# Tagged Questions

Manifolds are typically defined to be without boundaries (every point has a neighbourhood homeomorphic to an Euclidean open disc). Use this tag for the manifolds with boundaries, as well as manifolds with corners, and other such generalisations of the notion of a manifold.

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### Topological boundary of a specific manifold.

Let $\emptyset\neq X\subset\mathbb{R}^n$, $n\ge2$, be an $(n-1)$-dimensional differentiable submanifold, i.e. for every $p\in X$ there is an open $U_p\subset\mathbb{R}^n$ with $p\in U_p$, and a ...
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### On the trace theory and restrictions of Sobolev space functions

Consider a connected open bounded subset $D\subset \mathbb{R}^d$ with smooth boundary. It is easier to state for the disc so I will do so. I don't think it would change the argument (except for in our ...
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### Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$?

The first question is Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$? In other words, is it obvious? The question stems from a theoretical physics ...
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### Reference for theorems in Hirsch

In Hirsch's differential topology, we find the following theorems on page 31: 4.1 Theorem: Let $M$ be a $C^r$ $\partial$-manifold and $N$ a $C^r$ manifold, $r \geq 1$. Let $f : M \to N$ be a $C^r$ ...
I've just started to study differential geometry and I've some problems with the first definitions. We have defined a topological manifold with boundary of dimension n as a topological space $M$ such ...
### When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth?
When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth? I can't find a formal definition. I know, that we say, that $\partial\Omega$ has a $C^k$-...