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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Connectedness of manifolds lying in a cylinder with a hole

Problem setting: We define a cylinder with a hole \begin{equation} A_{\delta} = \{(x',x_N) \in \mathbb{R}^{N-1} \times \mathbb{R} \mid \delta \leq |x'| \leq 1 \} \end{equation} for some $\delta \in (0,...
IgotYourPoint's user avatar
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John Lee's ISM Problem 5-23 (smooth structure of regular level sets for manifolds with boundary)

The problem statement: Suppose $M$ is a smooth manifold with boundary, $N$ is a smooth manifold, and $F: M \to N$ is a smooth map. Let $S = F^{-1}(c)$, where $c \in N$ is a regular value for both $F$ ...
Tob Ernack's user avatar
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Is every smooth atlas for a manifold with boundary contained in a unique maximal smooth atlas?

I'm wondering if the following proposition (from Lee's Introduction to Smooth Manifolds) also applies to smooth manifolds with boundary: Definition: a map from an arbitrary subset $A\subseteq \mathbb{...
Sam's user avatar
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Converting a differential form to a measure

so today I was looking at the Generalized Stokes' Theorem: \begin{align} \intop_{\Omega} d\omega=\intop_{\partial\Omega}\omega\ \ , \end{align} where $\Omega$ is some region, and $\omega$ is a ...
Sora8DTL's user avatar
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What is the difference between a boundary *existing* (or rather, *not* existing) and a boundary *being empty*?

I'm confused about the concept of a "boundary" in topology – specifically, in studying manifolds. What is the difference between a boundary existing (or rather, not existing) and a boundary ...
The Pointer's user avatar
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5 votes
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The boundary of any manifold that is itself a boundary of a higher-dimensional manifold must always be empty?

I have seen it asserted several times that it is well-known that the boundary of any manifold that is itself a boundary of a higher-dimensional manifold must always be empty. Yet, despite it ...
The Pointer's user avatar
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3 votes
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Let $f:\mathbb{R}^m\to\mathbb{R}^n$ be continuous and $M\subseteq \mathbb{R}^m$ be an $m$-manifold. Is the graph of $f$ an $m$-manifold?

Below $M^\circ$ and $\partial M$ denote the manifold-interior and manifold boundary of the topological manifold $M$ respectively. Result (?): $f:\mathbb{R}^m\to\mathbb{R}^n$ be continuous and $M\...
Sam's user avatar
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2 votes
0 answers
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Proof verification: $dF_p$ nonsingular, then $F(p)\in \textrm{Int}N$

I am trying to solve Problem 4-2 on Lee's Introduction to Smooth Manifold. The problem states the following: Suppose $M$ is a smooth manifold (without boundary), $N$ is a smooth manifold with ...
Kaira's user avatar
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Why is $M_{f\geq 0}$ a manifold with boundary?

I'm studying Morse theory and I found this fact: Let be $f:M\rightarrow \mathbb{R} $ a Morse function on a m-manifold $M$. Suppose $0$ is not a critical value of $f$. Then $M_{f\geq 0}=\lbrace p\in M |...
3435's user avatar
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Issue about the definition on orientation on manifolds

In my class, orientation on a $C^\infty$ manifold is defined as follows: Let $\mathcal{O}_0$ be the canonical orientation on $\mathbb{R}^m$ (i.e. the equivalence class (with respect to orientation) ...
Luigi Traino's user avatar
1 vote
0 answers
38 views

Intersection number of submanifolds with boundary

Let $N$ be an $n$-dimensional compact oriented manifold without boundary, and let $X$, $Y$ be compact oriented submanifolds of complementary dimensions $k$ and $n - k$, with boundary but such that $X \...
NPG's user avatar
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2 votes
1 answer
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Cutting a disk with piecewise analytic curves

Let $B \subset \mathbb{C}$ be an open set and assume that the boundary of $B$ consists of finitely many piecewise analytic curves. Suppose $z_0 \in \partial B$ and assume the following: $U(z_0 , r) = ...
porridgemathematics's user avatar
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Boundary case of adapted orthonormal frame exercise in Professor Lee's Introduction to Riemannian Manifolds (IRM)

In the post Adapted Orthonormal frame on Riemann manifold, the poster offered the outline of a proof for IRM Proposition 2.14. I can fill in the details to make that proof work when the embedded ...
Jeff Rubin's user avatar
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tangent vector of a disk embedded in R^3

Consider in $\mathbb{R}^3$ the plane $N: x+y+z=0$ and the sphere $S: (x-a)^2+(y-b)^2+(z-c)=r$, where $r$ is fixed. The intersection $N\cap S$ is the disk embedded in $\mathbb{R}^3$. I would like to ...
Smilia's user avatar
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Let $M=(\mathbb{S}^2\times [0,1])/\sim$, where $(p,0)\sim (-p, 1)$. Prove that M is a compact, differentiable manifold. Is it orientable?

I think I don´t have trouble with the first part, because for the antipodal map $f:\mathbb{S}^2\rightarrow \mathbb{S}^2, \ \ p \mapsto -p$, and any differentiable atlas for the sphere $\mathcal{A}=\{(...
Nestor Anguiano's user avatar
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Finishing proof concerning manifold boundary

I am working on a proof that given some closed $Y\subseteq X$, where $X$ is a manifold with empty (manifold) boundary and $Y$ is a manifold of the same dimension as $X$ and $X\setminus Y$ is not ...
moboDawn_φ's user avatar
3 votes
0 answers
69 views

Gluing two disks to get an exotic sphere

I have found interest in the subject of exotic spheres. Particularly, the 7-sphere. To create such a sphere, you glue two copies $\mathbb{D}^4\times \mathbb{S}^3$ along the boundary identifying $(u,v)$...
Bessel's user avatar
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A Hilbert space which involves the boundary norm and interior norm?

Let $\Omega\subset \mathbb{R}^n$ be a bound domain with enough smooth boundary, does there exist a Hilbert space $H$ on $\Omega$ such that the norm is given by $$||f||^2=\int_{\Omega}|f|^2dx+\int_{\...
qinxs's user avatar
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4 votes
1 answer
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Confusion about notation for orientation of boundary

I am reading "All the math you missed" by Thomas A. Garrity, second edition. In the chapter on Divergence Theorem there is the following fragment: Here the derivative $\frac{df}{dx}$ is ...
deespodete's user avatar
1 vote
0 answers
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Is the boundary of a set defined using a smooth function also smooth?

Let $f:B_\delta(y)\rightarrow \mathbb{R}^n$ be a smooth function for some $\delta>0$, $y\in\mathbb{R}^m$. Let us define the set $G:= f(B_\delta(y)) + \mathbb{R}^n_\geq$ (using the Minkowski sum). I ...
Ina's user avatar
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2 answers
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Submanifold of ball is entire ball

I've seen the following used in 2 and 3 dimensions; I don't know how to prove it, and am wondering if it's true in all dimensions: If $M$ is an $n$-dimensional smooth compact manifold embedded in the ...
Hempelicious's user avatar
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1 answer
92 views

Boundary Orientation

Edits to the original post are in bold. I've been going through @Ted Shifrin's lectures on Stokes's Theorem, and I had a question relating to his choice of orientation of the tangent space as it comes ...
RHyp's user avatar
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2 votes
1 answer
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Intuitive understanding of inward and outward vector definitions

Let $M$ be a smooth manifold with boundary and $p \in \partial M$. In Lee's Introduction to Smooth Manifolds he gives the following definition: If $p \in \partial M$, a vector $v \in T_pM \setminus ...
CBBAM's user avatar
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3 votes
2 answers
142 views

Uniqueness of smooth structures on submanifolds with boundary

In Professor Lee's Introduction to Smooth Manifolds (Second Edition), he states and proves Theorem 5.31, which guarantees that the smooth structure on an embedded or immersed submanifold of a smooth ...
Jeff Rubin's user avatar
2 votes
1 answer
74 views

Results that hold on smooth manifolds, but not on smooth manifolds with boundary? [closed]

The only one I can think of is that the product of two smooth manifolds is a smooth manifold. This isn’t the case for smooth manifolds with boundary. Are there other results like this?
Chris's user avatar
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3 votes
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Closed Hilbert half-space

Suppose $H$ is the seperable infinite-dimensional real Hilbert space and $f$ is a continuous linear functional on it. Is the closed half-space $H_{f \ge 0} = \{ x \in H | f(x) \ge 0 \}$ homeomorphic ...
Zerox's user avatar
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Proof that $M(a)$ is a regular domain in $\mathrm{Int} M$ in Prof. Lee's Introduction to Smooth Manifolds errata for Theorem 9.26

In Professor John Lee's Introduction to Smooth Manifolds, Second Edition, Theorem 9.26 has, as its hypothesis, a smooth manifold with nonempty boundary $M$. In his errata for the book, there is an ...
Jeff Rubin's user avatar
1 vote
2 answers
129 views

On the homology of manifolds with boundary [closed]

Suppose that $M$ is a smooth manifold with a boundary. Let $\mathrm{int}(M):=M\setminus\partial M$ be its interior. Is there a relation between the (relative or absolute) homology of $M$ and the ...
brick's user avatar
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2 votes
0 answers
51 views

Riemannian manifold with almost nonnegative sectional curvature

A Riemannian manifold has almost nonnegative sectional curvature if it admits a sequence of Riemannian metric $g_{i}$ such that $sec(g_{i})\geq (-1/i)$ and $D(g_{i})\leq 1$ where $sec(g_{i})$ is the ...
Math's user avatar
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1 vote
1 answer
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Extending an embedding to the boundary

Let $f:int(M^k) \rightarrow N^n$, ($k\leq n$) be an embedding of the interior of a $k$-manifold $M$ into a closed (i.e. compact and without boundary) $n$-manifold $N$. Can I always extend this map to ...
Léo Mousseau's user avatar
1 vote
1 answer
101 views

Must the topological boundary of an embedded manifold be a set of Lebesgue measure zero? Why is this question closed?

Let $X\subset\mathbb{R}^n$ be a bounded connected $\mathcal{C}^1$ embedded k dimensional manifold (k<n ); i.e., for each x∈X , there exists an open (in the subspace topology) neighborhood $U_x$ of ...
John Johnson's user avatar
0 votes
1 answer
89 views

Smooth maps between manifolds with boundary: restricting the codomain to a submanifold with boundary

Let $M,N,S$ be smooth manifolds with boundary such that $S\subset M$ is an embedded submanifold. Problem 9-13 from J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., asks in particular to prove the ...
Elías Guisado Villalgordo's user avatar
2 votes
1 answer
30 views

Orthogonal geodesic chords

Let $(M^n,g)$ be a connected Riemannian manifold with boundary, $n \geq 2$. A chord of $M$ is a non constant geodesic $c : [a,b] \to M$ with endpoints $c(a), c(b) \in \partial M$ but $c(t) \in \...
Eduardo Longa's user avatar
2 votes
0 answers
27 views

A manifold whose boundary is a connected sum

Let $X$ be a manifold of dimension $n\geq 3$ whose boundary $\partial X$ is a disjoint union $Y_1 \cup Y_2$ of $(n-1)$-manifolds $Y_1,Y_2$. If we choose $y_1\in Y_1$, $y_2\in Y_2$, and an embedding $\...
blancket's user avatar
  • 2,076
2 votes
1 answer
142 views

Stuck trying to prove Theorem 5.53(b) in Professor Lee's Introduction to Smooth Manifolds

Theorem numbers are those of the second edition. This is the embedded submanifold with boundary version of Theorem 5.29. Its title is "Restricting Maps to Submanifolds with Boundary" and the ...
Jeff Rubin's user avatar
1 vote
0 answers
99 views

When does a covering map between boundaries extend to covering maps between interiors?

Given two smooth three-manifolds $(M,\partial M)$ and $(N,\partial N)$ with smooth boundary, if we know that there is a covering map $\Gamma:\partial M\to\partial N$, is there a simple criterion to ...
Bob Knighton's user avatar
0 votes
1 answer
75 views

Is the boundary of a smooth manifold with boundary the countable union of smooth manifolds?

Let $(\mathcal{M}, \mathcal{O}, \mathcal{A})$ be a smooth manifold with boundary, where $\mathcal{M}$ is a set, $\mathcal{O}$ is a topology and $\mathcal{A}$ is a smooth atlas. Under which conditions ...
Rafael Rojas's user avatar
3 votes
0 answers
114 views

Cohomology of submanifolds

Suppose I have a manifold $M$ and a submanifold or a boundary $N\subset M$. By the natural inclusion $\iota:N\hookrightarrow M$ we can easily see that $$\omega\in\mathrm{H}^k(M) \quad\implies\quad \...
brick's user avatar
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0 answers
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Prove that the disk $\left\{(x, y)\mid x^{2}+y^{2} \leq 1\right\}$ is a manifold with boundary. [duplicate]

A manifold with a boundary is a topological space $(X, T)$ whose open sets have continuous one-to-one maps to open sets in half space. Half space is the region of $\mathbb{R}^{n}$ for which $x^{1} \...
Zeph Rodriquez's user avatar
2 votes
0 answers
74 views

What is a Riemann surface with boundary?

My guess is that $X$ is a Riemann surface-with-boundary if it is a topological 2-manifold-with-boundary such that the transition charts are biholomorphic. Now what does biholomorphism mean for charts ...
Mohith Raju's user avatar
2 votes
0 answers
41 views

A $\pi_1$-neglibility criterion in 4-dimensional manifolds

I'm reading about the h-cobordism theorem in boundary dimension 4. Most of the steps are the same as in the classical statement, but finding whitney disks to homotope the 2- and 3-handles into ...
Timotheus Hauptinius's user avatar
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0 answers
64 views

Complement of a knot complement

Consider a knot K in the 3-sphere, its knot complement C(K) is obtained by removing a tubular neighborhood of K from the 3-sphere. What can I say about the complement of C(K) in the 3-sphere? Is it ...
Spinoro's user avatar
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1 vote
1 answer
99 views

Do we have $\int_{\partial M}f\langle X,N\rangle_g dV_\tilde{g}=0$ if both $f$ and $X$ are compactly supported?

In Euclidean spaces, if a partial integration involves a function with compact support, then we won't have to include the boundary integral because it contributes nothing. More precisely, if $U\...
Wombat's user avatar
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0 votes
0 answers
16 views

Drawing a triangulation

For K, a 2-complex that triangulates the closed disk, B. a and b are the interior vertices; u, v, w boundary vertices; ab, au, uv interior edges; and vw a boundary edge. To draw K such that it ...
Math Learner 24's user avatar
0 votes
0 answers
54 views

2-manifold with boundary l = 2

Can someone help me figure out what a 2-manifold with l=2 boundary would look like? Can it simply be a closed disk with double boundary? I'm new to topological data analysis and I know this is a very ...
Math Learner 24's user avatar
6 votes
1 answer
249 views

Boundary conditions for differential forms

I am trying to understand differential forms on manifolds with boundaries, and I am a bit confused with the boundary conditions. For the following, let $(M,g_M)$ be a smooth Riemannian manifold with ...
brick's user avatar
  • 183
0 votes
1 answer
31 views

Is $A=\{ x \in \mathbb{R}^2 : ||x|| \leq 1 \}$ a 1d or 2d manifold

Suppose the set $A=\{ x \in \mathbb{R}^2 : ||x|| \leq 1 \} $. Can A be a 1d or 2d or n-dimentional manifold ? My thought is that we can write A as $A=A_1\cup A_2 = \{ x \in \mathbb{R}^2 : ||x|| < ...
lebong66's user avatar
  • 437
0 votes
1 answer
77 views

$f(\partial K)\subset\partial f(K)$ and $f(\operatorname{Int}K)\subset\operatorname{Int}f(K)$ for a well-behaved $f:\mathbb R^2\to\mathbb R^2$?

Let $f:\mathbb{R}^2\to\mathbb{R}^2$ be a continuously differentiable bijection with nonzero Jacobian. Let $K\subset\mathbb{R}^2$ be a compact subset. Then $f(K)$ is compact. Does the boundary of $K$ ...
ashpool's user avatar
  • 6,560
1 vote
0 answers
62 views

Numerical PDE: How can I incorporate boundary conditions for the Laplace-Beltrami eigenvalue problem on a manifold?

Problem description: I have a 2D smooth manifold with boundary $M$ embedded in $\mathbb R^3$, discretized by a triangular mesh $T$, and need to find the eigenfunctions of the Laplace-Beltrami operator ...
trisct's user avatar
  • 5,131
2 votes
4 answers
281 views

boundary of a simply connected, compact manifold with boundary

For a simply connected, compact n-manifold with boundary (n > 1), is its boundary connected? It’s obviously false when n = 1, but how to prove or disprove the statement when n > 1? I’m ...
chaohuang's user avatar
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