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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.

28
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0answers
783 views

Why are functions with vanishing normal derivative dense in smooth functions?

Question Let $M$ be a compact Riemannian manifold with piecewise smooth boundary. Why are smooth functions with vanishing normal derivative dense in $C^\infty(M)$ in the $H^1$ norm? Here I define $...
16
votes
1answer
742 views

Can you hear the pins fall from bowling game scores?

Let $\mathbb T=\{1,\dotsc,10\}$ represent the ten pins in a standard game of bowling. Given two sets of pins $T\subseteq S\subseteq \mathbb T$, let's write $p_{S\to T}$ to represent the conditional ...
13
votes
2answers
5k views

The boundary of an $n$-manifold is an $n-1$-manifold

The following problem is from the book "Introduction to topological manifolds". Suppose $M$ is an $n$-dimensional manifold with boundary. Show that the boundary of $M$ is an $(n-1)$-dimensional ...
11
votes
1answer
458 views

Is the square pyramid a manifold with corners?

An n-manifold with corners is topologically an n-manifold with boundary, but with a smooth structure that makes it locally diffeomorphic to $[0,\infty)^n$ instead of $[0,\infty) \times \mathbb{R}^{n-1}...
10
votes
1answer
1k views

(Whitney) Extension Lemma for smooth maps

I am currently reading Lee's book "Introduction to Smooth Manifolds (2nd edition)". Corollary 6.27 in that book states that a smooth map $f\colon A \rightarrow M$ where $M$ is a smooth manifold ...
9
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0answers
191 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 ...
8
votes
0answers
254 views

Given a family of 2D curves, find a 3D manifold whose geodesics project to the plane curves

Below is an image of the family of 2D curves for reference. The image is arbitrary. Still trying to formulate a concise question. I know that the geodesics for flat Euclidean Space are straight lines....
8
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0answers
240 views

Homeomorphism between $[0,\infty)^n$ with Upper Half-Space $\mathbb{H}^n$ [duplicate]

I've been working through an exercise in Lee's Smooth Manifold. The author ask us to show that the space $\bar{\mathbb{R}}_+^n := [0,\infty)^n$ is homeomorphic to upper half-space $\mathbb{H}^n := \...
8
votes
0answers
89 views

Cartan geometry on manifolds with boundary

I was reading Sharpe's text on Cartan geometry, and I started to wonder: Does the theory change in any significant way if the base manifold for the Cartan principal bundle is allowed to be a smooth ...
6
votes
2answers
798 views

Any N dimensional manifold as a boundary of some N+1 dimensional manifold?

Is this statement true: Question: Can any N dimensional manifold be realized as a boundary of some N+1 dimensional manifold? If so/not, how to prove/disprove it? I read a TQFT paper from Edward ...
6
votes
1answer
180 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 ...
6
votes
1answer
792 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 ...
6
votes
1answer
300 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 ...
6
votes
1answer
400 views

The definition of smooth maps given in Introduction to Smooth manifolds by John M. Lee

I'm currently reading Introduction to Smooth Manifolds by John M. lee. I'm trying to understand his definition of smooth maps $F:A\subseteq M\to N$ given in page 45. Let's start from scratch to have ...
6
votes
0answers
294 views

Integration by parts on manifold with a boundary

Suppose $C$ is a 3-form, and $G$ is a 4-form defined by $G = dC$. Also, $M_{11}$ is an 11-dimensional manifold (without a boundary), $W_{6}$ is a 6-dimensional submanifold of $M_{11}$ and $D_{\epsilon}...
6
votes
1answer
1k views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves $\...
5
votes
2answers
103 views

Classification of surfaces

The Classification Theorem for surfaces says that a compact connected surface $M$ is homeomorphic to $$S^2\# (\#_{g}T^2)\# (\#_{b} D^2)\# (\#_{c} \mathbb{R}P^2),$$ so $g$ is the genus of the surface, $...
5
votes
1answer
146 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 $...
5
votes
2answers
249 views

Geometrically, what is the stereographic projection of a closed $n$-ball?

To show $\overline{B^n}$ is a $n$-manifold with boundary, apparently there is a trick to use stereographic projection after subtracting out the radius connecting $0$ to the north pole. I'm familiar ...
5
votes
1answer
54 views

Why do we sometimes call the boundary of the body by $\partial$?

I very often see the symbol "$\partial$" is used to define the boundary of a body in three-dimensional space. For example, if the body is called $\Omega$ the boundaries is labeled as $\partial\Omega$. ...
5
votes
1answer
74 views

$W$ spin implies $\partial W$ spin

Let $M$ be a compact orientable manifold with the first two Stieffel-Whitney numbers equal to zero (this is my definition of SPIN manifold). Let $B$ be the boundary of $M$; I want to show that $B$ is ...
5
votes
1answer
1k views

What does it mean for a manifold to be oriented?

I'm currently working through Spivak's Calculus on Manifolds. I've got to Stokes' Theorem, which is stated thus (the bold is my emphasis): Stokes' Theorem If $M$ is a compact oriented $k$-...
5
votes
1answer
111 views

Open subsets of the connected sum $M_1\# M_2$ [duplicate]

I'm trying to solve a problem in John Lee's ITM (Problem 4-19), but seems that i need helps now. Here's the problem : Let $M_1 \# M_2$ be a connected sum of $n$-manifolds $M_1$ and $M_2$. Show ...
5
votes
1answer
359 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 ...
5
votes
0answers
90 views

Is a manifold-with-boundary with given interior and non-empty boundary essentially unique?

Let $M$ be a compact connected manifold-with-boundary such that $\circ M \neq \emptyset$, where $\circ M$ is the boundary of $M$. Let $N$ be a compact connected manifold-with-boundary such that $\circ ...
4
votes
2answers
149 views

Is the closed hemisphere diffeomorphic to the closed disk?

This might be silly but it puzzles me. Let $M$ be the closed upper hemisphere of $\mathbb{S}^2$. This is a manifold with boundary. Is it diffeomorphic to a closed disk in $\mathbb{R}^2$? These ...
4
votes
2answers
858 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.
4
votes
1answer
1k views

Inverse Function Theorem for Manifolds with Boundary as the Domain

In Lee's Smooth Manifolds, it is written that the inverse function theorem can fail for manifolds with boundary. As hint, it is given the inclusion of half space into euclidean space $\iota:\mathbb{H}^...
4
votes
2answers
75 views

Show that ∂(M×N)=M × ∂N.

Let M smooth manifolds (without boundary) and N is a smooth manifold with boundary. Could someone help me to show that $∂(M × N) = M × ∂N$? I saw a suggestion here on the site how to do it, but I'm ...
4
votes
3answers
575 views

Confusion about definition of manifold with boundary

Consider this definition: A space $M$ is a manifold with boundary if each point $x\in M$ has a neighborhood $U_x$ that is homeomorphic to $\mathbb R^n$ or to $\mathbb R^n_+=\{(x_1,\cdots,x_n)\;|\; ...
4
votes
1answer
76 views

2-manifold with involution without fixed points is a boundary of a 3-manifold.

I am preparing for geometry/topology test and I can not handle the following problem from previous years: Let $F$ is closed orientable 2-manifold with involution without fixed points. Prove that $F$ ...
4
votes
1answer
185 views

Open neighborhood of a manifold boundary point

Manifold with boundary: An $n$-dimensional manifold with boundary is a second countable Hausdorff space in which every point has a neighborhood homeomorphic either to an open subset of $\mathbb{R^n}...
4
votes
2answers
99 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 ...
4
votes
1answer
163 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 ...
4
votes
0answers
88 views

Spivak Calculus on Manifolds - Tangent space on a boundary point of a manifold

I am an undergraduate student who is studying Spivak's calculus on manifolds. I have several questions in the pages 119 and 120 of the book, which are about the tangent space at a boundary point of a ...
4
votes
1answer
84 views

Definition of the preimage orientation

Guillemin and Pollack give quite confusing (at least for me) definition of the preimage orientation (see below). I don't understand the part starting from the last display. Namely: How exactly does ...
4
votes
0answers
263 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 ...
4
votes
0answers
361 views

Stokes' theorem: Induced orientation on the boundary of a manifold

The Question Let $K = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \geq 1\}$, where $K$ is oriented via the canonical volume form on $\mathbb{R}^3$: $dx \wedge dy \wedge dz$. Let $\mathbb{S}^2$ be ...
4
votes
0answers
742 views

Submanifold with boundary of a manifold with boundary

Let $M$ be a smooth manifold. (1) A subset $S$ of $M$ that with the subspace topology is a topological manifold (with or without boundary), together with a differential structure that makes the ...
3
votes
3answers
129 views

No where vanishing exact $1$-form on compact manifold.

I found several answers on the following question : Does there exists a no where vanishing exact $1$-form on a compact manifold without boundary? All answer says that certainly not. But I cannot ...
3
votes
2answers
263 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 ...
3
votes
2answers
150 views

$\mathbb S^2$ or $\mathbb RP^2$ on boundary of a 3-manifold

Let $M$ be a 3-manifold with boundary $\partial M$. Suppose $\partial M$ contains a sphere or a projective plane, which is contractable in $M$. Show that $M$ is also contractable. The above statement ...
3
votes
1answer
57 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 ...
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 ...
3
votes
1answer
62 views

Spin structure and bordism

I have some questions about bordism and spin structures on manifolds. If you have any examples or references I would appreciate it. Is there a 3-manifold $ M $, orientable, which does not support 3 ...
3
votes
1answer
52 views

2 -manifolds that can't be decomposed into two equal parts

The real projective plane has the property that if you divide into two different 2-manifolds, they will not be homeomorphic (i.e. one will be orientable, and the other non-orientable). The sphere ...
3
votes
1answer
329 views

The corner of the squre

A square is a topological manifold with boundary but not a smooth manifold with boundary because of its corners. But I am confused about it. I think since for a specific corner $p$, there is only one ...
3
votes
2answers
332 views

Invertibility theorem on the boundary for a function between two closed 2D manifolds

Assume a function $f:\mathbb{R}^2\to\mathbb{R}^2$ on a simply connected, closed domain $D\subset\mathbb{R}^2$ including its boundary $\partial D$. I am interested in the local invertibility of $f$ ...
3
votes
2answers
716 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. ...
3
votes
1answer
67 views

Is this conjecture about the boundary of a surface correct?

I came up with an intuitive conjecture about boundaries of surfaces based on the idea that at a boundary point we can wrap a string across the edge, and the two halves of the string (on opposite sides ...