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.

118 questions with no upvoted or accepted answers
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29
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835 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 $...
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194 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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269 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....
8
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92 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
36 views

What is the boundary of a tubular neighbourhood of the projective plane embedded in $\mathbb{R}^4$?

There are various ways to embed the projective plane into $\mathbb{R}^4$ very nicely, see e.g. Wikipedia. Suppose now that I take such an embedded projective plane $P \subset \mathbb{R}^4$ and fix a ...
6
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0answers
310 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}...
6
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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 $\...
5
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98 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 ...
5
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378 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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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 ...
4
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107 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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1answer
93 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 ...
4
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281 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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791 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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1answer
74 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 ...
3
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0answers
247 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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93 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 $...
3
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315 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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89 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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1answer
179 views

Long exact sequence for manifolds with boundary

Let $T^* S^n$ be the disk cotangent bundle on $S^n$ consisting of covectors with norm less than or equal to $1$. I am confused about the long exact sequence of the pair $(T^*S^n, \partial T^*S^n)$. ...
3
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82 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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20 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 ...
2
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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 ...
2
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0answers
32 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 ...
2
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0answers
41 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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93 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
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1answer
66 views

The riemannian metric of a neighborhood of the boundary of a compact manifold

Let $M$ be a compact riemannian manifold with boundary $\partial M$. We have that $\partial M$ is also compact and I was able to show that there is some $a>0$ such that the map $F:[0,a]\times \...
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24 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
76 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 ...
2
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94 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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1answer
151 views

How is the “surface measure” on a manifold defined?

Let $k,n\in\mathbb N$ with $k\le n$ $M$ be a $k$-dimensional $C^1$-submanifold of $\mathbb R^n$ $\Omega\subseteq\mathbb R^k$ be open, $\phi:\Omega\to M$ be a global chart of $M$ and $$g_\phi(x):=\det\...
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46 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 ...
2
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65 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, $U$...
2
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42 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 "...
2
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1answer
121 views

“Defining a smooth structure on a topological manifold with boundary”

This is a proposition for topological manifolds. However, I am wondering if this prop holds when $M$ is a topological manifold "with boundary." Thus, can the expression like "a smooth atlas on a ...
2
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0answers
110 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 ...
2
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0answers
69 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 ...
2
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0answers
129 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
13 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 ...
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25 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(...
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32 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:...
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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?
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32 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
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 $M$ ...
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0answers
46 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
36 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 ...
1
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1answer
370 views

Show that the closed $n$-ball $B^n(a)$ is a manifold

Show that the closed $n$-ball $B^n(a)$ is a manifold. I know how to show that $S^{n-1}(a)=\partial B^n(a)$ is an $n-1$ manifold without boundary. We consider the function $f(x)=a^2-\Vert x\Vert ^2$. ...
1
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0answers
71 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
43 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 ...