# 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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### Are there metrics of nonnegative Gaussian curvature on these surfaces?

Let $\Sigma$ be a compact surface of genus $g \geq 1$ and having $r \geq 1$ boundary components. Are there metrics of nonnegative Gaussian curvature on $\Sigma$? If $\Sigma$ were closed, then the ...
0answers
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### What is the orientation of the normalized boundary $\partial(M\times N)$ of product manifold?

Assume $M$ and $N$ are two oriented smooth manifold with or without boundaries. Then $M\times N$ is an oriented manifold with corners. Inspired by the theory of cobordism or differential forms, the ...
2answers
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### Is it possible to construct a 3D equivalent of Gabriel's Horn in a higher dimensional space?

Gabriel's Horn has the interesting property that it is an infinite surface area bound within a finite volume. I was wondering if there was an extension of this to 3D space in a higher dimensional ...
0answers
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### What is the boundary of a tubular neighbourhood of the projective plane embedded in $\mathbb{R}^4$?

There are various ways to embed the projective plane into $\mathbb{R}^4$ very nicely, see e.g. Wikipedia. Suppose now that I take such an embedded projective plane $P \subset \mathbb{R}^4$ and fix a ...
1answer
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### Do homeomorphic manifolds with boundary have homeomorphic interiors?

Let $M, N$ be manifolds with boundary and $f: M \rightarrow N$ be a homeomorphism. I want to show that $\text{Int}M$ is homeomorphic to $\text{Int}N$. I think I have most of the proof but it relies on ...
1answer
51 views

### If the interior of a manifold with boundary is smooth, is the whole manifold smooth?

Let $M$ be a topological manifold with boundary. Let Int$M$, its interior, be a smooth manifold. Is it a known result that $M$ itself will be a smooth manifold with boundary? Can we extend a smooth ...
0answers
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### Is $TN$ a smooth embedded submanifold with or without boundary in $TM$?

Let $M$ be a smooth manifold with or without boundary, and $N$ a smooth embedded submanifold with or without boundary in $M$, then the inclusion map $N\to M$ induces $TN\to TM$, is $TN$ a smooth ...
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1answer
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### If $M$ is a smooth manifold with boundary, $f \in C^\infty(M)$, $b$ a regular value of $f$, then is $f^{-1}((-\infty,b])$ a regular domain in $M$?

The above is taken from John Lee's Introduction to Smooth Manifolds (p. 121). In Proposition 5.47, one supposes that $M$ is a smooth manifold. Does Proposition 5.47 also hold if $M$ is changed to a ...
1answer
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### How to conclude each $E_i$ is a compact regular domain by Proposition 5.47?

In Proposition 5.47, $M$ is a smooth manifold, but in Theorem 6.15, $M$ is a smooth manifold with or without boundary, how to conclude each $E_i$ is a compact regular domain by Proposition 5.47?
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### Is any map $f:M\to N$ smooth?

Let $M$ be a 0-dimensional smooth manifold and $N$ a smooth manifold with or without boundary, is any map $f:M\to N$ smooth?
0answers
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### Is a compact connected manifold-with-boundary a CW complex?

Suppose $M$ is a compact connected manifold-with-boundary with non-empty boundary. What can be said on whether $M$ can be given a CW complex structure? A similar problem has been discussed for ...
1answer
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### How can we apply generalized Stokes' theorem to a non-oriented manifold with boundary?

I do not really know much about the boundary of non-oriented manifold. A boundary of oriented manifold, if it exists, has a sign. If you reverse the orientation, the boundary picks up an extra ...
1answer
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### Spin structure and bordism

I have some questions about bordism and spin structures on manifolds. If you have any examples or references I would appreciate it. Is there a 3-manifold $M$, orientable, which does not support 3 ...
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### geodesic balls in Riemannian manifolds with bounded geometry

Let $(M,g)$ be an open (:=complete, non-compact) Riemannian manifold with bounded geometry, in the sense that in some atlas of charts of radius $r_0>0$, the metric and all its derivatives are ...
0answers
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### Spivak manifolds - definition of $dw$ for a p-form $w$ on a manifold $M$

Spivak says the definition of $dw$ for a k-form $w$ does not make sense on a manifold because $D_j(w_{i_1, \dots , i_p})$ has no meaning. Does it have no meaning because the function w_{i_1, \dots , ...
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### Spivak Calculus on Manifolds - Tangent space on a boundary point of a manifold

I am an undergraduate student who is studying Spivak's calculus on manifolds. I have several questions in the pages 119 and 120 of the book, which are about the tangent space at a boundary point of a ...
1answer
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### Show that the intersection of two objects is a manifold with boundary

The question is to find $a$ for which the intersection of the solid hyperboloid $x^2+y^2-z^2\leq a$ with $x^2+y^2+z^2 = 1$ is a manifold with boundary. My attempt: Let $I$ be the intersection. ...
1answer
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### What is the ring and does $\phi^*$ preserves module operations

I have posted this problem recently and I think that as it was too long, not many people took interest in it. So I am reposting it also I am giving you an earlier link :I had a problem in manifold ...