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.

Filter by
Sorted by
Tagged with
2
votes
2answers
17 views

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 ...
0
votes
0answers
21 views

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 ...
1
vote
2answers
53 views

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 ...
6
votes
0answers
35 views

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 ...
1
vote
1answer
58 views

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 ...
5
votes
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 ...
1
vote
0answers
13 views

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 ...
1
vote
0answers
24 views

Tangent vectors at boundary points

It is well known that we can define the tangent space of a manifold $M$ at a point $p\in M$ as the set of speeds, at time $0$, of curves $\alpha : (-\varepsilon, \varepsilon) \to M$ such that $\alpha(...
0
votes
0answers
35 views

Counting closed orbits on a compact surface topologically equivalent to a sphere

Previously I asked a question about a surface formed by gluing together pieces of spheres, or, contracting overlayed copies of spheres, or configuring $4$ twice pointed spheres inside a region of ...
1
vote
1answer
30 views

Does there exist a compact smooth embedding submanifold $N\subset M$ with or without boundary such that $N\supset A$?

Let $M$ be a smooth manifold with or without boundary and $A$ a compact subset of $M$, does there exist a compact smooth embedding submanifold $N\subset M$ with or without boundary such that $N\...
1
vote
1answer
34 views

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 ...
0
votes
1answer
19 views

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?
1
vote
0answers
32 views

Why do Guillemin and Pollack avoid the term “submanifold with boundary”?

On p. 60 of the book "Differential Topology" by named authors they state: Theorem. Let $f$ be a smooth map of a manifold $X$ with boundary onto a boundaryless manifold $Y$, and suppose that both $f:...
2
votes
0answers
20 views

Extending a triangulation from a manifold boundary to the interior

Let $M$ be a (second-countable topological) compact connected manifold-with-boundary. Suppose $\partial M$ has a triangulation. Does there exist a triangulation of $M$ which extends the triangulation ...
2
votes
1answer
31 views

Does Proposition 5.18 also hold if $N$ is changed to a smooth manifold with boundary?

Does Proposition 5.18 also hold if $N$ is changed to a smooth manifold with boundary?
0
votes
1answer
16 views

What's the meaning of projection with kernel $\Bbb Rv$?

In Lemma 6.13, it says "For any $v\in\Bbb R^N\setminus \Bbb R^{N-1}$, let $\pi_v:\Bbb R^N\to \Bbb R^{N-1}$ be the projection with kernel $\Bbb Rv$". What's the meaning of projection with kernel $\Bbb ...
1
vote
2answers
63 views

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?
0
votes
0answers
16 views

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 ...
1
vote
1answer
18 views

Deformation retraction onto an open subset of manifold boundary

I'd like to prove the following result, perhaps with additional assumption if needed -- I don't know whether the claim holds. Let $M$ be a compact connected manifold with boundary $\partial M$. Let $U'...
0
votes
0answers
13 views

In- and out-boundaries of the unit interval

I am trying to understand orientation of one-point manifolds in order to figure out in- and out-boundaries of the unit interval. So far I have understood that an orientation of a zero-manifold is an ...
0
votes
0answers
11 views

What's the meaning of “critical value of $F|_{\overline{V_a}}$” and “critical value of $F_c$”?

In Lee's book "Introduction to Smooth Manifolds", "critical value" is defined for smooth maps between smooth manifolds with or without boundary. But in the Proof of Sard's Theorem, the author doesn't ...
2
votes
0answers
34 views

Gauss-Bonnet theorem for vector bundles on manifolds with boundary

I hope this question is not a duplicity, but I really failed to find a good reference for it. I am wondering whether there is a generalization of the Gauss-Bonnet theorem to real vector bundles on a ...
3
votes
1answer
74 views

Smooth map between smooth manifold and boundary of manifold

This is my first question here so I'm asking for your understanding. I also apologize in advance for my English. I'm beginner on smooth manifolds topics and I can't solve the following problem. Let ...
0
votes
0answers
15 views

Do we have $f(M)\subset \operatorname{Int}N$?

Let $M$ be a connected smooth manifold and $N$ a smooth manifold with boundary. If $f: M\to N$ is a smooth map of constant rank and there exists $p\in M$ such that $f(p)\in \operatorname{Int}N$, do we ...
1
vote
1answer
25 views

Can $dF_p:T_pM\to T_{F(p)}N$ be surjective?

Suppose $M$ is a smooth manifold and $N$ a smooth manifold with boundary, $F:M\to N$ a smooth map, if $F(p)\in \partial N$, can $dF_p:T_pM\to T_{F(p)}N$ be surjective?
2
votes
0answers
32 views

Are integral curves on a connected manifold always path-connected?

I am working out some math involving integral curves of a gradient field on a smooth, connected manifold with boundary. Let's assume the function that provides the gradient field is smooth or even ...
4
votes
0answers
129 views

Special Riemannian metric on the product

Let $M^2$ be an orientable surface with boundary endowed with a Riemannian metric $g$. We know that if we put in the manifold $M \times \mathbb{R}$ the product metric $\overline{g} = g + dt^2$, then ...
0
votes
0answers
29 views

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)....
1
vote
0answers
32 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 $\...
5
votes
2answers
117 views

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, $...
0
votes
0answers
45 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 ...
1
vote
1answer
51 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 ...
3
votes
1answer
73 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 ...
5
votes
0answers
98 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 ...
0
votes
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' ...
0
votes
1answer
29 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\...
0
votes
0answers
35 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 ...
0
votes
0answers
42 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 , ...
4
votes
0answers
106 views

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 ...
0
votes
1answer
69 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. ...
1
vote
1answer
33 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 \...
2
votes
2answers
97 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 ...
1
vote
0answers
101 views

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 $M$ ...
2
votes
1answer
54 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 ...
0
votes
0answers
26 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 ...
0
votes
0answers
49 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), ...
1
vote
0answers
46 views

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 ...
1
vote
1answer
57 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 ...
0
votes
1answer
62 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{...
0
votes
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 ...