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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Realizing singular homology boundaries as boundary restrictions of manifold maps, in smooth case

Let $X$ be a topological space. Let $\rho = \sum_j \sigma_j$, where the sum is finite, be a chain of singular $k$-simplices in $X$ (in the $k$th singular chain group of $X$, with $\mathbb{Z}$ ...
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Is the complement of a quadrant a manifold with corners?

The definition of a manifold with corners is analogous to that of a smooth manifold, except that, instead of being locally diffeomorphic to $\mathbb R^n$, is locally diffeomorphic to $[0,+\infty)^k\...
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2 answers
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An affine invariant notion of minimal surface?

The notion of minimal surface (i.e. having vanishing mean curvature) is not "affinely invariant" in the following sense: if $M\subset\Bbb R^n$ is an ($m$-dimensional) minimal surface, then $...
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Help proving statement in Lee's Introduction to Riemannian Manifolds about smooth curves into manifolds with nonempty boundary.

The statement appears on page 33 of the second edition of Professor Lee's Introduction to Riemannian Manifolds. It is in the section on Lengths and Distances in Riemannian manifolds, but I think the ...
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1 vote
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Lee Smooth Manifolds, why does the Whitney Approximation Theorem fail when the co-domain has non-empty boundary?

I am trying to study chapter 6 of Lee's Introduction to Smooth Manifolds. In a remark after the Whitney Approximation Theorem, Lee stated that this theorem do not hold because it might not be possible ...
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General condition for the surface element be the same as the volume element, up to a dt

The surface element in spherical coordinates is $r \sin \theta \mathrm{d}\theta \mathrm{d}\varphi$, and the volume element is $r \sin \theta \mathrm{d}\theta \mathrm{d}\varphi \mathrm{d}r$. We see ...
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Understanding difference between a distinguished boundary and 'normal' boundary in several complex variables

I am reading through Tasty Bits of Several Complex Variables and I come across the term distinguished boundary. It seems a distinguished boundary is different from a normal boundary as the author ...
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Normal derivative and Hodge star operator

Let $(M,g)$ be an oriented Riemannian manifold with boundary $\partial M$. Let $j\colon \partial M\to M$ denotes the natural embedding and $\mu_M$ the canonical volume form. If $\psi\in C^\infty(M)$ ...
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1 answer
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Does the 3-manifold $S^1\times S^2$ bound a smooth integral homology ball?

Does the 3-manifold $S^1\times S^2$ bound a smooth (integral) homology ball? The only 4-manifolds I know whose boundary is $S^1\times S^2$ are $S^1\times D^3$ and $D^2\times S^2$, and both are not ...
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Integration by parts on compact, non-orientable Riemannian manifold with boundary

Let $(M,g)$ be a compact Riemannian manifold, not necessarily orientable or without boundary. Let $\mu$ be a normalized volume measure on $M$ and $u$ be a smooth function on $M$. In some notes that I ...
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Flat Riemannian manifolds with boundary

The universal cover of a flat, connected, complete Riemannian manifold is Euclidean space, which allows for classification of all such manifolds as quotient spaces $\mathbb{R}^n/\Gamma$ for ...
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Preimage of a regular value by smooth function on bounded manifold

Suppose $f:M\to\mathbb{R}$ is a smooth function, where $M$ is a smooth, bounded manifold with boundary, $\partial M$. Suppose we knew $f_{|\partial M}=0$ and that some $c\in\mathbb{R}-\{0\}$ is a ...
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Stokes' theorem for a manifold without boundary

Stokes' theorem states that when $M$ is a compact oriented $m$-manifold with boundary, and $\omega$ is a $(m-1)$-form on $M$, we have $$\int_{\partial M}\omega = \int_{M} d\omega.$$ This is confusing ...
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Heat kernel expansion for space with boundaries but with no particular boundary conditions

In the document Heat kernel expansion, user's manual Vassilevich expresses in equations $(5.29-5.33)$ the coefficients of the heat kernel expansion up to the fourth one for a manifold with boundaries. ...
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2 votes
1 answer
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Top de Rham cohomology group for noncompact manifolds with boundary

Suppose that $M$ is a smooth, connected, oriented $m$-manifold with (empty or nonempty) boundary. I am aware that top de Rham cohomology group $H^m_{\mathrm{dR}}(M;\mathbb{R})$ is trivial for ...
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How to check that a tangent vector field is outward pointing at the boundary?

I'm reading a paper (https://www.jstor.org/stable/pdf/25151781.pdf) which gives a definition of a outward-pointing tangent vector field (see Assumption 1 below). Poincare-Hopf Theorem. Let $M \subset \...
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Confusion over boundary definition in topology

I am watching the below lecture by Dr Tadashi Tokieda on Topology: https://www.youtube.com/watch?v=J7vojBbvudQ&t=710s At around the 12:00 minute mark he states that the boundary ($\partial$) of an ...
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  • 565
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Can a non-orientable manifold have an orientable boundary?

It is true that if a Manifold $M$ with boundary is orientable then its boundary $\partial M$ is also orientable. I'm wondering if the converse is also true. Of fact, there are numerous minor examples ...
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  • 710
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What is meant by a manifold identified with the boundary of another manifold?

Recently, I've been looking into the topic on quasi-local mass. This is a topic related to general relativity, and I'd like to acquaint myself with it via the book titled Geometric Relativity and ...
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6 votes
1 answer
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Boundary of manifold with boundary has empty boundary: What about corners in the topological category?

It is a general fact that the boundary $\partial\mathcal{M}$ of any topological manifold with boundary $\mathcal{M}$ is itself a topological manifold and has empty boundary, i.e. $\partial (\partial\...
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The Submersion Level Set Theorem for Manifolds with Boundary

We have the following theorem. (Lee, Introduction to Smooth Manifolds). Theorem. Let $M$ and $N$ be smooth manifolds. Let $F : M \to N$ be a submersion. Then each level set of $F$ (i.e., $F^{-1}(c)$) ...
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  • 306
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Whitney's embedding theorem for manifolds with boundary.

I have been stuck with the following problem for a few days: Let $M$ be a manifold with boundary of dimension $n$ (not necessarily compact), then there exists a smooth embedding $h:M\to \mathbb{H}^{...
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Connect Sum is the same as Boundary Connect Sum with punctured manifold

I have a fact which I haven't been able to prove. I briefly touched upon it in The first Kirby move and $\mathbb{C}P^2$. Let $M$ be a manifold-with-boundary, and let $N$ be a closed manifold, both of ...
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Finding a Retraction for a Collar Neighborhood

In Hirsch, pp.113-114 there is a proof that in a manifold with boundary $M$, $\partial M$ has a collar neighborhood. I attach it here: I am not sure how he is gluing together two local retractions ...
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Is a closed/open nondegenerate triangle a submanifold (with boundary)?

Let $M\subseteq\mathbb R^3$ be a (nondegenerate) closed/open triangle spanned by $p_0,p_1,p_2\in\mathbb R^3$. Can we show that $M$ is a $2$-dimensional embedded $C^1$ of $\mathbb R^3$ (possibly with ...
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2 votes
1 answer
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How do we obtain a chart for the sphere $\{x:\|x\|=1\}$ from a chart of the ball $\{x:\|x|\le1\}$?

Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$, $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary, $(\Omega,\phi)$ be a $k$-dimensional $C^1$-chart of $M$ (i.e. $\Omega$ is ...
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Two definitions of Lens Spaces

I know two definitions of the Lens Space $L(p,q)$: Take $S^3\subset\mathbb{C}^2$, and consider the $\mathbb{Z}_p$ action $T(z_1,z_2)\mapsto (\omega z_1, \omega^q z_2)$, where $\omega = e^{\frac{2\pi ...
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Apparent contradiction with gluing 2-handle to a 4-manifold all being isotopic.

In Hirsch's Differential Topology Chapter 8 Theorem 2.3, it says : Let $f, g:\partial Q\approx \partial P$ be isotopic diffeomorphisms. Then $P\cup_f Q\approx P\cup_g Q$. Here $\approx$ means ...
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1 vote
1 answer
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Differentiable structure on the boundary induces differentiable structure on the entire manifold

I am trying to prove the uniqueness of smoothed corners, in detail. In Milnor's notes, https://www.maths.ed.ac.uk/~v1ranick/papers/homsph.pdf, there's an explanation in p.34-37 on how to do it. The ...
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0 answers
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Smooth maps on manifold with nonconstant push forward maps

I am a beginner in manifold. I am reading Push forward map ( or differential map at a point). I came across the following question: Give smooth maps $F: M \to \mathbb{R^3}$ and $G: N \to \mathbb{R^3}$ ...
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Tangent space at a boundary point of closed disc.

I am a beginner at Manifolds. I am reading Tangent space at a point of a manifold $M$. What I know: If I take my manifold $M$ to be a closed unit disk then for any point $a$ inside the disk I know ...
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  • 614
1 vote
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Relative homology of $M\times M$ to diagonal

Let $M$ be a compact manifold. I want to understand why $$ \beta_i(M\times M, M) = \beta_i(M\times M)-\beta_i(M), $$ where $\beta_i$ denotes the $i$th Betti number and we identify $M$ with the ...
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  • 335
1 vote
0 answers
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Fibrations of manifolds with boundary - reference request

A smooth map $f\colon M\to B$ between compact manifolds is a fibration if it is surjective and submersive. I would like to use the same definition when $M$ and $B$ have boundary. Write $$ \partial M = ...
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Analyticity of surface integrals

Let $\epsilon: \mathbb{T}^d \to \mathbb{R}$ be a "well-behaved" function on the compact $d$-dim torus and consider the integral $$ f:E\mapsto \int_{\epsilon(k) = E} g(k) \frac{dS_k}{|\nabla \...
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  • 2,428
2 votes
1 answer
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Definition of an inward-pointing tangent vector for smooth manifolds with corners

Recall first the definition of an inward-pointing tangent vector for a smooth manifold with boundary. Let $M$ be a smooth manifold with boundary. If $p \in \partial M$, a vector $v \in T_pM - T_p\...
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1 vote
0 answers
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construction of fibered $3$-manifold compatible with flat slices (fibered)

Consider a bounded (by a unit cube) real fibered $3$-manifold $M=(0,1)^3.$ Slicing $M$, with planes orthogonal to the faces of the cube, for example, $z=c$ for some positive constant $c\in(0,1)$ yield ...
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  • 841
2 votes
1 answer
96 views

Regularity of higher order elliptic problem on compact smooth manifolds with boundary

I have trouble in finding a source in the literature for the following result: Let $\overline{M}$ be a compact smooth manifold of dimension $n \in \mathbb{N}$ with interior $M$ and non-empty boundary $...
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0 votes
1 answer
27 views

The boundary of $\mathbb{S}^{2}$

I have read that $\mathbb{S}^{2} \subset \mathbb{R}^{3}$ does not have boundary. However, given a set $X \subset \mathbb{R}^{n}$ its boundary is the set $\partial X = \overline{X}\cap (\overline{\...
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2 votes
1 answer
193 views

Could someone explain how John M. Lee defines the integration over the boundary of a manifold with corners?

In the $10$-th chapter of the text Introduction to Smooth Manifolds written by John M. Lee it is written what put to follow. The boundary of a smooth manifold with corners, however, is in general not ...
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1 vote
0 answers
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Calculate the volume of a $k$-simplex in $\Bbb R^n$

Definition If $x_0,...,x_n$ are $(k+1)$ affinely indipendent point of $\Bbb R^n$ (which means that the vectors $(x_1-x_0),...,(x_k-x_0)$ are linearly independent) then simplex determined by them is ...
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1 vote
0 answers
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does there exist $\delta>0$ such that $B(0,\delta) \times \{0\} \subset \partial \Omega$

Let $\Omega $ be an open, connected, bounded subset of the Upper Half space $ \Bbb R^n_+$ with a $C^{2,\epsilon}$ boundary. Then if its boundary $\partial \Omega$ (taken with respect to $\Bbb R^n$, i....
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0 votes
1 answer
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Lemma about smooth functions on manifolds with boundary

Let $f:M\to N$ be a continuous map between smooth manifolds with boundary. Is it true that smoothness of both $f|_{\mathring{M}}:\mathring{M}\to N$ and $f|_{\partial M}:\partial M\to N$ implies that $...
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  • 7,888
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0 answers
68 views

Kirby Diagram for the Trefoil

I'm studying R.E. Gompf and A.I. Stipsticz, 4-Manifolds and Kirby Calculus and I got stuck with a question. Let $K$ be the right-handed trefoil embedded in $\partial \mathbb{D}^4$, we know that, ...
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1 vote
1 answer
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Question on Bredon's Theorem 11.9 about the duality of intersection product and cup product

Let $i_K^W:K^k\to W^w$, $i_N^W:N^n\to W^w$ be smooth embeddings of smooth manifolds with boundary and assume that $N$ resp. $K$ meet $\partial W$ transversely in $\partial N$ resp. $\partial K$. 11.8 ...
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1 vote
1 answer
139 views

If $M$ is smooth $k$-manifold with corners then is $M$ without corner points a smooth $k$-manifold with boundary?

First of all we remember some elementary definitions and results about manifolds with corners. Definition A function $f$ defined in a subset $S$ of $\Bbb R^k$ is said of class $C^r$ if it can be ...
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2 votes
1 answer
122 views

Is the boundary of a $k$-manifold with corners a $(k-1)$-manifold with corners too?

First of all we remember some elementary definitions and results about manifolds with corners. Definition A function $f$ defined in a subset $S$ of $\Bbb R^k$ is said of class $C^r$ if it can be ...
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0 answers
80 views

The notation for the boundary of a region

For a region $A$ in $n$space, we use the notation $\partial A$ to denote the boundary of the region $A$, but why we are using the symbol $\partial$, is there some connection between the partial ...
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  • 3,141
2 votes
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Manifold with corners, without its corners

I'm working on Lee's Introduction to Smooth Manifolds Problem 16-8: Suppose $M$ is a smooth manifold with corners, and let $\mathcal{C}$ denote the set of corner points of $M$. Show that $M \setminus \...
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7 votes
1 answer
191 views

Shortest paths on a manifold with boundary are composed solely of geodesics and boundary sections?

Is the following true? Proposition: Let $M$ be a manifold with boundary $\partial M$. For any $p, q \in M$ let $P$ be a shortest path from $p$ to $q$. Then $P = \bigcup_{k=1}^n P_k$ where: $P_k$ is ...
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1 vote
1 answer
84 views

Gauss-Bonnet on a finite, non-compact surface.

Let $D$ be a two dimensional (simply connected) compact surface with boundary and a consider a point $p\in D$. Since $D \backslash\{p\}$ is no longer compact, the Gauss-Bonnet identity does not ...
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