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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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1answer
153 views

Let $M$ be a smooth manifold and let $N$ be a manifold with boundary. If $dF_{p}$ is an isomorphism, then $F\left(p\right)\in\mbox{int}M $.

Let $M$ be a smooth manifold and let $N$ be a manifold with boundary, both with the same dimension $n$. If $dF_{p}$ is an isomorphism, then $F\left(p\right)\in\mbox{int}N $. I am trying to prove ...
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2answers
5k views

The boundary of an $n$-manifold is an $n-1$-manifold

The following problem is from the book "Introduction to topological manifolds". Suppose $M$ is an $n$-dimensional manifold with boundary. Show that the boundary of $M$ is an $(n-1)$-dimensional ...
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1answer
389 views

Proving that a regular value of a smooth function isn't in the boundary of the counter-domain

Suppose $X$ is a manifold without boundary and $Y$ is a manifold. Suppose there is a smooth function $f: X \rightarrow Y$ and we are given a $y \in Y$ such that $y$ is a regular value of $f$ and $f^{...
2
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1answer
288 views

Boundary of a topological manifold invariant?

Let $M=(X,\tau)$ be a topological manifold with boundary. One can proof that the interior $Int(M)$ and boundary $\partial M$ of the manifold are distinct sets. I was wondering if someone knows a ...
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0answers
178 views

Manifolds with boundaries and partitions of unity

How do I 1 show that $M=[0,3]\subset \mathbb{R}$ is a manifold with boundary? 2 find a $C^2$ partition of unity for the open cover $M=[0,2)\cup(1,3]$? 3 show that $\omega=(x-2)dx$ is/is not an ...
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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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2answers
799 views

Any N dimensional manifold as a boundary of some N+1 dimensional manifold?

Is this statement true: Question: Can any N dimensional manifold be realized as a boundary of some N+1 dimensional manifold? If so/not, how to prove/disprove it? I read a TQFT paper from Edward ...
6
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1answer
400 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 ...
11
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1answer
458 views

Is the square pyramid a manifold with corners?

An n-manifold with corners is topologically an n-manifold with boundary, but with a smooth structure that makes it locally diffeomorphic to $[0,\infty)^n$ instead of $[0,\infty) \times \mathbb{R}^{n-1}...
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1answer
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(Whitney) Extension Lemma for smooth maps

I am currently reading Lee's book "Introduction to Smooth Manifolds (2nd edition)". Corollary 6.27 in that book states that a smooth map $f\colon A \rightarrow M$ where $M$ is a smooth manifold ...
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1answer
585 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 ...
6
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1answer
302 views

Isometric immersions between manifolds with boundary are locally distance preserving?

Let $M$ be a compact, connected, oriented smooth Riemannian manifold with non-empty boudary. Let $f:M \to M$ be a smooth orientation preserving isometric immersion. Is it true that $f$ is locally ...
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1answer
146 views

Smoothness of the boundary is the only obstruction for being a submanifold with boundary?

Let $M$ be a smooth manifold, and let $S$ be an open smooth submanifold of $N$. Assume the topological boundary of $S$, $\partial S :=\bar S \setminus S$ is a smooth submanifold of codimension 1 in $...
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1answer
1k views

Inverse Function Theorem for Manifolds with Boundary as the Domain

In Lee's Smooth Manifolds, it is written that the inverse function theorem can fail for manifolds with boundary. As hint, it is given the inclusion of half space into euclidean space $\iota:\mathbb{H}^...
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1answer
285 views

Are there Kirby diagrams for manifolds with boundaries?

There are Kirby diagrams for 3- and 4-manifolds which consist of framed links corresponding to 1- and 2-handles attached to a single 0-handle. Any such diagram will give a unique closed manifold since ...
4
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2answers
150 views

Is the closed hemisphere diffeomorphic to the closed disk?

This might be silly but it puzzles me. Let $M$ be the closed upper hemisphere of $\mathbb{S}^2$. This is a manifold with boundary. Is it diffeomorphic to a closed disk in $\mathbb{R}^2$? These ...
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3answers
256 views

Are topological manifolds with boundary metrizable?

It is standard that topological manifolds (without boundary) are metrizable. Is the same true for manifolds with boundary?. I'm using the following definition: Let $\mathbb{R}^n_{x_n\ge 0}=\{x\in \...
2
votes
3answers
556 views

Manifold Boundary versus Topological Boundary.

Let $M$ a $n$-manifold whit boundary, i.e., for each $x\in M$, there exist $U_x\subseteq M$ open in the topology of $M$ such that $U_x$ is homeomorphic to $\mathbb{R}^n$ or homeomorphic to $\mathbb{H}^...
1
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1answer
145 views

Is a map with invertible differential that maps boundary to boundary a local diffeomorphism?

Let $M,N$ be smooth manifolds with boundary (of the same dimension). Let $f:M \to N$ be a smooth map satisfying $(1) \, \,f(\partial M)=\partial N,f(\operatorname{Int} M)=\operatorname{Int} N$. $(...
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1answer
84 views

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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0answers
362 views

Stokes' theorem: Induced orientation on the boundary of a manifold

The Question Let $K = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \geq 1\}$, where $K$ is oriented via the canonical volume form on $\mathbb{R}^3$: $dx \wedge dy \wedge dz$. Let $\mathbb{S}^2$ be ...
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2answers
83 views

Topological boundary of a specific manifold.

Let $\emptyset\neq X\subset\mathbb{R}^n$, $n\ge2$, be an $(n-1)$-dimensional differentiable submanifold, i.e. for every $p\in X$ there is an open $U_p\subset\mathbb{R}^n$ with $p\in U_p$, and a ...
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2answers
207 views

Is every flat manifold with boundary locally isometric to the Euclidean half-space?

Let $M$ be a smooth manifold with boundary, endowed with a smooth Riemannian metric $g$. Suppose $g$ is flat, and let $p \in \partial M$. Is there an open neighbourhood of $p$ which is isometric to ...
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1answer
46 views

Can we choose the anti exterior derivative in a way that respect the behaviour on the boundary?

Let $M$ be an oriented, compact smooth $d$-dimensional manifold with boundary, and let $\omega_t$ be a smooth family of $d$-forms that agree on some open neighbourhood of $\partial M$. (i.e $\...
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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
1k views

The definition of submanifold of a manifold with boundary

What is the exact definition of submanifold of a manifold with boundary? For example, when $H$ is the half space of the plane and S is a cycle which intersects with the origin in the half plane. Then ...
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1answer
155 views

Boundary points of a manifold

I'm reading about Riemannian Geometry and my question is regarding Manifolds with Boundary. I want to show a point of a manifold with boundary is either an interior point or a boundary point, so no ...
0
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1answer
69 views

Does the intrinsic and extrinsic distances coincide on the interior?

Let $M$ be a Riemannian manifold with boundary. Consider the interior of $M$ (which we denote by $M^\circ$). $M^\circ$ is an open submanifold of $M$. Let $d_M$ be the induced distance function (...
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0answers
192 views

Is the boundary of every 2-manifold the disjoint union of circles?

Note that in the title I am being sloppy and should say "Is the boundary of every 2-manifold with boundary the disjoint union of circles?". Also when I say "disjoint union of circles" I mean up to ...