# 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 -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 ...
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### Manifolds with boundary and foliations

Is there a theory of foliations by manifolds with boundary? Particularly, Is there a generalization of the Frobenius theorem and the Stefan-Sussmann theorem in which the leaves are manifolds with ...
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### A question about the proof of Extension Lemma for Smooth functions

I am trying to understand the proof, but am stuck at the last line. Why does the first equality of the picture below holds? The codomain of $f$ is $R^k$, not some the set of nonnegative numbers. Is ...
1answer
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### Giving a counterexample for the extension lemma of smooth functions

I am supposed to give a counterexample showing the conclusion is false when $A$ is not closed. I tried to find one when $M$ is Euclidean space but kept failing... Could anyone please show me a ...
1answer
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### Showing that a diffeomorphism preserves the boundary

I am supposed to use the fact presented below to show that the diffeomorphism $F$ in the theorem 2.18 preserves the boundary. I found a way to prove it but it does not use the fact below. I am curious ...
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### Integration of one-form

I am trying to compute $$\int_C i^*\eta$$ $\eta=(x^2+y^2)dz$ and $C=\{(x,y,-1): x^2+y^2=1\}$ and $i$ is the inclusion map This is what I did $$\int_{-1}^{-1}(x^2+y^2) dz=0$$ Is this correct?
1answer
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### “Defining a smooth structure on a topological manifold with boundary”

This is a proposition for topological manifolds. However, I am wondering if this prop holds when $M$ is a topological manifold "with boundary." Thus, can the expression like "a smooth atlas on a ...
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### Show that if $dF_p$ is nonsingular, then $F(p)\in \text{Int}N$ : Lee's Smooth Manifolds

Suppose $M$ is a smooth manifold without boundary and $N$ is a smooth manifold with boundary and let $F : M \rightarrow N$ is a smooth map. Show that if $p \in M$ is a point such that $dF_p$ is ...
2answers
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### What are the possible boundaries of connected compact manifolds?

I'm trying to understand what types of manifolds can occur as the boundary of a compact manifold $M$. When $M$'s dimension $d$ is 1, the boundary is always either empty or two points (corresponding to ...
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### Stokes' theorem proof without FTC

To present time I have not found a proof of Stokes' Theorem for manifolds that does not involve the Fundamental Theorem of Calculus in some way. Is it possible to prove the general theorem without ...
1answer
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### The set that we add to compactify is itself compact.

Let $X$ be a topological space. Let $X_c$ a compactification of $X$. I want to know if $X_c\setminus X$ is always compact, or maybe more simply whether it is always closed in $X_c$. I think it is true ...
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### Are there topological manifolds with boundary that are not compact?

Following this question Are there compact manifolds without boundary? I'm asking if there is any example of a manifold with boundary that is not compact, Is the interval $X=[0,1)$ a counterexample ? ...
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### integrating a 2-form over the sphere

I need to integrate the following form over the 2-sphere: $$w=\frac{xdy\wedge dz-ydx\wedge dz+zdx\wedge dy}{(x^2+y^2+z^2)^\frac{3}{2}}$$ now a direct calculation shows that $dw=0$, thus from stokes' ...
1answer
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### 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 $\partial\Omega$. ...
2answers
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### Suppose $M$ is a connected manifold and $B \subseteq M$ is a regular coordinate ball, is $M \setminus B$ connected?

Suppose $M$ is a connected manifold of dimension $n > 1$ and $B \subseteq M$ is a regular coordinate ball, is $M \setminus B$ connected? I think the answer to this is definitely yes, but I'm ...
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### 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 $F$ ...
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### $M\setminus\partial M$ open and proof of dimension

Let $M$ be an $n$-dimensional manifold with boundary. Show that: $M\setminus\partial M$ is an open subset of $M$ and it is an $n$-dimensional manifold, and that $\partial M$ is a closed subset of $M$...
1answer
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### Coordinate chart of 1-dimensional manifold with boundary

Let $M$ be a 1-dimensional manifold with boundary, $x_0\in M$, and $\varphi:U\rightarrow V$ be a local coordinate chart such that $x_0\in U$. Take $x_0$ such that $\varphi(x_0)=0\in \partial\mathbb{H}$...
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### $\mathbb S^2$ 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 ...
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### Calculus of variations when target space has boundary

Let $M,N$ be smooth oriented Riemannian manifolds; $M$ closed and $N$ with non-empty boundary. Let $f:M \to N$ be smooth and suppose the image of $f$ intersects $\partial N$. Let W:=\{V \in \Gamma(...
1answer
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### 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 ...
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### Immersive limit of embeddings is injective on the interior?

$\newcommand{\pl}{\partial}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be smooth compact, connected, oriented manifolds of the same dimension. $\M$ can have a boundary....
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### Volume of a manifold

Throughout this post, I am presuming $M$ to be an $2$-dimensional manifold that is parametrized by one chart $\varphi$, and I presume $\omega$ be a $2$-form on $M$. Apparently, there is no natural ...
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### How to show reflection of $\mathbb{S}^n$ and the identity are not homotopic?

Let $N=\mathbb{S}^n$, $M \subseteq \mathbb{S}^n$ be a hemisphere, including the equator (I am considering $M$ with boundary). Let $f_1:M \to N$ be the inclusion map, and let $f_2:M \to N$ be the ...
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### Do trivial homotopy groups imply existence of boundary preserving homotopies?

Let $N$ be a smooth $d$-dimensional connected orientable manifold $N$ which have the following property: For every smooth $d$-dimensional manifold $M$ with non-empty boundary, and for every smooth ...
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### Is uniform limit of embeddings injective?

$\newcommand{\pl}{\partial}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be smooth compact, connected, oriented manifolds of the same dimension with non-empty boundaries....
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### Boundary of a submanifold

Let be $M$ a smooth manifold with boundary and $N \subset M$ a submanifold with boundary, such $N$ is a closed subset of $M$ in the topological sense. Denote $\partial N$ be the boundary of $N$ as a ...
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