# 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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### $(x + y + xy)/2 = f( f^{[-1]}(x) + f^{[-1]}(y) )$

Consider $(x + y + xy)/2 = f( f^{[-1]}(x) + f^{[-1]}(y) )$ Where $f^{[-1}]$ denotes the functional inverse of $f$. How to find $f$ ? How about the more General idea of finding $f$ for a given $g$? ...
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### Extending Riemannian Manifold to Boundary

If you have a Riemannian manifold $(M,g)$ (maybe with other assumptions as need), is there a natural way to extend it to a smooth manifold with boundary? For example, the Lobachevsky space viewed as ...
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### Compact 3-manifolds with boundary an orientable surface

The usual embedding of the closed orientable surface $M_g$ of genus $g$ in $\mathbb{R}^3$ bounds a compact orientable $3$-manifold which is homotopically equivalent to a bouquet of $g$ circles. If a ...
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### Can every Riemmanian Manifold be completed?

I had two trails of though.. is either of them fruitful? I know every metric space can be completed, my question is: can a Riemmanian manifold $M$ be embedded smoothly and isometrically into it's ...
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### Topological boundary as a submanifold

Let $U$ be an open subset of a smooth $n$-manifold. Consider $\partial U$ the topological boundary of $U$. Is the following true ? : If $\partial U$ is a smooth $n-1$ submanifold without boundary, ...
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### Is $\overline{u^{-1}((-\infty,a))}$ a $C^k$-manifold with boundary for a $C^k$ function $u$?

Let $u\in C^k(\mathbf{R}^d)$, $M:=\overline{u^{-1}((-\infty,a))}=\overline{\{x\in\mathbf{R}^d \mid u(x)<a\}}$, where $a$ is a constant such that $M\neq\emptyset$. Is $M$ a $C^k$-manifold with ...
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### Induced orientation on manifold

Suppose $M$ is a naturally oriented $n$-manifold embedded in $\mathbb{R}^n$ with non-empty boundary. Give $\partial M$ the induced orientation. The induced orientation defines a unit normal vector ...
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