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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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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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 ...
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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 ...
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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 ...
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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 ...
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Is there any variation of Lipschitz continuity, where one can bound difference between value of 2 functions which act on different space?

Lipschitz continuity can be used to bound the difference between value of a function at two different points. Is there any variation of this, where one can bound difference between value of 2 ...
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Boundary of the boundary of a manifold with corners

A point of a manifold with corners is a boundary point by definition if one of its coordinates is $0$ by some (hence in all) chart with corners (see here). In the same page one can read: The ...
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Boundary of a topological manifold invariant?

Let $M=(X,\tau)$ be a topological manifold with boundary. One can proof that the interior $Int(M)$ and boundary $\partial M$ of the manifold are distinct sets. I was wondering if someone knows a ...
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39 views

Smooth mapping between manifold such that $\text{Im}(f) \subset \partial N$

Let $f:M \to N$ be smooth such that $\text{Im}(f) \subset \partial N$. Prove that $f$ as mapping $f:M \to \partial N$ is smooth. I've tried to write down $f:M \to \partial N$ as composition of two ...
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What does it mean for a manifold to be oriented?

I'm currently working through Spivak's Calculus on Manifolds. I've got to Stokes' Theorem, which is stated thus (the bold is my emphasis): Stokes' Theorem If $M$ is a compact oriented ...
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38 views

Jacobian on manifolds

I'm trying to make sense of integrals of the form $$\int_\Omega L[D\psi_1(x), D\psi_2(x)]\ \mathrm dx$$ Where $\Omega \subset \mathbb R^d$ is a $p$-dimensional compact manifold with boundary. ...
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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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67 views

Differential-form version of Cauchy-Schwarz on manifold boundary

Consider a manifold $M \subset \mathbb{R}^2$ with boundary $\partial M$. Then, consider a zero-form field, $\phi^{(0)}$, defined on the whole of $M$ (including the boundary). Then, the following ...
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$G$ is an $(n-1)$-manifold without boundary and is the topological boundary to an open $K\subset \mathbb{R}^n$. Prove $G \cup K$ is an $n$-manifold.

All manifolds are smooth. Let $M = G \cup K$. The interior of $M$ is an open set in $\mathbb{R}^n$ and can be given a global coordinate by the identity map. The points in $M$ not on the interior of ...
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Invertibility theorem on the boundary for a function between two closed 2D manifolds

Assume a function $f:\mathbb{R}^2\to\mathbb{R}^2$ on a simply connected, closed domain $D\subset\mathbb{R}^2$ including its boundary $\partial D$. I am interested in the local invertibility of $f$ ...
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Boundary points of a manifold

I'm reading about Riemannian Geometry and my question is regarding Manifolds with Boundary. I want to show a point of a manifold with boundary is either an interior point or a boundary point, so no ...
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214 views

Is the complex projective plane a compact manifold with or without boundary (closed manifold)?

my question is the one in the title. (My motivation is to understand in which way Freedman's classification of compact simply-connected 4-manifolds implies the Poincare conjecture for 4-manifolds, as ...
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157 views

Gauss-Green Theorem from generalized Stoke's Theorem.

I am trying to deduce the next identity (Green-Gauss theorem) $$\int_\Omega \dfrac{\partial u}{\partial x_i} dx = \int_{\partial \Omega} uv_i dS$$ from the generalized Stoke's theorem for manifolds. ...
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rudin's principles of mathematical analysis 10.31

I'm working on rudin's principles of mathematical analysis(3rd edition). There is problem too complex for me to solve.Please help me. The problem is on p270-271 of text. "Let T be 1-1 mapping of ...
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The space of collars of a manifold is contractible

Theorem: Let $M$ be a smooth manifold with boundary $\partial M$. Let $e_0,e_1 : \partial M\times [0,1]\rightarrow M$ be collars of $M$, i.e. $e_i$ are embeddings such that $e_i(x,0)=x$ for each ...
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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 ...
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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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Definition of boundary in a topological invariant way

I'm reading through Aguilar & Prieto lecture notes "Fiber bundles" (available online by googling it, ...
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A singular $n-$cube and a circumference defined the border than 2-cube

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
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257 views

(Whitney) Extension Lemma for smooth maps

I am currently reading Lee's book "Introduction to Smooth Manifolds (2nd edition)". Corollary 6.27 in that book states that a smooth map $f : A \rightarrow M$ where $M$ is a smooth manifold ...
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53 views

Orientation of manifolds with boundary

I have an ambiguity about how to orient the boundary of a manifold. In particular : Consider the example $M=B^2 \subset \mathbb{R^2}$ be the manifold with boundary. suppose positive orientation for M ...
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135 views

Inverse Function Theorem for Manifolds with Boundary

In Lee SM it is written that the inverse function theorem can fail for manifolds with boundary.As hint it is given the inclusion of half space into euclidean space $\iota:\mathbb{H}^n\to\mathbb{R}^n$ ...
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1answer
62 views

Proving that a regular value of a smooth function isn't in the boundary of the counter-domain

Suppose $X$ is a manifold without boundary and $Y$ is a manifold. Suppose there is a smooth function $f: X \rightarrow Y$ and we are given a $y \in Y$ such that $y$ is a regular value of $f$ and ...
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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 ...
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Any N dimensional manifold as a boundary of some N+1 dimensional manifold?

Is this statement true: Question: Can any N dimensional manifold be realized as a boundary of some N+1 dimensional manifold? If so/not, how to prove/disprove it? I read a TQFT paper from Edward ...
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On the “regularity” of the boundary of an open set

Let $M = \mathbb{R}^2$ (or more generally, let $M$ be a topological manifold) and let $\Omega$ be an open set in $M$. I'm considering the following regularity conditions for the boundary of $\Omega$: ...
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Boundary of product cartesian

What's the boundary of $\Omega\times (a,b)$, where $\Omega$ is an open bounded subset of $\mathbb R^n$ ?
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The boundary of an $n$-manifold is an $n-1$-manifold

The following problem is from the book "Introduction to topological manifolds". Suppose $M$ is an $n$-dimensional manifold with boundary. Show that the boundary of $M$ is an $(n-1)$-dimensional ...
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Is the square pyramid a manifold with corners?

An n-manifold with corners is topologically an n-manifold with boundary, but with a smooth structure that makes it locally diffeomorphic to $[0,\infty)^n$ instead of $[0,\infty) \times ...
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Are there Kirby diagrams for manifolds with boundaries?

There are Kirby diagrams for 3- and 4-manifolds which consist of framed links corresponding to 1- and 2-handles attached to a single 0-handle. Any such diagram will give a unique closed manifold since ...
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Manifold Boundary versus Topological Boundary.

Let $M$ a $n$-manifold whit boundary, i.e., for each $x\in M$, there exist $U_x\subseteq M$ open in the topology of $M$ such that $U_x$ is homeomorphic to $\mathbb{R}^n$ or homeomorphic to ...
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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 ...
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787 views

The definition of submanifold of a manifold with boundary

What is the exact definition of submanifold of a manifold with boundary? For example, when $H$ is the half space of the plane and S is a cycle which intersects with the origin in the half plane. Then ...