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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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What is an orbifold with corners

Can one have a formal definition of orbifold with corners? note that it is not parallel to the definition of manifold with corners, as a manifold with boundary is already an orbifold(not with boundry)....
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26 views

Embedding a Riemannian manifold with boundary in a closed manifold

Let $M$ be a complete Riemannian manifold of finite volume, with sectional curvatures bounded by some $K>0$ in absolute value. Let $M_{\geq R}$ be the set of points in $M$ with injectivity radius $\...
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2answers
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Classification of surfaces

The Classification Theorem for surfaces says that a compact connected surface $M$ is homeomorphic to $$S^2\# (\#_{g}T^2)\# (\#_{b} D^2)\# (\#_{c} \mathbb{R}P^2),$$ so $g$ is the genus of the surface, $...
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1answer
584 views

Showing that $\bar{\mathbb{B}}^n$ is a manifold with boundary (Lee ITM Probelm 3-4)

"Show that every closed ball in $\mathbb{R}^n$ is an $n$-dimensional manifold with boundary, as is the complement of every open ball. Assuming the theorem on the invariance of the boundary, show that ...
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39 views

Uniqueness of integral curves starting on boundary?

I'm only aware of ODE uniqueness on open subsets of $R^n$, which is useful when we want to prove uniqueness of integral curves on a smooth manifold. Suppose we have an interal curve $f:[0,\delta]\to ...
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1answer
41 views

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 ...
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90 views

Is a manifold-with-boundary with given interior and non-empty boundary essentially unique?

Let $M$ be a compact connected manifold-with-boundary such that $\circ M \neq \emptyset$, where $\circ M$ is the boundary of $M$. Let $N$ be a compact connected manifold-with-boundary such that $\circ ...
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1answer
87 views

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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1answer
1k views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves $\...
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1answer
62 views

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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1answer
20 views

Is there Method to visualize the object $Disc \times Disc$?

For me its clear how to build up the object $S^1 \times S^1$ , who is our old friend torus, but the product of 'interior'of these objetcs doesn't look clear to me how to build up I tried 'forget' ...
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1answer
28 views

Question about manifolds with boundary

Prove that if $f:X\to Y$ is a diffeomorphism of manifolds with boundary, then $f$ maps $\partial X$ to $\partial Y$ diffeomorphically. Answer: Let $U\subset H^k$ be an open subset and let $\phi:U\...
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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 ...
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2answers
95 views

Searching for a connected neighborhood of fixed radius for every point of a submanifold

Let $\Gamma=\gamma(\partial D)\subset \mathbb R^3$ be an image of $C^1$-curve, where $D$ is an open ball in $\mathbb R^2$. My question is whether it is possible to find a fixed $\epsilon >0$ such ...
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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 ...
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35 views

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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1answer
36 views

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. ...
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1answer
21 views

A question about diffeomorphism between manifolds with boundary

I don't understand a couple of things about the following proof: Statement: Suppose $f:X\rightarrow Y$ is a diffeomorphism of manifolds with boundary. Show that $\partial f:\partial X\rightarrow \...
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2answers
150 views

$\mathbb S^2$ or $\mathbb RP^2$ on boundary of a 3-manifold

Let $M$ be a 3-manifold with boundary $\partial M$. Suppose $\partial M$ contains a sphere or a projective plane, which is contractable in $M$. Show that $M$ is also contractable. The above statement ...
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0answers
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Integration of forms over manifolds

Spivak start the section 'Stokes theorem on manifolds' of his book 'Calculus on manifolds' with the following definition: If $\omega$ is a $p$-form on a $k$-dimensional manifold with boundary $M$ ...
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1answer
41 views

Question on the definition of tangent space at boundary points

I'm going thought Spivak's 'Calculus On Manifolds' and I am bit confused with certain things in the definition of outward normal unit vector. For the notation. With $v_{p}$ Spivak denotes the tangent ...
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23 views

Manifold Optimization in Human understandable language

I am coming across a project that involves with "manifold optimization". I don't know what that is. After Googling for a bit, my understanding is that we use a low dimensional surface to approximate ...
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39 views

Several questions in Lee's <Introduction of smooth manifolds>, chapter 1,2

(1), In the proof of Lemma 1.10: Every topological manifold has a countable basis of precompact coordinate balls, what is the function of $B_{r^{'}}(x)$? Why do we need it? Proof of Lemma 1.10 (2), ...
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Deduce the curvature of the manifold by probing it with geodesics

Suppose in order to probe the curvature of a manifold in $\Bbb R^2$, you decide to shoot off geodesics from the four corners of the square. Consider a set of geodesics, $K,$ each symmetrical about ...
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1answer
56 views

Why does any connected closed $m$-manifold that can be embedded in $E^{m+1}$ bounds a compact connected $(m+1)$-manifold?

I am reading Sheila Carter and S.A. Robertson's paper Relations Between a Manifold and its Focal Set. In this paper, they use the following facts: Any closed $m$-manifold $M$ that can be embedded ...
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1answer
60 views

Existence of a special homeomorphism on $\mathbb{T}^2$.

Let $A, B$ be closed topological subspaces of $\mathbb{T}^2$. Suppose that $A$ and $B$ are homeomorphic as topological spaces. My Question: Is it possible to construct a homeomorphism $h: \mathbb{...
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0answers
30 views

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 ...
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2answers
62 views

Retract of noncompact surface to its boundary?

Suppose $M$ is a connected, noncompact 2-manifold, and its boundary $\partial M$ is a circle. What's the simplest way to show there is a retraction $r: M\rightarrow \partial M$? Here are some ...
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0answers
254 views

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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32 views

The orientation induced on the boundary of a manifold.

I just learned about the notion of orientability of a manifold which is difficult and abstract for me. If we consider all basis of a vector space, the matrix that transforms one basis in another basis ...
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Homotopy type of smooth manifold with boundary

It seems very likely to me that a $n$-dimensional smooth manifold with boundary has the homotopy type of a $(n-1)$-dimensional CW complex. Is that true? Does the manifold need to be compact? What ...
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Are the two definitions of that a map $F:A\to N$ is smooth on $A$ equivalent?

In the above pictures, we can see two definitions of that a map $F:A\to N$ is smooth on $A$, are they equivalent?
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1answer
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Is this conjecture about the boundary of a surface correct?

I came up with an intuitive conjecture about boundaries of surfaces based on the idea that at a boundary point we can wrap a string across the edge, and the two halves of the string (on opposite sides ...
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1answer
188 views

rudin's principles of mathematical analysis 10.31

I'm working on rudin's principles of mathematical analysis(3rd edition). There is problem too complex for me to solve.Please help me. The problem is on p270-271 of text. "Let T be 1-1 mapping of $Q^n$...
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1answer
74 views

$W$ spin implies $\partial W$ spin

Let $M$ be a compact orientable manifold with the first two Stieffel-Whitney numbers equal to zero (this is my definition of SPIN manifold). Let $B$ be the boundary of $M$; I want to show that $B$ is ...
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215 views

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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1answer
57 views

Proof of that if $M$ is a $n$-manifold oriented, then $\partial M$ is a $(n-1)$-manifold oriented

I'm trying understand the result described on the title of the topic by the book "Differential forms and Applications" by do Carmo. The proof given by author can be found below $\textbf{Proposition ...
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1answer
34 views

A difficulty in understanding a part of a paragraph in Guillemin & Pollack p.60

I do not understand the highlighted part of the paragraph given below: Could anyone explain it for me please?
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1answer
24 views

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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1answer
41 views

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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147 views

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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3answers
129 views

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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1answer
77 views

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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30 views

preserves or reverse orientation of sphere surface

Let $\varphi: (0, \infty) \times (0, \pi) \times (0, 2 \pi) \to \mathbb{R}^3 \setminus \{(x,y,z) \in \mathbb{R}^3| y=0, x \geq 0 \}$ $$(p,\phi,\theta) \mapsto (p \sin \phi \cos \theta, p \sin \phi \...
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1answer
742 views

Can you hear the pins fall from bowling game scores?

Let $\mathbb T=\{1,\dotsc,10\}$ represent the ten pins in a standard game of bowling. Given two sets of pins $T\subseteq S\subseteq \mathbb T$, let's write $p_{S\to T}$ to represent the conditional ...
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71 views

Boundary of the image is the image of the boundary.

Suppose that $f:O\to \mathbb R^n$ is a smooth change of variables on the open subset O of $\mathbb R^n$. Let D be an open subset of $\mathbb R^n$ such that $D\cup \partial D$ is contained in O. Show ...
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1answer
51 views

The riemannian metric of a neighborhood of the boundary of a compact manifold

Let $M$ be a compact riemannian manifold with boundary $\partial M$. We have that $\partial M$ is also compact and I was able to show that there is some $a>0$ such that the map $F:[0,a]\times \...
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1answer
51 views

Existence of boundary cylindrical neighborhood for a compact manifold

Let $M$ be a compact riemannian manifold with boundary $\partial M\neq \varnothing$. I would like to show that there is some neighborhood $U$ of $\partial M$ which is diffeomorphic to $[0,a)\times \...
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0answers
35 views

Reference request Eigenspace decomposition Hodge Laplacian on forms on manifolds with boundary

I know that on a connected, compact, oriented Riemannian manifold without boundary the Hodge Laplacian $\Delta_k=(d+\delta)^2$ (acting on $k$-forms) admits an orthogonal eigenspace decomomposition of $...
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1answer
44 views

Prove there exists a outward unit normal field on the boundary this manifold

Let $M$ be a compact subset of $\mathbb{R}^3 $ with the standard orientation $\mu =[e_1,e_2,e_3] $ and let $S = \partial{M}$ is its smooth boundary with the induced orientation from $M$. Prove there ...