 views

A problem about diffeomorphism of two components of the boundary of a manifold.

My geometry professor said that the following statement is true: Let $M$ be a compact smooth manifold such that $\partial M = M_0 \cup M_1$. Suppose that there exist a smooth function $f:M \to \mathbb ...
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1answer
164 views

There is no diffeomorphism between one quadrant in the plane and the half plane

I am trying to prove rigourosly that the unit square $[0,1]\times [0,1]\subset \mathbb R^2$ is not a differentiable manifold with boundary (I've searched for a rigouros proof, but haven't found anyone)...
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1answer
130 views

Is there any necessarily NON-simply-connected bordism between compact 1-manifolds?

As I seem to have confirmed in this previous question, any compact 1-manifold is homeomorphic to the (possibly empty) disjoint union of circles. So my question reduces to essentially: Does there ...
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205 views

Is the boundary of every 2-manifold the disjoint union of circles?

Note that in the title I am being sloppy and should say "Is the boundary of every 2-manifold with boundary the disjoint union of circles?". Also when I say "disjoint union of circles" I mean up to ...
3
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1answer
179 views

Long exact sequence for manifolds with boundary

Let $T^* S^n$ be the disk cotangent bundle on $S^n$ consisting of covectors with norm less than or equal to $1$. I am confused about the long exact sequence of the pair $(T^*S^n, \partial T^*S^n)$. ...
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0answers
196 views

Manifolds with boundaries and partitions of unity

How do I 1 show that $M=[0,3]\subset \mathbb{R}$ is a manifold with boundary? 2 find a $C^2$ partition of unity for the open cover $M=[0,2)\cup(1,3]$? 3 show that $\omega=(x-2)dx$ is/is not an ...
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1answer
131 views

Does Stokes hold? (Manifolds)

Let $M$ be the manifold with boundary $ M=\mathbb{R}_{\geq0}$ and $\omega\in\Omega^0(M)$ a 0-form. Suppose both $\int_M d\omega$ and $\int_{\partial M}\omega$ are finite. Does Stokes theorem hold? ...
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0answers
104 views

Mobius band gluing construction

Can somebody help me understand how the mobius band can be viewed as $\mathbb{R}\times [0,1]/\sim$ where $(x,y)\sim (x+1,1-y)$? (also, if somebody can help me with the appropriate tags)
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2answers
64 views

Does smoothness on the interior and on the boundary (separately) imply smoothness?

Let $M,N$ be smooth manifolds with boundary. Suppose we have a map $\phi:M \to N$ which satisfies the following properties: $$ (1) \, \, \phi(\operatorname{int}M)=\operatorname{int}N,\phi(\partial M)...
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0answers
231 views

Submersion Theorem for manifold with boundary

I've got a question about a lemma in Milnors "TOPOLOGY FROM THE DIFFERENTIABLE VIEWPOINT". LEMMA 4: If $y\in N$ is a regular value, both for $f$ and for the restriction $f|_{\partial X}$, then $f^{-1}...
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1answer
150 views

Is a map with invertible differential that maps boundary to boundary a local diffeomorphism?

Let $M,N$ be smooth manifolds with boundary (of the same dimension). Let $f:M \to N$ be a smooth map satisfying $(1) \, \,f(\partial M)=\partial N,f(\operatorname{Int} M)=\operatorname{Int} N$. $(...
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2answers
898 views

A local diffeomorphism can map a boundary point to an interior point

I would like to find an example for a local diffeomorphism between smooth manifolds with boundary which maps some boundary point to an interior point. I am quite sure such an example exists.
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3answers
269 views

Are topological manifolds with boundary metrizable?

It is standard that topological manifolds (without boundary) are metrizable. Is the same true for manifolds with boundary?. I'm using the following definition: Let $\mathbb{R}^n_{x_n\ge 0}=\{x\in \...
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0answers
61 views

$T_P(M)$ vs. $T_P(\partial M)$

I tried to understand in what way an orientation on $M$ induces an orientation on $\partial M$. Note that I am learning in the context of the extrinsic definition of manifolds, i.e. everything is ...
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1answer
117 views

When is a continuous map $f: M \longrightarrow N$ between smooth manifolds homotopic to a smooth one?

I know there have been similar questions here, but I haven't been able to completely pin down the precise conditions on $M$ and $N$. I have seen one proof of this that uses tubular neighborhood ...