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
1
vote
0answers
11 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
19 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
24 views

Counting closed orbits on a compact surface topologically equivalent to a sphere but on one that is not everywhere smooth

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
27 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
29 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
15 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
29 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:...
1
vote
1answer
80 views

Manifolds with corners

Here is what I'm experiencing. A text introduces manifolds. Then, manifolds with boundary is introduced. After all, manifolds with corners is introduced. First of all, I checked whether Implicit ...
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?
2
votes
0answers
15 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 ...
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 ...
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 ...
1
vote
1answer
17 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 ...
3
votes
1answer
72 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 ...
2
votes
0answers
90 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 ...
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 ...
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
30 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 ...
7
votes
1answer
444 views

The definition of smooth maps given in Introduction to Smooth manifolds by John M. Lee

I'm currently reading Introduction to Smooth Manifolds by John M. lee. I'm trying to understand his definition of smooth maps $F:A\subseteq M\to N$ given in page 45. Let's start from scratch to have ...
0
votes
0answers
28 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
29 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
115 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, $...
2
votes
1answer
624 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 ...
0
votes
0answers
43 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
45 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 ...
5
votes
0answers
96 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 ...
2
votes
1answer
92 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 ...
6
votes
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 $\...
3
votes
1answer
72 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 ...
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
31 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 ...
2
votes
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 ...
4
votes
0answers
101 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
0answers
40 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 , ...
0
votes
1answer
64 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
30 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 \...
3
votes
2answers
156 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 ...
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
51 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
25 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
43 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
45 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
61 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{...