Manifolds are typically defined to be without boundaries (every point has a neighbourhood homeomorphic to an Euclidean open disc). Use this tag for the manifolds with boundaries, as well as manifolds with corners, and other such generalisations of the notion of a manifold.

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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: $(M)$: $\forall x \in M, \...
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$(x + y + xy)/2 = f( f^{[-1]}(x) + f^{[-1]}(y) )$

Consider $(x + y + xy)/2 = f( f^{[-1]}(x) + f^{[-1]}(y) )$ Where $f^{[-1}]$ denotes the functional inverse of $f$. How to find $f$ ? How about the more General idea of finding $f$ for a given $g$? ...
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Extending Riemannian Manifold to Boundary

If you have a Riemannian manifold $(M,g)$ (maybe with other assumptions as need), is there a natural way to extend it to a smooth manifold with boundary? For example, the Lobachevsky space viewed as ...
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Compact 3-manifolds with boundary an orientable surface

The usual embedding of the closed orientable surface $M_g$ of genus $g$ in $\mathbb{R}^3$ bounds a compact orientable $3$-manifold which is homotopically equivalent to a bouquet of $g$ circles. If a ...
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60 views

Can every Riemmanian Manifold be completed?

I had two trails of though.. is either of them fruitful? I know every metric space can be completed, my question is: can a Riemmanian manifold $M$ be embedded smoothly and isometrically into it's ...
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38 views

An orientation on a $(n-1)$-dimensional submanifold.

This question goes on where this question ended. Given is an non-empty $(n-1)$-dimensional ($n\ge2$) differentiable submanifold $X\subset\mathbb{R}^n$ such that there exists an open $U\subset\mathbb{R}...
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Topological boundary of a specific manifold.

Let $\emptyset\neq X\subset\mathbb{R}^n$, $n\ge2$, be an $(n-1)$-dimensional differentiable submanifold, i.e. for every $p\in X$ there is an open $U_p\subset\mathbb{R}^n$ with $p\in U_p$, and a ...
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37 views

On the trace theory and restrictions of Sobolev space functions

Consider a connected open bounded subset $D\subset \mathbb{R}^d$ with smooth boundary. It is easier to state for the disc so I will do so. I don't think it would change the argument (except for in our ...
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62 views

Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$?

The first question is Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$? In other words, is it obvious? The question stems from a theoretical physics ...
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1answer
42 views

$\sin(\frac{1}{x})$ on $\Bbb R^2$ interior, cluster points, boundary

Given $$M:=\{(x_1, x_2) \in \Bbb R^2 : x_2 =\sin(\frac 1 {x_1}), x_1\in(0,\frac 1 \pi)\}$$ , a subset of the normed Space $(\Bbb R^2,||\cdot||_2)$. $M$ is the above Curve. (the commen $\sin(\frac{1}...
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Topological boundary as a submanifold

Let $U$ be an open subset of a smooth $n$-manifold. Consider $\partial U$ the topological boundary of $U$. Is the following true ? : If $\partial U$ is a smooth $n-1$ submanifold without boundary, ...
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41 views

Is $\overline{u^{-1}((-\infty,a))}$ a $C^k$-manifold with boundary for a $C^k$ function $u$?

Let $u\in C^k(\mathbf{R}^d)$, $M:=\overline{u^{-1}((-\infty,a))}=\overline{\{x\in\mathbf{R}^d \mid u(x)<a\}}$, where $a$ is a constant such that $M\neq\emptyset$. Is $M$ a $C^k$-manifold with ...
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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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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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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 ...
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The boundary $\partial S$ of the square in $\mathbb{R}^2$ has no topology and smooth structure which makes it an immersed submanifold.

This is a problem in Smooth Manifolds by Lee, but it seems like there is an obvious decomposition of the $\partial S$ into a manifold with 4 components. The top and bottom edges including the corners ...
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44 views

Lenght of the curve in Riemannian metric.

Let $M^{k}$ a submanifold, $h:U\to M^{k}$ a chart, and $\gamma:[a,b]\to h(U)\subset M^{k}$ a curve in $M^{k}$. Represent the curve in coordinates $(h,U)$ as $h^{-1}\circ\gamma(t)=(y^{1}(t),...,y^{k}(t)...
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39 views

$M$ be an $R$-orientable compact (topological) n-manifold with boundary $\partial M$. Then $\partial M$ is $R$-orientable.

I have a question of a proof of the following proposition. Proposition: Let $R$ be a commutative ring with unit $1_R$, $M$ be an $R$-orientable compact (topological) n-manifold with boundary $\...
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Proving that the following is a submanifold with boundary

Let $\mathbb{H}^{k} = \{ \ \mathbb{x} = (x_{1}, ..., x_{k}) \in \mathbb{R}^{k}\ | \ x_{k} \geq 0\ \}$. Definition: If the following holds, then $\mathcal{M}$ is a $k$-submanifold of $\mathbb{R}^{n}$ ...
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41 views

Stokes Theorem for non compact subsets of $\mathbb{C}.$

Consider a regular compact subset $K \subset \mathbb{C}$. If $f \in C^1(\mathbb{C})$ one has $$\int_{K} \frac{\partial f}{\partial \overline{z}} = \int_{\partial K} f dz.$$ Now I ask the question for ...
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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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33 views

Partitions of unity subordinate to open cover of manifolds with boundary?

I am attempting to adapt Lee's proof of the fact that open covers of manifolds without boundary always admit smooth partitions of unity to the case in which the manifold does have boundary. The ...
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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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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 ...
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87 views

Are $\mathbb{C}^2$ and $\mathbb{C}^2/(x,y)\sim(y,x)$ homeomorphic?

Let $A$ be the set of monic quadratics over $\mathbb C$ and let $B$ be the set of unordered pairs over $\mathbb C$ where possibly the two elements of the pair may be the same. Then the map which takes ...
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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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Find a direct basis for boundary orientation

Let M in R4 be a manifold defined by equation d= $x^2$ + $y^2$ +$z^2$ and oriented by sgn $dx_1$^$dx_2$^$dx_3$. Consider the subset where d<=1. Show that it is a piece with boundary. Let x be a ...
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29 views

Open neighborhood of a manifold boundary point

Manifold with boundary: An $n$-dimensional manifold with boundary is a second countable Hausdorff space in which every point has a neighborhood homeomorphic either to an open subset of $\mathbb{R^...
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In finding boundary of the product of two half-lines, shall homeomorphism be global?

Lets $ \mathbb{R_{+}^{n}} = \mathbb{R^{n-1}} \times [0;+\infty[ $ Basically in my course I have this statement within the definition of a manifold with boundary: $ \forall x \in M, \exists U_x $ an ...
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42 views

The corner of the squre

A square is a topological manifold with boundary but not a smooth manifold with boundary because of its corners. But I am confused about it. I think since for a specific corner $p$, there is only one ...
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32 views

Topological Manifold is Manifold with Empty Boundary

I want to show that every $n$ topological manifold $M$ is an $n$ Manifold with boundary where $\partial M=\emptyset$. i.e. every chart $(U,\phi)$ maps to an open set $V\subseteq\mathbb{H}^{n\circ}$ (...
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The nature of components in a certain manifold

Let $N$ be a smooth, connected manifold and $f:N \to \mathbb R$ a smooth, proper and surjective map, transverse to some $k \in \mathbb N$. This means that $M:=f^{-1}(k) \subset N$ is a finite ...
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Tangent vectors in $T_p\partial M$

I know that if $M$ is a smooth $n$-dimensional manifold with boundary, then $\partial M$ is a smooth $(n-1)$-dimensional manifold. So for $p\in\partial M$, we have $T_p\partial M\cong\mathbb{R}^{n-1}\...
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Computing parametrizations for a differentiable $2$-manifold with boundary

Consider the following subset of $\mathbb{R}^{3}$ \begin{equation} C=\{(x,y,z)\in\mathbb{R}^{3}\:|\:0\leq x\leq 1,\:0\leq y\leq 1,\:z=x^{2}+y^{2} \}. \end{equation} Intuitively, this looks like a ...
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72 views

Topological Boundary vs Manifold Boundary

Let $A$ be the open unit disc in $\mathbb{R}^2$ and $B$ be the closed unit disc in $\mathbb{R}^2$. The toplogical boundary of $A$ and $B$ is $S^1$. This I understand. The manifold boundary of $A$ ...
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About the number of minimum parametrizations of a $1$-smooth manifold compact w/ boundary in $\mathbb{R}^{3}$

Let $C$ be a $1$-dimensional compact differentiable manifold with boundary in $\mathbb{R}^{3}$. In easy examples, it looks like we can always parametrize such a manifold with only two charts: usually ...
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56 views

Universal cover of boundary

Let $M$ be a compact manifold-with-boundary and $B$ a component of $\partial M$. Let $\tilde{M}$ be the univeral cover of $M$ with infinite-sheeted covering map $p:\tilde{M} \to M$. I wonder about the ...
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property of sum of coefs of a chain

Suppose c is a k+1 chain in U(open set in space R^n), then boundary of c (a k chain) can be expressed as a linear combination of k-cubes, using boundary operator: $$∂c=∑_ia_ic_i$$, where a_i are the ...
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Euler Characteristic of a boundary of a Manifold

I need some guidance in understanding a specific passage of the following result taken from [tom Dieck Algebraic Topology, page 456] Proposition 18.6.2. Let $B$ be a compact $(n+1)$-manifold with ...
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How to qualify a N dimensional manifold as Compact under following condions?

Suppose a manifold of N dimensions is closed and bounded in a dimension but it remained unbounded in all other dimensions, so how to categorize the manifold. For example, in simpler form how to ...
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Is there any shorter term for manifolds with boundary?

The “with boundary” does get a bit unwieldy when you have to write it more than a couple of times. I can't seem to find any alternative term on Wikipedia or elsewhere, but surely someone ...
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Reference on manifolds with boundary

I am here because I want to know if someone knows of some good e fast books or references about manifolds with boundary. Help me please.
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Charts in an oriented manifold with boundary

Let $M$ be an oriented manifold (with boundary) with $dim (M)\ge 2$. Show that there exists an atlas $\{(U_{\alpha},\phi_\alpha)\}_{\alpha\in I}$ for the chosen orientation such that $\forall\...
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Tangent space of manifold has two unit vectors orthogonal to tangent space of its boundary

I'm reading spivak calculus on manifolds and got stuck. Let M be a k-dimensional manifold with boundary in $\mathbb{R^{n}}$, and $M_{x}$ is the tangent space of M at x with dimension k, then $\partial ...
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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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Reference for theorems in Hirsch

In Hirsch's differential topology, we find the following theorems on page 31: 4.1 Theorem: Let $M$ be a $C^r$ $\partial$-manifold and $N$ a $C^r$ manifold, $r \geq 1$. Let $f : M \to N$ be a $C^r$ ...
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84 views

Basic examples topological manifolds with boundary

I've just started to study differential geometry and I've some problems with the first definitions. We have defined a topological manifold with boundary of dimension n as a topological space $M$ such ...
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Maximally symmetric manifold with boundary and non-vanishing extrinsic curvature?

I was wondering if the following requirements are compatible: Given a $d$-dimensional manifold with boundary $M$ with $\partial M\neq \emptyset$ endowed with a metric $g$. The following conditions ...
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46 views

When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth?

When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth? I can't find a formal definition. I know, that we say, that $\partial\Omega$ has a $C^k$-...
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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 ...