# 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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### Converting a differential form to a measure

so today I was looking at the Generalized Stokes' Theorem: \begin{align} \intop_{\Omega} d\omega=\intop_{\partial\Omega}\omega\ \ , \end{align} where $\Omega$ is some region, and $\omega$ is a ...
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### What is the difference between a boundary *existing* (or rather, *not* existing) and a boundary *being empty*?

I'm confused about the concept of a "boundary" in topology – specifically, in studying manifolds. What is the difference between a boundary existing (or rather, not existing) and a boundary ...
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### The boundary of any manifold that is itself a boundary of a higher-dimensional manifold must always be empty?

I have seen it asserted several times that it is well-known that the boundary of any manifold that is itself a boundary of a higher-dimensional manifold must always be empty. Yet, despite it ...
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### Issue about the definition on orientation on manifolds

In my class, orientation on a $C^\infty$ manifold is defined as follows: Let $\mathcal{O}_0$ be the canonical orientation on $\mathbb{R}^m$ (i.e. the equivalence class (with respect to orientation) ...
1 vote
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### Boundary case of adapted orthonormal frame exercise in Professor Lee's Introduction to Riemannian Manifolds (IRM)

In the post Adapted Orthonormal frame on Riemann manifold, the poster offered the outline of a proof for IRM Proposition 2.14. I can fill in the details to make that proof work when the embedded ...
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### tangent vector of a disk embedded in R^3

Consider in $\mathbb{R}^3$ the plane $N: x+y+z=0$ and the sphere $S: (x-a)^2+(y-b)^2+(z-c)=r$, where $r$ is fixed. The intersection $N\cap S$ is the disk embedded in $\mathbb{R}^3$. I would like to ...
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### What is a Riemann surface with boundary?

My guess is that $X$ is a Riemann surface-with-boundary if it is a topological 2-manifold-with-boundary such that the transition charts are biholomorphic. Now what does biholomorphism mean for charts ...
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### A $\pi_1$-neglibility criterion in 4-dimensional manifolds

I'm reading about the h-cobordism theorem in boundary dimension 4. Most of the steps are the same as in the classical statement, but finding whitney disks to homotope the 2- and 3-handles into ...
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### Complement of a knot complement

Consider a knot K in the 3-sphere, its knot complement C(K) is obtained by removing a tubular neighborhood of K from the 3-sphere. What can I say about the complement of C(K) in the 3-sphere? Is it ...
1 vote
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### $f(\partial K)\subset\partial f(K)$ and $f(\operatorname{Int}K)\subset\operatorname{Int}f(K)$ for a well-behaved $f:\mathbb R^2\to\mathbb R^2$?

Let $f:\mathbb{R}^2\to\mathbb{R}^2$ be a continuously differentiable bijection with nonzero Jacobian. Let $K\subset\mathbb{R}^2$ be a compact subset. Then $f(K)$ is compact. Does the boundary of $K$ ...
1 vote
Problem description: I have a 2D smooth manifold with boundary $M$ embedded in $\mathbb R^3$, discretized by a triangular mesh $T$, and need to find the eigenfunctions of the Laplace-Beltrami operator ...