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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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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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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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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
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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25 views

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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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 ...
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1answer
35 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
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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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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
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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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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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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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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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0answers
32 views

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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0answers
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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
67 views

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
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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1answer
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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
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
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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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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
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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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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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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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0answers
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
50 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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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 ...
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1answer
59 views

Manifold with boundary and topological boundary

Why aren't the topological boundary of a manifold with boundary and the boundary of the manifold the same sets? We defined the boundary of a manifold as follows: $\partial M:= \lbrace p \in M : \...
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Lipschitz and / or piecewise smooth boundary

Let $\Omega_3 = \overline{Q_1 \cup Q_2}$ with $Q_1 = (-1, 3) \times (0,2) \times (-1, 0)$ and $Q_2 = (0,2) \times (-1, 3) \times (0,1)$ Here is what I've done until now: $\Omega_3$ can be imagines as ...
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Image of smooth map lies in interior [duplicate]

Problem 4.2, pg 98, John Lee's smooth manifold: Suppose $M$ is a smooth manifold (without boundary), $N$ is a smooth manifold with boundary, $F:M \rightarrow N$ is smooth. Show that if $p \in M$ is a ...
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0answers
22 views

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 $\...
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0answers
59 views

Boundary of a one-dimensional manifold - choosing an oriented atlas

I am trying to understand how to assign an orientation to the boundary of one-dimensional manifolds using Loring Tu's book on manifolds. This is what I got so far: Let $M$ be an oriented manifold. If ...
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1answer
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When both $U$ and $W$ are open in $\mathbb{H}^k$ and $\mathbb{H}^l$, respectively, then why $U\times W$ cannot be open in $\mathbb{H}^{k+l}$

In the book of Analysis on Manifolds by Munkres, at page 202 question 5, it is asked that Show that if $M$ is a k-manifold without boundary in $\mathbb{R}^m$, and if $N$ is an l-manifold in $\...
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Boundaries of manifolds and algebraic topology

I would like to know how to tackle questions of the following type: Show that $\mathbb{CP}^{2n}$ is not the boundary of any manifold. Another such question would be: Let $\iota: S^1 \to S^3$ be ...