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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2
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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 $\...
2
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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? ...
2
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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|_{\...
1
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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 ...
0
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0answers
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 ...
0
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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 ...
0
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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 ...
4
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2answers
104 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 ...
0
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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,\...
2
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1answer
653 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 ...
3
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0answers
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 ...
1
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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 ...
0
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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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0answers
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 ...
0
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0answers
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). ...
0
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1answer
979 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 ...
5
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1answer
375 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 ...
3
votes
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 ...
2
votes
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 ...
6
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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 ...
0
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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 (...
2
votes
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 ...
0
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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 ...
1
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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 ...
2
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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 ...
4
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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 ...
1
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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)...
0
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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 ...
0
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0answers
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 ...
1
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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 ...
3
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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? ...
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 $(T^*S^n, \partial T^*S^n)$. ...
0
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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)
0
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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}...
2
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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)...
1
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1answer
151 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 $(1) \, \,f(\partial M)=\partial N,f(\operatorname{Int} M)=\operatorname{Int} N$. $(...
4
votes
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.
2
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3answers
270 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 \...
0
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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 ...
1
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1answer
83 views

Manifolds with corners

Here is what I'm experiencing. A text introduces manifolds. Then, manifolds with boundary is introduced. After all, manifolds with corners is introduced. First of all, I checked whether Implicit ...
1
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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 ...
5
votes
1answer
155 views

Smoothness of the boundary is the only obstruction for being a submanifold with boundary?

Let $M$ be a smooth manifold, and let $S$ be an open smooth submanifold of $N$. Assume the topological boundary of $S$, $\partial S :=\bar S \setminus S$ is a smooth submanifold of codimension 1 in $...
2
votes
1answer
350 views

If a continuous function is locally injective and injective on the boundary, is it injective?

Let $M$ be a compact connected manifold with boundary whose interior has dimension $n\geq 2$. Suppose that we have a map $f:M\rightarrow S^n$ which is continuous and such that the restriction of $f$ ...
0
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2answers
78 views

Munkres Problem: Define a $C^{\infty}$ map $f: \mathbb{R}^9 \to \mathbb{R}^6$ such that $O(3)$ is the solution of $f(x)=0$.

On Munkres's book analysis on manifold chap "the boundary of manifold", question 3, says: let $O(3)$ the set of orthogonal matrices, as a subspace of $\mathbb{R}^9$. a) define a $C^{\infty}$ map $f: \...
0
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1answer
239 views

If the boundary of an open connected set $\Omega$ of class $C^1$ is bounded $\Rightarrow$ $\Omega$ is bounded?

Let $\Omega \subset \mathbb{R}^N$ be an open set. I know bounded boundary doesn't imply bounded set, but what if we consider the boundary of an open connected set of class $C^1$ (i.e. the boundary $\...
1
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0answers
57 views

Clarification for Manifold and Manifold with Boundary Definitions

I am reading Spivak's Calculus on Manifolds, and just need a bit of clarification when it comes to definitions. He defines a manifold as some space $M$ that satisfies: $(M)$: $\forall x \in M, \...
0
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2answers
54 views

$(x + y + xy)/2 = f( f^{[-1]}(x) + f^{[-1]}(y) )$

Consider $(x + y + xy)/2 = f( f^{[-1]}(x) + f^{[-1]}(y) )$ Where $f^{[-1}]$ denotes the functional inverse of $f$. How to find $f$ ? How about the more General idea of finding $f$ for a given $g$? ...
2
votes
1answer
176 views

Extending Riemannian Manifold to Boundary

If you have a Riemannian manifold $(M,g)$ (maybe with other assumptions as need), is there a natural way to extend it to a smooth manifold with boundary? For example, the Lobachevsky space viewed as ...
3
votes
2answers
272 views

Compact 3-manifolds with boundary an orientable surface

The usual embedding of the closed orientable surface $M_g$ of genus $g$ in $\mathbb{R}^3$ bounds a compact orientable $3$-manifold which is homotopically equivalent to a bouquet of $g$ circles. If a ...