Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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

558 questions
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Why do we sometimes call the boundary of the body by $\partial$?

I very often see the symbol "$\partial$" is used to define the boundary of a body in three-dimensional space. For example, if the body is called $\Omega$ the boundaries is labeled as $\partial\Omega$. Is it any reason using this symbol? What can…
Reza
  • 247
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Understanding the definition of boundary points of a manifold

Let $M$ be a topological manifold. We call it a $n$-manifold with boundary if for each $x\in M$, there is a chart $(U,\phi)$ at $x$ such that $\phi$ is a homeomorphism from $U$ to an open subset of $\mathbb{H}^n$, where…
Ribhu
  • 742
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topological isotopy of "corner"

Let $C$ be a compact manifold and $f,g$ two open, continuous embeddings $f,g: C\times [0,1)^r \to M$ where $M$ is a topological manifold with boundary. If $f$ and $g$ are equal on $C\times \{0\}$, can I say there is an isotopy $h$ from $f$ to $g$…
Tom Leness
  • 43
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Possible to describe random 3D surfaces (geograhical height over limited area) by formula?

Coming from a geographic computer sciences background and working with 3D terrain (so please forgive if my terminology is inappropriate), I was always wondering if it is possible to describe the 3D surface of a limited area with a single…
geozelot
  • 111
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Prove that the disk $\left\{(x, y)\mid x^{2}+y^{2} \leq 1\right\}$ is a manifold with boundary.

A manifold with a boundary is a topological space $(X, T)$ whose open sets have continuous one-to-one maps to open sets in half space. Half space is the region of $\mathbb{R}^{n}$ for which $x^{1} \geq 0$ (in Cartesian coordinates). The topology on…
Zeph Rodriquez
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