# 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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### Finding the optimal surface enclosing a given volume

I would like to find, over the set of continuous surfaces that enclose a volume $V$, the one(s) that lead to the maximal value of a certain cost function. I'm working on a physics problem which ...
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### Smoothness of a cylinder

I´m trying to check the smoothness property of a cylinder $D:=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2\leq 1, z\in [0, H]\}$, but I´m having a problem understanding the definition of Lipschitz continuous ...
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### Product of manifold with boundary

I have a question about the Cartesian product of two manifold with boundary $M$ and $N$. I am working with topological manifolds instead of smooth manifold. I know that $M\times N$ is still a manifold ...
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### Guillemin's proof of manifold boundary dimension

I'm having trouble understanding Guillemin's proof that given a k-dimensional manifold $X$ with a boundary, $\partial X$'s a $(k-1)$ dimensional manifold without boundary. I understand up to the ...
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### Disjoint curves connecting specified pairs of points on the edge of a Möbius strip

Suppose $2n$ points along the edge of a Möbius strip are labeled by some given permutation of $[1,1,2,2,\dots,n,n]$. Is there a criterion or algorithm to determine whether we can draw $n$ non-...
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### Names of two manifold-like structures

I'm looking for the canonical names of (and optionally a reference for) the following structures: The result of cutting out, from a differentiable manifold homeomorphic to $\mathbb{R}^n$, an "...
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### Why can we replace the euclidean space with the upper half space as model space for a manifold

Reading the Tu.'s book "introduction to manifolds" it first defines manifolds as topological spaces which are locally euclidean (which means locally they are homeomorphic to the euclidean ...
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### A prime orientable 3-manifold containing a nonseparating 2-sphere is homeomorphic to $S^2\times S^1$.

I am reading the notes, trying to understand the proof of Proposition 3.5: I have no idea why $X$ is homeomorphic to $S^2\times S^1$ with the interior 3-ball removed. I think by the "obvious ...
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### John Lee's ISM Problem 5-23 (smooth structure of regular level sets for manifolds with boundary)

The problem statement: Suppose $M$ is a smooth manifold with boundary, $N$ is a smooth manifold, and $F: M \to N$ is a smooth map. Let $S = F^{-1}(c)$, where $c \in N$ is a regular value for both $F$ ...
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### Proof verification: $dF_p$ nonsingular, then $F(p)\in \textrm{Int}N$

I am trying to solve Problem 4-2 on Lee's Introduction to Smooth Manifold. The problem states the following: Suppose $M$ is a smooth manifold (without boundary), $N$ is a smooth manifold with ...
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### Finishing proof concerning manifold boundary

I am working on a proof that given some closed $Y\subseteq X$, where $X$ is a manifold with empty (manifold) boundary and $Y$ is a manifold of the same dimension as $X$ and $X\setminus Y$ is not ...
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### Gluing two disks to get an exotic sphere

I have found interest in the subject of exotic spheres. Particularly, the 7-sphere. To create such a sphere, you glue two copies $\mathbb{D}^4\times \mathbb{S}^3$ along the boundary identifying $(u,v)$...
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### Confusion about notation for orientation of boundary

I am reading "All the math you missed" by Thomas A. Garrity, second edition. In the chapter on Divergence Theorem there is the following fragment: Here the derivative $\frac{df}{dx}$ is ...
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### Is the boundary of a set defined using a smooth function also smooth?

Let $f:B_\delta(y)\rightarrow \mathbb{R}^n$ be a smooth function for some $\delta>0$, $y\in\mathbb{R}^m$. Let us define the set $G:= f(B_\delta(y)) + \mathbb{R}^n_\geq$ (using the Minkowski sum). I ...
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### Submanifold of ball is entire ball

I've seen the following used in 2 and 3 dimensions; I don't know how to prove it, and am wondering if it's true in all dimensions: If $M$ is an $n$-dimensional smooth compact manifold embedded in the ...
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### Boundary Orientation

Edits to the original post are in bold. I've been going through @Ted Shifrin's lectures on Stokes's Theorem, and I had a question relating to his choice of orientation of the tangent space as it comes ...
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Let $M$ be a smooth manifold with boundary and $p \in \partial M$. In Lee's Introduction to Smooth Manifolds he gives the following definition: If $p \in \partial M$, a vector \$v \in T_pM \setminus ...