# 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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### Is the geodesic diameter on simply connected domains with boundary always realized by points on the boundary?

Assuming standard euclidean metric, the geodesic diameter, of simply connected polygons in the plane is realized by a shortest connection between two vertex points. This result is e.g. referenced in "...
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### 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 ...
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### compute integral of function of distance to the boundary only

How does one compute the integral $\int_{\Omega_{\epsilon}} f( d(x))\ dx = ?$ where $d(x)$ is the distance to the boundary and $\Omega_\epsilon := \{ x\in \Omega: d(x)<\epsilon\}$, supposing ...
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### Showing that a neighborhood is not homeomorphic to the open subset of $R^n$

This is an exercise from Boothby that I am stuck at. It is trivial that the boundary of $H^n$ is a manifold of dimnension $n-1$ because it is homeomorphic to $R^{n-1}$. However I cannot show that no ...
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### Let $M$ be a smooth manifold and let $N$ be a manifold with boundary. If $dF_{p}$ is an isomorphism, then $F\left(p\right)\in\mbox{int}M$.

Let $M$ be a smooth manifold and let $N$ be a manifold with boundary, both with the same dimension $n$. If $dF_{p}$ is an isomorphism, then $F\left(p\right)\in\mbox{int}N$. I am trying to prove ...
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### Take a regular coordinate ball and you get a manifold with boundary.

Suppose $M$ is a (topological) manifold of dimension $n \geq 1$ and $B$, is a regular coordinate ball in $M$. Show that $M\backslash B$ is an $n$-manifold with boundary and whose boundary is ...
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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 ...
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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' ...
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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 $\partial\Omega$. ...
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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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### Volume form on a disk and induced orientation 1-form

I'm having trouble with this question: Choose a volume form $\zeta$ on the disk $D(r) = \{(x,y,0)\in \mathbb{R}^3:x^2+y^2\leq r^2\}$ and describe what the induced orientation 1-form looks like in the ...
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### A crucial step in showing that the boundary of a $n-$manifold is a $(n-1)-$manifold

Consider $M$ an $n-$manifold with boundary $\partial M$. In showing that $\partial M$ is an $(n-1)-$manifold a crucial step is not clear to me: Let $x\in\partial M$, then there exists an open ...
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### Understanding a conclusion through “elementary considerations about bilinear forms”

I am having a hard time to understand how "elementary considerations about bilinear forms" can imply the following result: Let $E$ be a function space, 1 the constant function $x\mapsto 1$ and let \$...