# Questions tagged [diffeomorphism]

This tag is for questions regarding to "diffeomorphisms", a map between manifolds which is differentiable and has a differentiable inverse.

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### Local diffeomorphism between a disk and a sphere

This may be a silly question, but I’ll make it anyway. Let $f: D^2 \to S^2$ be a local diffeomorphism between the closed unit disk and the unit sphere. Is it necessarily injective?
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### Diffeomorphism between circle and square clarification

I refer to the top answer on the following post: No diffeomorphism that takes unit circle to unit square. If we assume that there is a diffeomorphism $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$, we want ...
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### Showing that a function in $\Bbb{R}^{2}$ is a diffeomorphism.

Let $f:\Bbb{R}\rightarrow\Bbb{R}$ a function of class $C^{1}$ such that $|f'(t)|\leq k < 1\, \forall\, t\in\Bbb{R}$. Define $\phi:\Bbb{R}^{2}\rightarrow\Bbb{R}^2$ by $$\phi(x,y)=(x+f(y),y+f(x)).$$...
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### Homotopy equivalence in relative Diffeomorphism group

I am currently working on a project where one studies a smooth bordered compact $3$-manifold $M$ with some properly embedded essential surface(s) $S \subset M$. More precisely, I am interested in the ...
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### non-trivial example of a map $h:K \to K$?

Consider a surface $U=(0,1)^2$ in the real plane. Decompose $U$ into an infinite set of real analytic functions which form a family, $F_s=\{f_{s}(x):s\in \Bbb R_{>0}\},$ with real parameter $s$, s....
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### Let $\phi: O_1 \subset \mathbb{R}^3 \to O_2\subset \mathbb{R}^3$ be a diffeomorphism and $S$ be a surface. Then $\phi:S \to \phi(S)$ is a diffeo.

I am having some trouble with the following problem (Exercise 2.45 of the book Curves and Surfaces, by Montiel and Ros): Let $\phi: O_1 \subset \mathbb{R}^3 \to O_2 \subset \mathbb{R}^3$ be a ...
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### The relationship between the tubular neighbourhoods of two diffeomorphic manifolds

I'm a beginner of this complex area and want to use the differential geometry as a tool to solve some control problems. So my statement might be a little bit inaccurate...I will try my best. There ...
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### Properties of $\mathcal{C}^1$-diffeomorphisms which keep invariant the uniform distribution on the n-cube?

Let us consider the n-cube (n-dimensional hypercube) $H_P={]0,\,1[}^P$ and let $\psi:\,H_P\rightarrow{}H_P$ be a $\mathcal{C}^1$-diffeomorphism which keep the uniform distribution (with respect to the ...
### How to prove $\mathbb R/(\mathbb Z+\alpha \mathbb Z) \simeq \mathbb T^2/\Delta_\alpha$, the irrational torus, in diffeology?
I was reading this introductory text on diffeology by Patrick Iglesias-Zemmour and it is claimed in page $18$ that $T_\alpha$ is diffeomorphic $\mathbb R/(\mathbb Z + \alpha \mathbb Z)$, where $\alpha$...