# 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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### 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 ...
1answer
451 views

### 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 ...
1answer
321 views

### 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 ...
1answer
51 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 ...
1answer
1k views

### 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 $k$-...
0answers
124 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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### 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 ...
1answer
763 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 ...
1answer
147 views

2answers
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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 ...
1answer
213 views

### 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 $\Omega$: ...
1answer
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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 $\mathbb R^n$ ?
2answers
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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 ...
1answer
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### 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 ...
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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 ...