# 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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### (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 $f\colon A \rightarrow M$ where $M$ is a smooth manifold ...
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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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### 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....
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### Is the closed hemisphere diffeomorphic to the closed disk?

This might be silly but it puzzles me. Let $M$ be the closed upper hemisphere of $\mathbb{S}^2$. This is a manifold with boundary. Is it diffeomorphic to a closed disk in $\mathbb{R}^2$? These ...
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### A local diffeomorphism can map a boundary point to an interior point

I would like to find an example for a local diffeomorphism between smooth manifolds with boundary which maps some boundary point to an interior point. I am quite sure such an example exists.
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### 2-manifold with involution without fixed points is a boundary of a 3-manifold.

I am preparing for geometry/topology test and I can not handle the following problem from previous years: Let $F$ is closed orientable 2-manifold with involution without fixed points. Prove that $F$ ...
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### 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 ...
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### 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 ...
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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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### 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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### 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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### No where vanishing exact $1$-form on compact manifold.

I found several answers on the following question : Does there exists a no where vanishing exact $1$-form on a compact manifold without boundary? All answer says that certainly not. But I cannot ...
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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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### $\mathbb S^2$ or $\mathbb RP^2$ on boundary of a 3-manifold

Let $M$ be a 3-manifold with boundary $\partial M$. Suppose $\partial M$ contains a sphere or a projective plane, which is contractable in $M$. Show that $M$ is also contractable. The above statement ...
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### Are two compact $n$-dim submanifolds with boundary of $\mathbb{R}^n$ with identical boundaries coincide?

Let $U,V$ be the interiors of codimension $0$ compact embedded submanifolds with boundary of $\mathbb{R}^n$. Suppose that $\partial U=\partial V$. Is it true that $U=V$? (In other words, I am asking ...
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### 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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### Spin structure and bordism

I have some questions about bordism and spin structures on manifolds. If you have any examples or references I would appreciate it. Is there a 3-manifold $M$, orientable, which does not support 3 ...
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### 2 -manifolds that can't be decomposed into two equal parts

The real projective plane has the property that if you divide into two different 2-manifolds, they will not be homeomorphic (i.e. one will be orientable, and the other non-orientable). The sphere ...
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### 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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### 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 $f$ ...
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. ...