# 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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### Manifolds with Boundary and Maximal Atlas

I was reading Tu's An Introduction to Manifolds and learned about the notion of manifolds with boundary. But there was a point which was not clear to me. Here are the definitions(I will use the word ...
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### What's the meaning of being expressible as a convergent power series in a neighborhood of each point?

The following pictures are from Lee's "Introduction to Smooth Manifolds". What's the meaning of being expressible as a convergent power series in a neighborhood of each point? However, I only know ...
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### Constant Rank Theorem for Manifolds with Boundary

I'm trying to answer problem 4-3 from Lee's Introduction to Smooth Manifolds, 2nd edition. The problem says: Formulate and prove a version of the rank theorem for a map of constant rank whose ...
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### if $f: X \rightarrow Y$ is a diffeomorphism of manifolds with boundary, then $\partial f$ maps $\partial X$ diffeomorphically onto $\partial Y$.

Prove that if $f: X \rightarrow Y$ is a diffeomorphism of manifolds with boundary, then $\partial f$ maps $\partial X$ diffeomorphically onto $\partial Y$. Could anyone give me a hint for the proof ...
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### The boundary is disjoint from the interior in 2d manifold

I'm self-studying on Lee's Introduction to Topological Manifolds, and I'm stuck on a problem (8-6). I'm missing something really basic, I fear. I must prove that the set of the boundary points of a 2-...
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### No where vanishing exact $1$-form on compact manifold.

I found several answers on the following question : Does there exists a no where vanishing exact $1$-form on a compact manifold without boundary? All answer says that certainly not. But I cannot ...
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### Orientable manifold $M$ ,then $\partial M$ is orientable

Let $M$ a topological manifold of dimension $n$ with boundary $\partial M$. We define $M$ to be orientable if $M- \partial M$ is orientable. Here when I say orientable, I mean there is a locally ...
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### Dirichlet principle on compact manifolds with boundary

How is it the Dirichlet's principle on compact manifolds with boundary $\partial M= \mbox{manifold boundary}$?. I've just found the Dirichlet's principle on domains $U \subset \mathbb{R}^{n}$, i.e, $U$...
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### Given a family of 2D curves, find a 3D manifold whose geodesics project to the plane curves

Below is an image of the family of 2D curves for reference. The image is arbitrary. Still trying to formulate a concise question. I know that the geodesics for flat Euclidean Space are straight lines....
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### Geodesic curvature change under conformal metrics

Suppose that $\sigma_0$ is a fixed metric on a compact riemannian 2-manifold $M$ with boundary $\partial M$. Let $\sigma=\rho \sigma_{0}$, where $\rho=e^{2\varphi}$ with $\varphi \in C^{\infty}(M)$, ...
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### Is the geodesic diameter on simply connected domains with boundary always realized by points on the boundary?

Assuming standard euclidean metric, the geodesic diameter, of simply connected polygons in the plane is realized by a shortest connection between two vertex points. This result is e.g. referenced in "...
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### Definition of the preimage orientation

Guillemin and Pollack give quite confusing (at least for me) definition of the preimage orientation (see below). I don't understand the part starting from the last display. Namely: How exactly does ...
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### Constructing boundary charts for the n-dim closed balls

This is an exercise from John Lee's book. I am having extreme difficulty with constructing explicit boundary charts for $M$. I have no idea at all how to explicitly express the boundary charts for $M$....
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### Boundary of Seifert Fibered Space is a $T^2$ or $K^2$

I'm getting my feet wet with Seifert Fibered Spaces in Hatcher's 3-manifold papers. Elsewhere, it is said that this follows easily from the definition. I am not seeing it. I think we would need to ...
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### Defining an open submanifold with boundary - John Lee book's proposition and exercise

Above is from p.13 of John Lee's Introduction to Smooth Manifolds. I am curious if this proposition also holds when $M$ is a topological manifold with boundary, thus making it into a smooth manifold ...
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### Product of manifolds with or without boundary

Let $M_1, M_2, ... M_k$ be smooth manifolds with or without boundary such that at most one of these has nonempty boundary, say $M_a$ has nonempty boundary where $a$ is in $\{1,2, ..., k\}$. Then it ...