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.
542
questions
1
vote
1
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82
views
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,...
0
votes
0
answers
34
views
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 $F$ ...
- 4,275
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0
answers
25
views
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{...
- 4,708
0
votes
0
answers
64
views
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 ...
- 65
1
vote
0
answers
40
views
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 ...
- 4,480
5
votes
2
answers
156
views
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 ...
- 4,480
3
votes
1
answer
60
views
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\...
- 4,708
2
votes
0
answers
74
views
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 ...
- 1,381
1
vote
1
answer
51
views
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 |...
- 373
2
votes
2
answers
81
views
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) ...
- 713
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 \...
- 23
2
votes
1
answer
82
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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) = ...
- 2,302
0
votes
0
answers
97
views
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 ...
- 707
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votes
0
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24
views
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 ...
- 1,942
2
votes
1
answer
63
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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}=\{(...
0
votes
0
answers
41
views
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 ...
- 707
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 $(u,v)$...
- 120
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0
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13
views
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_{\...
- 316
4
votes
1
answer
75
views
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 ...
- 43
1
vote
0
answers
47
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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 ...
- 31
4
votes
2
answers
111
views
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 ...
- 805
0
votes
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 ...
- 93
2
votes
1
answer
86
views
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 ...
- 4,323
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 ...
- 707
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?
- 2,073
3
votes
0
answers
46
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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 ...
- 1,895
1
vote
1
answer
56
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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 ...
- 707
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 ...
- 183
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 ...
- 31
1
vote
1
answer
56
views
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 ...
- 326
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 ...
- 23
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 ...
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 \...
- 5,145
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 $\...
- 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
...
- 707
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 ...
- 424
0
votes
1
answer
75
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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 ...
- 681
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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 \...
- 183
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0
answers
78
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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} \...
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 ...
- 719
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 ...
0
votes
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 ...
- 93
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\...
- 313
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 ...
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 ...
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 ...
- 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|| < ...
- 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 $K$ ...
- 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 ...
- 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 ...
- 6,117