# 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.

258 questions
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### 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 ...
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### Regular values for maps between manifolds with boundary

Let $h:X \to Y$ be a smooth map between manifolds with boundaries. How does one characterize a regular value for $h$ ? I am sifting through Milnor's Topology from the Differentiable Viewpoint and ...
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### Is every flat manifold with boundary locally isometric to the Euclidean half-space?

Let $M$ be a smooth manifold with boundary, endowed with a smooth Riemannian metric $g$. Suppose $g$ is flat, and let $p \in \partial M$. Is there an open neighbourhood of $p$ which is isometric to ...
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### 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 ...
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### Does the intrinsic and extrinsic distances coincide on the interior?

Let $M$ be a Riemannian manifold with boundary. Consider the interior of $M$ (which we denote by $M^\circ$). $M^\circ$ is an open submanifold of $M$. Let $d_M$ be the induced distance function (...
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### What is the boundary of two manifolds with boundary?

I know that if $M$ is a manifold without boundary and $N$ a manifold with boudary, then $\partial(M\times N)=M\times \partial N$, but, if I have the product of two manifolds with boudary, it's known ...
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### Manifold with boundary and boundary points?

I was curious that since the boundary of a manifold with boundary is boundary-less, $\left(\partial (\partial M)=\emptyset\right)$, then whether the following example of a disc with an open interval ...
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### Is there a (what is the) intrinsic definition of boundary?

It is asked to show that the closed disk $\overline{D}^2=\{(x,y)\in \Bbb{R}^2:x^2+y^2\leq 1\}$ (with the topology induced from $\Bbb{R}^2$) is not a regular surface. It seems obvious that we have a ...
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### Why are those two manifolds with boundary diffeomorphic to $D^2\times S^1$?

I have this problem and I don't know why I can't finish it: Let $S=\{x\in \mathbb{R}^4\mid \vert\vert x \vert\vert=2\}$ the sphere of dimension $3$ and radius $2$. Let $T_+$ (resp. $T_-$) the set ...
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### 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 ...
### 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 ...
### Show that $\partial(M\times N)=M\times\partial(N)$
Let M a $k$-dimensional manifold without boundary of $\mathbb{R}^{n}$ and N a $l$-dimensional manifold of $\mathbb{R}^{m}$ with or withour boundary. Show that $\partial(M\times N)=M\times\partial(N)$ ...