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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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786 views

Why are functions with vanishing normal derivative dense in smooth functions?

Question Let $M$ be a compact Riemannian manifold with piecewise smooth boundary. Why are smooth functions with vanishing normal derivative dense in $C^\infty(M)$ in the $H^1$ norm? Here I define $...
10
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0answers
192 views

Codimension $1$ Embedding into $\mathbb{R}^{n+1}$

I am trying to determine which homotopy types can be realized by $n$-manifolds that have codimension one embeddings into $\mathbb{R}^{n+1}$. Suppose I have $X^n \subset \mathbb{R}^{n+1}$ and an ...
8
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0answers
257 views

Given a family of 2D curves, find a 3D manifold whose geodesics project to the plane curves

Below is an image of the family of 2D curves for reference. The image is arbitrary. Still trying to formulate a concise question. I know that the geodesics for flat Euclidean Space are straight lines....
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89 views

Cartan geometry on manifolds with boundary

I was reading Sharpe's text on Cartan geometry, and I started to wonder: Does the theory change in any significant way if the base manifold for the Cartan principal bundle is allowed to be a smooth ...
6
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0answers
294 views

Integration by parts on manifold with a boundary

Suppose $C$ is a 3-form, and $G$ is a 4-form defined by $G = dC$. Also, $M_{11}$ is an 11-dimensional manifold (without a boundary), $W_{6}$ is a 6-dimensional submanifold of $M_{11}$ and $D_{\epsilon}...
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0answers
90 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 ...
4
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0answers
89 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 ...
4
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0answers
263 views

Induced orientation on manifold

Suppose $M$ is a naturally oriented $n$-manifold embedded in $\mathbb{R}^n$ with non-empty boundary. Give $\partial M$ the induced orientation. The induced orientation defines a unit normal vector ...
4
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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 ...
4
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0answers
742 views

Submanifold with boundary of a manifold with boundary

Let $M$ be a smooth manifold. (1) A subset $S$ of $M$ that with the subspace topology is a topological manifold (with or without boundary), together with a differential structure that makes the ...
3
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0answers
217 views

Constant Rank Theorem for Manifolds with Boundary

I'm trying to answer problem 4-3 from Lee's Introduction to Smooth Manifolds, 2nd edition. The problem says: Formulate and prove a version of the rank theorem for a map of constant rank whose ...
3
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0answers
84 views

Lemma 3 of the Milnor's Topology from the differentiable viewpoint.

Lemma 3 in Milnor's "Topology from the differentiable Viewpoint" (p.12) is stated as below; Let $M$ be a manifold without boundary and let $g:M \to \mathbb{R}$ have 0 as regular value. The set of $...
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0answers
286 views

The boundary of a manifold is a closed subset.

We want to show that the boundary $\partial M$ of an $n-$manifold M is a closed subset of the manifold. We show that its complement $M\setminus\partial M$ is open in $M$. Indeed, each point $x \in M\...
3
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0answers
84 views

Intersection of a minimal surface and a closed ball

Let $u:\Omega\subset\mathbb{R}^2\to\mathbb{R}$ such that $u$ satisfies the differential equation of minimal surfaces on the region $\Omega$. Let $p\in(\Omega\times\{0\})\subset\mathbb{R}^3$ and a ...
3
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0answers
80 views

A $k+1$ Manifold whose boundary is the solution set to the equation $f(\vec{x})=\vec{0}$

Let $f: \mathbb{R}^{n+k} \to \mathbb{R}^n$ be of class $C^r$. Let $f_1, ..., f_n$ be components of $f$. Define $$M=\{\vec{x} | f(\vec{x})=\vec{0}\},$$ $$N=\{\vec{x} | f_1(\vec{x})=0, ..., f_{n-1}(\...
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0answers
33 views

Homotopy type of smooth manifold with boundary

It seems very likely to me that a $n$-dimensional smooth manifold with boundary has the homotopy type of a $(n-1)$-dimensional CW complex. Is that true? Does the manifold need to be compact? What ...
2
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0answers
22 views

Understanding the definition of boundary points of a manifold

Let $M$ be a topological manifold. We call it a $n$-manifold with boundary if for each $x\in M$, there is a chart $(U,\phi)$ at $x$ such that $\phi$ is a homeomorphism from $U$ to an open subset of $\...
2
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0answers
85 views

Boundaries of manifolds and algebraic topology

I would like to know how to tackle questions of the following type: Show that $\mathbb{CP}^{2n}$ is not the boundary of any manifold. Another such question would be: Let $\iota: S^1 \to S^3$ be ...
2
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0answers
42 views

Extending functions on boundary into M as a harmonic function

I am trying to show that if $\varphi \in C^{\infty}(\partial M)$ then there is $\psi \in C^{\infty}(M)$ with $\psi|_{\partial M}=\varphi$ e $\Delta \psi=0$, where $M$ is a compact riemannian manifold ...
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0answers
41 views

Is the geodesic diameter on simply connected domains with boundary always realized by points on the boundary?

Assuming standard euclidean metric, the geodesic diameter, of simply connected polygons in the plane is realized by a shortest connection between two vertex points. This result is e.g. referenced in "...
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0answers
64 views

Show that a given set is a manifold with boundary

given $A:= \{ (x_1,x_2,x_3) \in \mathbb{R} : x_1^2 + x_2^2 + x_3^2 = 18, x_3 \leq 2\}$ show that $A$ is a manifold with boundary and calculate $ \delta A$ where $\delta A$ is the boundary of A. I ...
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0answers
123 views

A question about the boundary points of manifolds.

A topological $n$-manifold is a second countable Hausdorff space such that every point has a neighbourhood which is homeomorphic to an open ball centred at the origin in $\Bbb{R}^n$. The "centred at ...
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0answers
18 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 ...
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0answers
26 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 $\...
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0answers
96 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 $M$ ...
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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 ...
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0answers
32 views

The orientation induced on the boundary of a manifold.

I just learned about the notion of orientability of a manifold which is difficult and abstract for me. If we consider all basis of a vector space, the matrix that transforms one basis in another basis ...
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0answers
74 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 ...
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0answers
60 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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0answers
59 views

Dirichlet principle on compact manifolds with boundary

How is it the Dirichlet's principle on compact manifolds with boundary $\partial M= \mbox{manifold boundary}$?. I've just found the Dirichlet's principle on domains $U \subset \mathbb{R}^{n}$, i.e, $U$...
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0answers
55 views

About a regular value of a manifold with boundary

I don't understand the proof of the following theorem. Thorem: Let $f:X→N$ be a smooth mapping, where $X$ is $m$ dimensional smooth manifold with boundary and $N$ is $n$ dimensional smooth manifold. ...
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0answers
14 views

Possible to describe random 3D surfaces (geograhical height over limited area) by formula?

Coming from a geographic computer sciences background and working with 3D terrain (so please forgive if my terminology is inappropriate), I was always wondering if it is possible to describe the 3D ...
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0answers
39 views

Manifolds with boundary and foliations

Is there a theory of foliations by manifolds with boundary? Particularly, Is there a generalization of the Frobenius theorem and the Stefan-Sussmann theorem in which the leaves are manifolds with ...
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0answers
38 views

A question about the proof of Extension Lemma for Smooth functions

I am trying to understand the proof, but am stuck at the last line. Why does the first equality of the picture below holds? The codomain of $f$ is $R^k$, not some the set of nonnegative numbers. Is ...
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0answers
103 views

Stokes' theorem proof without FTC

To present time I have not found a proof of Stokes' Theorem for manifolds that does not involve the Fundamental Theorem of Calculus in some way. Is it possible to prove the general theorem without ...
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0answers
86 views

Are there topological manifolds with boundary that are not compact?

Following this question Are there compact manifolds without boundary? I'm asking if there is any example of a manifold with boundary that is not compact, Is the interval $X=[0,1)$ a counterexample ? ...
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0answers
135 views

integrating a 2-form over the sphere

I need to integrate the following form over the 2-sphere: $$w=\frac{xdy\wedge dz-ydx\wedge dz+zdx\wedge dy}{(x^2+y^2+z^2)^\frac{3}{2}}$$ now a direct calculation shows that $dw=0$, thus from stokes' ...
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0answers
175 views

Embedded extension of embedded submanifolds with boundary

Let $M$ be a smooth manifold (without boundary) and $S \subseteq M$ be a smooth embedded submanifold with non-empty boundary $\partial S$. Does there exist a smooth, embedded extension $\tilde S$ of $...
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0answers
51 views

Calculus of variations when target space has boundary

Let $M,N$ be smooth oriented Riemannian manifolds; $M$ closed and $N$ with non-empty boundary. Let $f:M \to N$ be smooth and suppose the image of $f$ intersects $\partial N$. Let $$W:=\{V \in \Gamma(...
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0answers
28 views

Immersive limit of embeddings is injective on the interior?

$\newcommand{\pl}{\partial}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be smooth compact, connected, oriented manifolds of the same dimension. $\M$ can have a boundary....
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0answers
33 views

Homotopy groups of the boundary of a projectively or conformally compactified hyperbolic manifold

Say that we have a projectively or conformally compactified hyperbolic manifold, and we know the homotopy groups (so just the fundamental group) of the hyperbolic manifold. Can we say anything about ...
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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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0answers
66 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 ...
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0answers
54 views

Clarification for Manifold and Manifold with Boundary Definitions

I am reading Spivak's Calculus on Manifolds, and just need a bit of clarification when it comes to definitions. He defines a manifold as some space $M$ that satisfies: $(M)$: $\forall x \in M, \...
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0answers
78 views

Integration over manifold

Let $M$ be a smooth 2-manifold in $\mathbb{R}^3$ such that $$4x^2+y^2+4z^2 = 4, y \ge 0$$ The boundary of $M$ is the set of points where $$x^2 + z^2 = 1, y = 0$$ Let $\alpha(u,v) = (u,2\sqrt{(1-u^2-...
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0answers
49 views

How to define an atlas on this manifold with boundary?

Consider the set $\mathcal{M} = \{\ \mathbf{x} \in \mathbb{R}^{3}\ | \ 1 \leq ||\mathbf{x}|| \leq 2 \ \}$. This is a $3$-submanifold with boundary. Obviously, we have $\partial \mathcal{M} = \{\ \...
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0answers
45 views

Show that $\partial(M\times N)=M\times\partial(N)$

Let M a $k$-dimensional manifold without boundary of $\mathbb{R}^{n}$ and N a $l$-dimensional manifold of $\mathbb{R}^{m}$ with or withour boundary. Show that $\partial(M\times N)=M\times\partial(N)$ ...
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0answers
149 views

Closure of a Manifold is a Manifold with Corners?

Is there a general theorem that shows that if you have a manifold $S$ then its closure $\overline{S}$ is a manifold with corners? I am dealing with a specific set $S$ (I would rather not say which ...
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0answers
115 views

maximum principle on compact manifolds with boundary

Let us consider the equation $Lu + f(u) = 0$ on a compact manifold $\overline{M} = M \cup \partial M$ with boundary, with Dirichlet boundary conditions. $L$ is a linear elliptic operator, and $f$ ...
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0answers
110 views

Can the closure of a geodesic chart of a manifold without boundary (and with bounded geometry) be defined as a compact manifold with boundary?

Let $(M,g)$ be a $d$-dimensional smooth, riemannian manifold without boundary, which is geodesically complete, connected, has positive injectivity radius and a bounded geometry (i.e. all derivatives ...