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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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Finding the optimal surface enclosing a given volume

I would like to find, over the set of continuous surfaces that enclose a volume $V$, the one(s) that lead to the maximal value of a certain cost function. I'm working on a physics problem which ...
amrit 's user avatar
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Smoothness of a cylinder

I´m trying to check the smoothness property of a cylinder $D:=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2\leq 1, z\in [0, H]\}$, but I´m having a problem understanding the definition of Lipschitz continuous ...
oli H.'s user avatar
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2 answers
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A troublesome boundary term in an integration-by-parts ... how to see it vanishes?

I'm re-wording a previous question that I had tried to simplify but inadvertently made more confusing. Here are the facts. $M$ is a 4-dimensional volume. I have a routine integration by parts: $$\...
Khun Chang's user avatar
1 vote
1 answer
64 views

Gauss–Bonnet Theorem for rectangular surface with rectangular holes: is the right hand side always $0$?

I want to use the Gauss–Bonnet theorem for a non-Euclidean rectangular 2D surface with discrete curved boundaries that has rectangular holes with discrete curves. I know that: $$ \oint k_g \, ds + \...
archrook's user avatar
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31 views

Product of manifold with boundary

I have a question about the Cartesian product of two manifold with boundary $M$ and $N$. I am working with topological manifolds instead of smooth manifold. I know that $M\times N$ is still a manifold ...
Frozer Clark's user avatar
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1 answer
54 views

On mod 2 intersection number

On page 78 of the book Differential Topology by Guillemin and Pollack is presented the following Theorem: If $f_0 , f_1 : X\to Y$ are homotopic and both transversal to $Z$, then $I_{2}(f_0 , Z)=I_{2}(...
user 987's user avatar
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Justification of the induced orientation on a sub-manifold with boundary

In my study of orientation of sub-manifold, I tried to construct the induced orientation on the boundary of a sub-manifold and I would like to have your advice because in the book I have (Milnor’s ...
G2MWF's user avatar
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3 votes
1 answer
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Stokes theorem for currents on manifolds with corners

Let $M\subset\mathbb R^N$ be a compact oriented $n$-(sub)manifold with corners and $\omega$ be an $(n-1)$-form on it. The usual statement of Stokes theorem $$\int_M d\omega=\int_{\partial M}\omega$$ ...
Derivative's user avatar
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1 vote
1 answer
116 views

Reflective Dodecadron

I'm trying to figure out a condensed (ideally, closed form) formula for a potentially nested set of vector reflections with unusual boundary conditions, which I define as follows: For some real A > ...
Breaking Bioinformatics's user avatar
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When is the map sending a boundary point to its outer normal vector continuous?

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and $\partial \Omega$ its boundary. Let $\nu$ be the map sending each point $x \in \partial\Omega$ to the unique outer normal vector at this point....
CBBAM's user avatar
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Relationship between the boundaries of two convex sets

Let $S\subset\mathbb{R}^3$ be a closed convex set contained in a plane $H$. (The plane $H$ passes through the point $\mathbf{0}$ and is orthogonal to the unit vector $\hat{\mathcal{r}}\in \mathbb{R}^...
Gino's user avatar
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1 answer
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Möbius strip is a smooth submanifold

I want to find a description of the Möbius strip without boundary as a submanifold. To be more specific, what I mean with "description". I know the following proposition: For a subset $M \...
Philip's user avatar
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1 answer
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"Boundary" of a Lie group

From my understanding of Lie groups, they can have multiple disconnected components. For example $\operatorname{GL}(n,\mathbb{R})$ has 2 components which can be seen since $\det$ is a Lie group ...
John Doe's user avatar
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Please explain Figure 24.4 in "Analysis on Manifolds" by James R. Munkres.

I am reading "Analysis on Manifolds" by James R. Munkres. I am reading the proof of Theorem 24.4. I don't understand Figure 24.4. Is the solid enclosed by the red closed curve in Figure 24.4 ...
佐武五郎's user avatar
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Whether $\mathbb{T}\times [0,1]$ is diffeomorphic to $\mathbb{D}^2$?

Here $\mathbb{T}$ denotes the torus $\mathbb{R}\backslash \mathbb{Z}$, and $\mathbb{D}^2$ the closed unit ball in $\mathbb{R}^2$. Since $\mathbb{T}\times [0,1]$ and $\mathbb{D}^2$ are both smooth ...
Jiawen Zhang's user avatar
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1 answer
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Does collar restrict to a closed neighbourhood?

Suppose $X$ is a (smooth) manifold with boundary and $f: [0,\infty)\times \partial X \rightarrow X$ is an open embedding such that $f(0,x) = x (\forall x \in \partial X) $ i.e., a collar. Is $f([0,1] \...
Skyskie's user avatar
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1 vote
1 answer
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Guillemin's proof of manifold boundary dimension

I'm having trouble understanding Guillemin's proof that given a k-dimensional manifold $X$ with a boundary, $\partial X$'s a $(k-1)$ dimensional manifold without boundary. I understand up to the ...
user1143399's user avatar
-1 votes
1 answer
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Disjoint curves connecting specified pairs of points on the edge of a Möbius strip

Suppose $2n$ points along the edge of a Möbius strip are labeled by some given permutation of $[1,1,2,2,\dots,n,n]$. Is there a criterion or algorithm to determine whether we can draw $n$ non-...
Karl's user avatar
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Names of two manifold-like structures

I'm looking for the canonical names of (and optionally a reference for) the following structures: The result of cutting out, from a differentiable manifold homeomorphic to $\mathbb{R}^n$, an "...
Jens's user avatar
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1 answer
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Why can we replace the euclidean space with the upper half space as model space for a manifold

Reading the Tu.'s book "introduction to manifolds" it first defines manifolds as topological spaces which are locally euclidean (which means locally they are homeomorphic to the euclidean ...
Andrea's user avatar
  • 77
2 votes
1 answer
60 views

A prime orientable 3-manifold containing a nonseparating 2-sphere is homeomorphic to $S^2\times S^1 $.

I am reading the notes, trying to understand the proof of Proposition 3.5: I have no idea why $X$ is homeomorphic to $S^2\times S^1$ with the interior 3-ball removed. I think by the "obvious ...
Kat's user avatar
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1 answer
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Is a $C^p$ ($p \geq 1$) orientable closed $(k-1)$-manifold always the boundary of a $C^p$ $k$-manifold-with-boundary in $\mathbb{R}^n$?

By manifold i mean an embedded submanifold of $\mathbb{R}^n$. The question is exactly the one in the title. A manifold M is closed if $\partial M = \emptyset$. I tried to prove it in the case of $k = ...
Lorenzo Vanni's user avatar
1 vote
0 answers
42 views

Defining the Tangent space to the boundary of a manifold $T_p(\partial S)$

While studying manifolds I am having some problem with definition of manifold with boundary.Let $S$ be a regular $n$-level surface in $\mathbb R^{n+1}$ with boundary defined by $S=f^{-1}(0)\cap (\...
Kishalay Sarkar's user avatar
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70 views

Defining function and $C^k$ boundaries

In the textbook I'm currently studying, it has somewhat concise detail about $C^k$ boundaries. The definitions it uses are as follows. Definition: A domain (connected open set) $\Omega$ in $\mathbb{R}...
Maths Matador's user avatar
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1 answer
97 views

Orientability of top-dimensional manifolds (with boundary)

Is every top-dimensional manifold-with-boundary orientable? If this is not true, is there an easy-to-understand counterexample? EDIT: Some context in case this question was unclear. The hypotheses of ...
Jeffrey Wu's user avatar
1 vote
1 answer
19 views

topological isotopy of "corner"

Let $C$ be a compact manifold and $f,g$ two open, continuous embeddings $f,g: C\times [0,1)^r \to M$ where $M$ is a topological manifold with boundary. If $f$ and $g$ are equal on $C\times \{0\}$, ...
Tom Leness's user avatar
1 vote
1 answer
76 views

How to prove subbundle example (10.33(c) in Lee’s Intro to Smooth Manifolds) in boundary case

Prove that $TS$ is subbundle of $TM|_S$ for immersed submanifold asked how to show that $TS$ is a subbundle of $TM|_S$ when $S\subseteq M$ is an immersed submanifold. However, the origin of this ...
Jeff Rubin's user avatar
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Boundary of the boundary of an oriented compact manifold-with-corners

Suppose I have an oriented, compact manifold-with-corners $M$ (see Jack Lee’s Introduction to Smooth Manifolds, pages 417-419). It is not hard to see (using the results of these pages) that its ...
Cronus's user avatar
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2 votes
0 answers
115 views

Homology of a Polyhedral Complex

I am new to algebraic topology and have no experience at all in this field. I started reading Hatchers Book on algebraic topology, however i am still not able to fully understand the following paper ...
CookieMonster98's user avatar
1 vote
1 answer
118 views

Connectedness of manifolds lying in a cylinder with a hole

Problem setting: We define a cylinder with a hole \begin{equation} A_{\delta} = \{(x',x_N) \in \mathbb{R}^{N-1} \times \mathbb{R} \mid \delta \leq |x'| \leq 1 \} \end{equation} for some $\delta \in (0,...
IgotYourPoint's user avatar
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0 answers
158 views

John Lee's ISM Problem 5-23 (smooth structure of regular level sets for manifolds with boundary)

The problem statement: Suppose $M$ is a smooth manifold with boundary, $N$ is a smooth manifold, and $F: M \to N$ is a smooth map. Let $S = F^{-1}(c)$, where $c \in N$ is a regular value for both $F$ ...
Tob Ernack's user avatar
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1 vote
0 answers
36 views

Is every smooth atlas for a manifold with boundary contained in a unique maximal smooth atlas?

I'm wondering if the following proposition (from Lee's Introduction to Smooth Manifolds) also applies to smooth manifolds with boundary: Definition: a map from an arbitrary subset $A\subseteq \mathbb{...
Sam's user avatar
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0 answers
139 views

Converting a differential form to a measure

so today I was looking at the Generalized Stokes' Theorem: \begin{align} \intop_{\Omega} d\omega=\intop_{\partial\Omega}\omega\ \ , \end{align} where $\Omega$ is some region, and $\omega$ is a ...
Sora8DTL's user avatar
  • 107
1 vote
0 answers
45 views

What is the difference between a boundary *existing* (or rather, *not* existing) and a boundary *being empty*?

I'm confused about the concept of a "boundary" in topology – specifically, in studying manifolds. What is the difference between a boundary existing (or rather, not existing) and a boundary ...
The Pointer's user avatar
  • 4,322
5 votes
2 answers
173 views

The boundary of any manifold that is itself a boundary of a higher-dimensional manifold must always be empty?

I have seen it asserted several times that it is well-known that the boundary of any manifold that is itself a boundary of a higher-dimensional manifold must always be empty. Yet, despite it ...
The Pointer's user avatar
  • 4,322
3 votes
1 answer
68 views

Let $f:\mathbb{R}^m\to\mathbb{R}^n$ be continuous and $M\subseteq \mathbb{R}^m$ be an $m$-manifold. Is the graph of $f$ an $m$-manifold?

Below $M^\circ$ and $\partial M$ denote the manifold-interior and manifold boundary of the topological manifold $M$ respectively. Result (?): $f:\mathbb{R}^m\to\mathbb{R}^n$ be continuous and $M\...
Sam's user avatar
  • 5,166
2 votes
0 answers
114 views

Proof verification: $dF_p$ nonsingular, then $F(p)\in \textrm{Int}N$

I am trying to solve Problem 4-2 on Lee's Introduction to Smooth Manifold. The problem states the following: Suppose $M$ is a smooth manifold (without boundary), $N$ is a smooth manifold with ...
Kaira's user avatar
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1 vote
1 answer
58 views

Why is $M_{f\geq 0}$ a manifold with boundary?

I'm studying Morse theory and I found this fact: Let be $f:M\rightarrow \mathbb{R} $ a Morse function on a m-manifold $M$. Suppose $0$ is not a critical value of $f$. Then $M_{f\geq 0}=\lbrace p\in M |...
3435's user avatar
  • 385
2 votes
2 answers
103 views

Issue about the definition on orientation on manifolds

In my class, orientation on a $C^\infty$ manifold is defined as follows: Let $\mathcal{O}_0$ be the canonical orientation on $\mathbb{R}^m$ (i.e. the equivalence class (with respect to orientation) ...
Luigi Traino's user avatar
2 votes
0 answers
69 views

Intersection number of submanifolds with boundary

Let $N$ be an $n$-dimensional compact oriented manifold without boundary, and let $X$, $Y$ be compact oriented submanifolds of complementary dimensions $k$ and $n - k$, with boundary but such that $X \...
NPG's user avatar
  • 33
2 votes
1 answer
89 views

Cutting a disk with piecewise analytic curves

Let $B \subset \mathbb{C}$ be an open set and assume that the boundary of $B$ consists of finitely many piecewise analytic curves. Suppose $z_0 \in \partial B$ and assume the following: $U(z_0 , r) = ...
porridgemathematics's user avatar
0 votes
0 answers
113 views

Boundary case of adapted orthonormal frame exercise in Professor Lee's Introduction to Riemannian Manifolds (IRM)

In the post Adapted Orthonormal frame on Riemann manifold, the poster offered the outline of a proof for IRM Proposition 2.14. I can fill in the details to make that proof work when the embedded ...
Jeff Rubin's user avatar
2 votes
1 answer
74 views

Let $M=(\mathbb{S}^2\times [0,1])/\sim$, where $(p,0)\sim (-p, 1)$. Prove that M is a compact, differentiable manifold. Is it orientable?

I think I don´t have trouble with the first part, because for the antipodal map $f:\mathbb{S}^2\rightarrow \mathbb{S}^2, \ \ p \mapsto -p$, and any differentiable atlas for the sphere $\mathcal{A}=\{(...
Nestor Anguiano's user avatar
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0 answers
43 views

Finishing proof concerning manifold boundary

I am working on a proof that given some closed $Y\subseteq X$, where $X$ is a manifold with empty (manifold) boundary and $Y$ is a manifold of the same dimension as $X$ and $X\setminus Y$ is not ...
moboDawn_φ's user avatar
3 votes
0 answers
124 views

Gluing two disks to get an exotic sphere

I have found interest in the subject of exotic spheres. Particularly, the 7-sphere. To create such a sphere, you glue two copies $\mathbb{D}^4\times \mathbb{S}^3$ along the boundary identifying $(u,v)$...
Bessel's user avatar
  • 120
4 votes
1 answer
97 views

Confusion about notation for orientation of boundary

I am reading "All the math you missed" by Thomas A. Garrity, second edition. In the chapter on Divergence Theorem there is the following fragment: Here the derivative $\frac{df}{dx}$ is ...
deespodete's user avatar
1 vote
0 answers
52 views

Is the boundary of a set defined using a smooth function also smooth?

Let $f:B_\delta(y)\rightarrow \mathbb{R}^n$ be a smooth function for some $\delta>0$, $y\in\mathbb{R}^m$. Let us define the set $G:= f(B_\delta(y)) + \mathbb{R}^n_\geq$ (using the Minkowski sum). I ...
Ina's user avatar
  • 31
4 votes
2 answers
117 views

Submanifold of ball is entire ball

I've seen the following used in 2 and 3 dimensions; I don't know how to prove it, and am wondering if it's true in all dimensions: If $M$ is an $n$-dimensional smooth compact manifold embedded in the ...
Hempelicious's user avatar
0 votes
1 answer
228 views

Boundary Orientation

Edits to the original post are in bold. I've been going through @Ted Shifrin's lectures on Stokes's Theorem, and I had a question relating to his choice of orientation of the tangent space as it comes ...
RHyp's user avatar
  • 155
2 votes
1 answer
278 views

Intuitive understanding of inward and outward vector definitions

Let $M$ be a smooth manifold with boundary and $p \in \partial M$. In Lee's Introduction to Smooth Manifolds he gives the following definition: If $p \in \partial M$, a vector $v \in T_pM \setminus ...
CBBAM's user avatar
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