# 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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### The Transverality Theorem in Differentiable Topology by Guillemin and Pollack

In Chapter 2, Section 3 of the book, most of the theorems requires the codomain $Y$ to be a manifold without the boundary and the submanifold $Z$ to be boundaryless as well. But I don't see why the ...
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### Unsound definition of interior and boundary points of a submanifold of $\mathbb R^N$

In my lecture notes $M$ is said to be a $n$-dimensional submanifold of $\mathbb R^N$ if for all $p\in M$ there is a homeomorphism $\psi$ from an open neighborhood $\Omega_1\subseteq\mathbb R^N$ of $p$ ...
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### When is a chart of a submanifold not only a homeomorphism, but a diffeomorphism?

I've got trouble to understand the concept of a "smooth structure" associated to a submanifold. Let $\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$. Say $M\subseteq\mathbb R^d$ is a $k$-...
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### Show that these two diffeomorphisms cannot exist simultaneously

Let $d\in\mathbb N$, $x\in M\subseteq\mathbb R^d$ and $\psi^{(i)}:\Omega_i\to\psi^{(i)}(\Omega_i)$ be a diffeomorphism with $x\in\Omega_i$, \psi^{(1)}(M\cap\Omega_1)=\psi^{(1)}(\Omega_1)\cap(\mathbb ...
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### Definition of a submanifold with boundary

I'm really struggling to understand the definition of a "submanifold with boundary". Until now, I'm only familiar with the notion of a "submanifold of $\mathbb R^d$. I've defined this ...
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### Fréchet manifold structure on space of sections

I know that the space $\mathsf{C}^\infty(M;N)$ of smooth maps from a closed (smooth) manifold $M$ to a (smooth) manifold $N$ is a Fréchet manifold. I have been looking for a more general version of ...
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### Stokes' Theorem general case

With the following lemma : Lemma : Let $f_{+},f_{-} : U \longmapsto \mathbb{R}$ be $C^{1}$ maps with $f_{-} \leq 0 \leq f_{+}$ with $U \subset \mathbb{R}^{n}$ open and bounded with $C^{1}$ boundary. ...
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### Is there an example of reducible compact 3-manifold with boundary that has no embedded incompressible two-sided surface?

There is a theorem stating that for irreducible compact manifolds with non-empty boundary there always exists such an embedded surface and I'm trying to understand why the irreducibility condition ...
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### Why does a certain integral on a 3-manifold depend only on its boundary data?

I am reading Dan Freed's lectures on Quantum Groups on Path Integrals. I am picking up the required math as I go along and I am finding certain calculations hard to follow. This is regarding the ...
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### Structure preserving maps between manifolds with boundary

While learning the basics about smooth manifolds with boundary in this semesters' course about analysis on manifolds, there's a seemingly basic property I didn't find anywhere. Namely I want to ...
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### Name of the set for which a given set is the boundary of

Consider that in a space S there are sets A and B, where B is the boundary of the compact and simply connected set A. What assumptions are required to define a unique A (or "A like" set) with respect ...
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