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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.

5
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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 ...
3
votes
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 ...
2
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2answers
211 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
votes
1answer
305 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
71 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
121 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
126 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
249 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
45 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
votes
1answer
168 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
vote
1answer
156 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
125 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
198 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 ...
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0answers
187 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
125 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
votes
1answer
176 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
103 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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0answers
223 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
votes
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
vote
1answer
147 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
874 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
votes
3answers
262 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
votes
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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0answers
67 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
vote
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
150 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
338 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
votes
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
votes
1answer
235 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 $\...
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0answers
56 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, \...
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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
155 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
267 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 ...
6
votes
1answer
185 views

Can every Riemmanian Manifold be completed?

I had two trails of though.. is either of them fruitful? I know every metric space can be completed, my question is: can a Riemmanian manifold $M$ be embedded smoothly and isometrically into it's ...
2
votes
1answer
206 views

An orientation on a $(n-1)$-dimensional submanifold.

This question goes on where this question ended. Given is an non-empty $(n-1)$-dimensional ($n\ge2$) differentiable submanifold $X\subset\mathbb{R}^n$ such that there exists an open $U\subset\mathbb{R}...
2
votes
2answers
84 views

Topological boundary of a specific manifold.

Let $\emptyset\neq X\subset\mathbb{R}^n$, $n\ge2$, be an $(n-1)$-dimensional differentiable submanifold, i.e. for every $p\in X$ there is an open $U_p\subset\mathbb{R}^n$ with $p\in U_p$, and a ...
0
votes
1answer
477 views

On the trace theory and restrictions of Sobolev space functions

Consider a connected open bounded subset $D\subset \mathbb{R}^d$ with smooth boundary. It is easier to state for the disc so I will do so. I don't think it would change the argument (except for in our ...
0
votes
1answer
106 views

Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$?

The first question is Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$? In other words, is it obvious? The question stems from a theoretical physics ...
1
vote
1answer
102 views

$\sin(\frac{1}{x})$ on $\Bbb R^2$ interior, cluster points, boundary

Given $$M:=\{(x_1, x_2) \in \Bbb R^2 : x_2 =\sin(\frac 1 {x_1}), x_1\in(0,\frac 1 \pi)\}$$ , a subset of the normed Space $(\Bbb R^2,||\cdot||_2)$. $M$ is the above Curve. (the commen $\sin(\frac{1}{...
2
votes
1answer
60 views

Topological boundary as a submanifold

Let $U$ be an open subset of a smooth $n$-manifold. Consider $\partial U$ the topological boundary of $U$. Is the following true ? : If $\partial U$ is a smooth $n-1$ submanifold without boundary, ...
1
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1answer
57 views

Is $\overline{u^{-1}((-\infty,a))}$ a $C^k$-manifold with boundary for a $C^k$ function $u$?

Let $u\in C^k(\mathbf{R}^d)$, $M:=\overline{u^{-1}((-\infty,a))}=\overline{\{x\in\mathbf{R}^d \mid u(x)<a\}}$, where $a$ is a constant such that $M\neq\emptyset$. Is $M$ a $C^k$-manifold with ...
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0answers
78 views

Integration over manifold

Let $M$ be a smooth 2-manifold in $\mathbb{R}^3$ such that $$4x^2+y^2+4z^2 = 4, y \ge 0$$ The boundary of $M$ is the set of points where $$x^2 + z^2 = 1, y = 0$$ Let $\alpha(u,v) = (u,2\sqrt{(1-u^2-...
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0answers
49 views

How to define an atlas on this manifold with boundary?

Consider the set $\mathcal{M} = \{\ \mathbf{x} \in \mathbb{R}^{3}\ | \ 1 \leq ||\mathbf{x}|| \leq 2 \ \}$. This is a $3$-submanifold with boundary. Obviously, we have $\partial \mathcal{M} = \{\ \...
4
votes
0answers
271 views

Induced orientation on manifold

Suppose $M$ is a naturally oriented $n$-manifold embedded in $\mathbb{R}^n$ with non-empty boundary. Give $\partial M$ the induced orientation. The induced orientation defines a unit normal vector ...
1
vote
1answer
92 views

Lenght of the curve in Riemannian metric.

Let $M^{k}$ a submanifold, $h:U\to M^{k}$ a chart, and $\gamma:[a,b]\to h(U)\subset M^{k}$ a curve in $M^{k}$. Represent the curve in coordinates $(h,U)$ as $h^{-1}\circ\gamma(t)=(y^{1}(t),...,y^{k}(t)...
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0answers
65 views

Show that a given set is a manifold with boundary

given $A:= \{ (x_1,x_2,x_3) \in \mathbb{R} : x_1^2 + x_2^2 + x_3^2 = 18, x_3 \leq 2\}$ show that $A$ is a manifold with boundary and calculate $ \delta A$ where $\delta A$ is the boundary of A. I ...
3
votes
0answers
80 views

A $k+1$ Manifold whose boundary is the solution set to the equation $f(\vec{x})=\vec{0}$

Let $f: \mathbb{R}^{n+k} \to \mathbb{R}^n$ be of class $C^r$. Let $f_1, ..., f_n$ be components of $f$. Define $$M=\{\vec{x} | f(\vec{x})=\vec{0}\},$$ $$N=\{\vec{x} | f_1(\vec{x})=0, ..., f_{n-1}(\...
10
votes
0answers
192 views

Codimension $1$ Embedding into $\mathbb{R}^{n+1}$

I am trying to determine which homotopy types can be realized by $n$-manifolds that have codimension one embeddings into $\mathbb{R}^{n+1}$. Suppose I have $X^n \subset \mathbb{R}^{n+1}$ and an ...
3
votes
2answers
107 views

Are $\mathbb{C}^2$ and $\mathbb{C}^2/(x,y)\sim(y,x)$ homeomorphic?

Let $A$ be the set of monic quadratics over $\mathbb C$ and let $B$ be the set of unordered pairs over $\mathbb C$ where possibly the two elements of the pair may be the same. Then the map which takes ...
1
vote
0answers
46 views

Show that $\partial(M\times N)=M\times\partial(N)$

Let M a $k$-dimensional manifold without boundary of $\mathbb{R}^{n}$ and N a $l$-dimensional manifold of $\mathbb{R}^{m}$ with or withour boundary. Show that $\partial(M\times N)=M\times\partial(N)$ ...