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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21 views

Convexity condition for smooth submanifold-with-boundary in $\mathbb{R}^n$

One can prove that a differentiable function $f\colon \mathbb{R}\to \mathbb{R}$ is strictly convex if and only if for each $a\in\mathbb{R}$ we have $$f'(a)(x-a)+f(a)\leq f(x)\qquad\text{for all }x\in\...
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29 views

Lee Smooth Manifolds Problem 1-11 - $\overline{\mathbb B}^n$ is a smooth manifold with boundary

The following is Problem 1-11 in Lee's Introduction to Smooth Manifolds, 2nd Edition: Let $M = \overline{\mathbb B}^n$, the closed unit ball in $\mathbb R^n$. Show that $M$ is a topological manifold ...
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1answer
52 views

If $\partial\Omega$ is of class $C^1$, does it need to be the boundary of a $C^1$-regular domain?

Let $d\in\mathbb N$ and $\Omega\subseteq\mathbb R^d$. In Theoretical Numerical Analysis, the notion of $\partial\Omega$ being "of class $C^1$" is defined in the following way: On the other ...
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1answer
40 views

Show that for a properly embedded submanifold the manifold and topogoical boundary coincide

Let $d\in\mathbb N$ and $M\subseteq\mathbb R^d$ be a $d$-dimensional properly embedded $C^1$-submanifold of $\mathbb R^d$. Let $\partial M$ and $M^\circ$ denote the manifold boundary and interior and $...
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1answer
38 views

Definition of the tangent space for a submanifold with boundary in terms of regular curves

Let $d\in\mathbb N$, $M\subseteq\mathbb R^d$ and $x\in M$. $\gamma$ is called curve on $M$ through $x$ if $\gamma:I\to M$ for some nontrivial interval $I\subseteq\mathbb R$ with $0\in I$ and $\gamma(0)...
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37 views

Characterization of $C^\alpha$-differentiability on a submanifold with boundary

Let $d\in\mathbb N$, $M\subseteq\mathbb R^d$, $x\in M$, $E$ be a $\mathbb R$-Banach space, $f:M\to E$ and $\alpha\in\mathbb N_0\cup\{\infty\}$. I was able to show the following: If $$f\circ\phi^{-1}\...
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1answer
69 views

How can we show that this normal field is “outward pointing”?

Let $d\in\mathbb N$, $\alpha\in\mathbb N$ and $M$ be a $d$-dimensional embedded $C^\alpha$-submanifold of $\mathbb R^d$ with boundary. How can we show that there is a unique $\nu_M:\partial M\to\...
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25 views

Extending the normal field of a $d$-dimensional submanifold $M$ of $\mathbb R^d$ with boundary to an open neighborhood of $\partial M$

Let $d\in\mathbb N$ $\alpha\in\mathbb N$ $M$ be a $d$-dimensional embedded $C^\alpha$-submanifold of $\mathbb R^d$ with boundary $\operatorname{Bd}(M)$ and $\partial M$ denote the topological and ...
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1answer
36 views

What can we say if the gradient at the boundary has constant norm?

Let $(M^n,g)$ be a Riemannian manifold and consider $\Omega$ a smooth and bounded domain in $M$. Let $u : \overline{\Omega} \to \mathbb{R}$ be a smooth function that satisfies both $u = 0$ and $\Vert \...
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44 views

The Transverality Theorem in Differentiable Topology by Guillemin and Pollack

In Chapter 2, Section 3 of the book, most of the theorems requires the codomain $Y$ to be a manifold without the boundary and the submanifold $Z$ to be boundaryless as well. But I don't see why the ...
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1answer
48 views

Definition and intuition of a tubular neighborhood of a submanifold

Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary$^1$. Now let $$T_xM:=\left\{v\in\mathbb R^d\mid\exists\varepsilon>0,\...
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56 views

Shape derivative of a boundary integral with a continuously differentiable function on a “tubular neighborhood”

Let $d\in\mathbb N$. I want to compute the shape derivative of a shape functional$^1$ $$\mathcal F(\Omega):=\int_{\partial\Omega}f\:{\rm d}\sigma_{\partial\Omega}\;\;\;\text{for }\Omega\in\mathcal A$$ ...
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0answers
18 views

boundary and interior of an embedded submanifold with boundary described by a single chart

Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$, $\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary$^1$. Assume, for ...
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38 views

A simple shape derivative example

Let $d\in\mathbb N$, $\tau>0$, $U\subseteq\mathbb R^d$ be open and $T_t$ be a $C^1$-diffeomorphism from $U$ onto an open subset of $\mathbb R^d$ for $t\in[0,\tau)$ with $T_0=\operatorname{id}_U$. ...
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36 views

If $T$ is a continuous bijection on $\mathbb R^d$, and $M$ is a $C^1$-submanifold with boundary, is $T(\partial M)=\partial T(M)$?

Let $d\in\mathbb N$ and $T:\mathbb R^d\to\mathbb R^d$ be bijective and continuous. I'm not sure how to approach this, but if $M\subseteq\mathbb R^d$, does $T$ map the topological boundary/interior of $...
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44 views

A chart of a $k$-dimensional submanifold maps interior points to $(\mathbb H^k)^\circ$ and boundary points to $\partial\mathbb H^k$

Let $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary$^1$, $M^\circ$ and $\partial M$ denote the manifold interior and boundary of $M$, respectively, and $(\Omega,\phi)...
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1answer
94 views

Find a specific countable atlas for a smooth submanifold with boundary

Let $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary.$^1$ We know that there is a countable family $((\Omega_i,\phi_i))_{i\in I}$ of $k$-dimensional $C^1$-charts$^2$ ...
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91 views

Show this identity for the surface measure on the boundary of a submanifold

Let $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary$^1$ and $(\Omega,\phi)$ be a $k$-dimensional boundary $C^1$-chart$^2$ of $M$. We know that$^3$ $(\tilde\Omega,\...
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1answer
60 views

Why is it important the manifold has codimension $1$ in order to prove this identity for $\operatorname{div}fV$ on $\partial M$?

I've seen the following claim in some lectures notes which let me think that I might have a major misunderstanding: The claim is that if $M$ is an embedded submanifold of $\mathbb R^d$ with boundary ...
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0answers
34 views

Show an identity for the Laplace-Beltrami operator

Let $\partial M$ denote the boundary of a $k$-dimensional embedded $C^1$-submanifold $M$ of $\mathbb R^d$, $T_x(\partial M)$ and $N_x(\partial M)$ denote the tangent and normal field of $\partial M$ ...
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0answers
59 views

Construct interior chart of a manifold boundary $\partial M$ from a boundary chart of the manifold $M$

Let $M$ be a $k$-dimensional embedded $C^\alpha$-submanifold of $\mathbb R^d$ with boundary, i.e. $M$ is locally $\mathcal C^\alpha$-diffeomorphic$^1$ to $\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$....
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1answer
115 views

Characterization of the tangent space of the boundary of an embedded submanifold of $\mathbb R^d$ with boundary

Let $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary, i.e. $M$ is locally $\mathcal C^1$-diffeomorphic$^1$ to $\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$, $$T_xM:=\...
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0answers
49 views

Unsound definition of interior and boundary points of a submanifold of $\mathbb R^N$

In my lecture notes $M$ is said to be a $n$-dimensional submanifold of $\mathbb R^N$ if for all $p\in M$ there is a homeomorphism $\psi$ from an open neighborhood $\Omega_1\subseteq\mathbb R^N$ of $p$ ...
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1answer
76 views

When is a chart of a submanifold not only a homeomorphism, but a diffeomorphism?

I've got trouble to understand the concept of a "smooth structure" associated to a submanifold. Let $\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$. Say $M\subseteq\mathbb R^d$ is a $k$-...
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2answers
66 views

Show that these two diffeomorphisms cannot exist simultaneously

Let $d\in\mathbb N$, $x\in M\subseteq\mathbb R^d$ and $\psi^{(i)}:\Omega_i\to\psi^{(i)}(\Omega_i)$ be a diffeomorphism with $x\in\Omega_i$, $$\psi^{(1)}(M\cap\Omega_1)=\psi^{(1)}(\Omega_1)\cap(\mathbb ...
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1answer
40 views

Definition of a submanifold with boundary

I'm really struggling to understand the definition of a "submanifold with boundary". Until now, I'm only familiar with the notion of a "submanifold of $\mathbb R^d$. I've defined this ...
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0answers
139 views

Fréchet manifold structure on space of sections

I know that the space $\mathsf{C}^\infty(M;N)$ of smooth maps from a closed (smooth) manifold $M$ to a (smooth) manifold $N$ is a Fréchet manifold. I have been looking for a more general version of ...
14
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252 views

Stokes' Theorem general case

With the following lemma : Lemma : Let $f_{+},f_{-} : U \longmapsto \mathbb{R}$ be $C^{1}$ maps with $f_{-} \leq 0 \leq f_{+}$ with $U \subset \mathbb{R}^{n}$ open and bounded with $C^{1}$ boundary. ...
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0answers
20 views

Is there an example of reducible compact 3-manifold with boundary that has no embedded incompressible two-sided surface?

There is a theorem stating that for irreducible compact manifolds with non-empty boundary there always exists such an embedded surface and I'm trying to understand why the irreducibility condition ...
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1answer
31 views

Rank of mapping betweeen two manifolds

Let $M$ be a $7$-dimentional manifold with nonempty boundary $\partial M$ and let $f:M\rightarrow N$ be a smooth mapping where $N$ is $5$-dimentional. Assume tham for some $p\in\partial M$ we have $...
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0answers
27 views

Curve that connects two points on the boundary of a manifold

So we have a manifold $M$ with nonempty boundary. Let $p,q\in\partial M$. I need to prove that I can connect $p$ and $q$ with a curve $\gamma:[a,b]\rightarrow M$ (such that $\gamma(a)=p$ and $\gamma(...
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3answers
151 views

Is $\mathbb{D} = [-1,1]^3$ a compact manifold?

Today I read about a generalization of the no-retraction theorem here which states the following: Then there is no smooth mapping $f:M\to \partial M$ such that the restriction $f_{|\partial M}: \...
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1answer
99 views

Isn't $[0,1]$ orientable?

It's not possible to put an atlas (of manifold with boundary) on $[0,1]$ because any chart whose domain contains $0$ has to have a positive derivative while any chart whose domain contains $1$ has to ...
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0answers
27 views

$M$ is a topological manifold if and only if $\partial M = \emptyset$

This question is an exercise from Lee's book Introduction to smooth manifold. Given $M$ is a manifold with boundary ( that is $M$ is Hausdorff, second countable and every point of $M$ has a ...
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1answer
58 views

Partition of unity and existance of function on manifold with nonempty boundary

Let the $M$ be a smooth manifold with nonempty boundary. I need to prove that there exists a smooth function $f:M\rightarrow \mathbb{R}$ such that: $\forall x\in\partial M$ we have $f(x)=0$ and $rank(...
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0answers
13 views

Why does a certain integral on a 3-manifold depend only on its boundary data?

I am reading Dan Freed's lectures on Quantum Groups on Path Integrals. I am picking up the required math as I go along and I am finding certain calculations hard to follow. This is regarding the ...
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0answers
13 views

Structure preserving maps between manifolds with boundary

While learning the basics about smooth manifolds with boundary in this semesters' course about analysis on manifolds, there's a seemingly basic property I didn't find anywhere. Namely I want to ...
1
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1answer
58 views

Name of the set for which a given set is the boundary of

Consider that in a space S there are sets A and B, where B is the boundary of the compact and simply connected set A. What assumptions are required to define a unique A (or "A like" set) with respect ...
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0answers
23 views

Book definition verification of boundary manifolds

Let $M$ be an n-dim topological manifold with boundary, A chart for M is a pair $(U,\phi)$ where $\phi: U\rightarrow \mathbb{R}^n$ is a map such that $\phi$ is a homeomorphism onto an open subset of $\...
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0answers
20 views

Prove that the boundary orientation of $S^k = \partial B^{k+1}$ is the same as its preimage orientation

I would like to verify if my approach to this problem is the correct one or not. This problem is from "Differential topology" by Victor Guillemin and Allan Pollack . More specifically is the problem 3....
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0answers
22 views

Extending triangulations of compact closed orientable surfaces to the handlebody they bound?

Given a triangulation (as in, simplicial complex) of a compact closed orientable surface, is it always possible to extend this triangulation to the “bounded” handlebody? If yes, is there any control ...
2
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0answers
19 views

Degree of vertices in a barycentric subdivision

Consider the barycentric subdivision of a triangulation of a manifold $X$, possibly with boundary. I am interested in the degree of vertices after subdivision mod $2$. I think it is true that for ...
4
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0answers
76 views

Prove that $M - \partial M \simeq M$.

Let $M$ be a topological manifold with boundary. I'm trying to prove that $M - \partial M \simeq M$, where "$\simeq$" denotes homotopy equivalence, using the collar neighbourhood theorem. This states ...
2
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1answer
38 views

Why is this set open in a proof of a manifold with boundary?

I try to understand the prove given here. There we need to find an open neighbourhood for all points on $\partial \overline{\Bbb B^n}= \Bbb S^{n-1}$ which is homeomorphic with an open subset in $\Bbb ...
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2answers
51 views

The unit Ball is a manifold with boundary

I try to prove that $\overline{\Bbb B}:=\{x \in \Bbb R^n: \|x\| \leq 1\}$ is a manifold with boundary. For this I have to show that for all $x \in \Bbb B$ there exist an open subset $U$ s.t. $U \...
3
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1answer
32 views

Extending a map of a surface's boundary

I'm reading the section in McDuff and Salamon's Introduction to Symplectic Topology on Chern numbers on surfaces. I've run into an elementary-looking detail in a proof that I don't see a clear reason ...
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0answers
14 views

A set union it's topological boundary as a manifold with boundary

Here is an intresting question about manifold with boundary I couldnt find an answer: Let M be a smooth k-dimensional manifold in Rn and N in M an open subset. Assume that ∂N, the boundary of N inM, ...
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0answers
23 views

intersection number on the boundary of a manifold

Let $F: W \to N$ be a smooth map, where $W$ is a compact manifold with boundary, $Z \subset N$ is closed and all manifolds are oriented. Also $\partial F \pitchfork Z$ and $F^{-1}(Z)$ is a compact, ...
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2answers
13 views

Non--homeomorphic manifolds-with-boundary having homeomorphic boundaries?

What is a simple example of two topological $n$-manifolds-with-boundary $M$ and $N$ that are not homeomorphic yet whose boundaries $\partial M$ and $\partial N$ are homeomorphic?
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1answer
53 views

Defining an outward unit normal and orientation for the boundary of a manifold (Spivak, Calculus on Manifolds)

I have a couple questions on how Spivak defines an outward unit normal and orientation for the boundary of a manifold. I've included the relevant section of Calculus on Manifolds below. I should ...

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