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Questions tagged [diffeomorphism]

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

A diffeomorphism is a bijective differentiable map such that the inverse is differentiable. They are the isomorphisms of differentiable manifolds.

i.e., A one-to-one continuously-differentiable mapping $~f:M\to N~$ of a differentiable manifold $~M~$ (e.g. of a domain in a Euclidean space) into a differentiable manifold $~N~$ for which the inverse mapping is also continuously differentiable. If $~f(M)=N~$, one says that $~M~$ and $~N~$ are diffeomorphic.

  • Every diffeomorphism is a homeomorphism, and the converse is false.

Reference:

https://en.wikipedia.org/wiki/Diffeomorphism

496 questions
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Affine function is a diffeomorphism?

Given an affine function $f:\mathbb{R}^n \to \mathbb{R}^n$ defined for all $x\in \mathbb{R}^n$ by $$f(x)=T(x)+a$$ such that $T$ is an invertible Linear map and $a\in \mathbb{R}^n$, is $f$ a diffeomorphism?
h3110w0r1d
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When is a multi-variable function a one-to-one?

For a function of one variable just looking at the graph I can see that the condition for a one-to-one analytic function is: $$y=f(x)$$ is one-to-one if we have the condition: $f'(x)>0$ for all $x$. i.e. the function is always increasing. Then one…
zooby
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Can we extend a diffeomoprhis of surface to symplectomorphism?

I think surface diffeomorphism can be extend to a symplectomorphism but i can't describe this. Is there some reference? If this is not turue, please tell me.
masao
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Diffeomorphism $\varphi:(0,1)^2\rightarrow U$. $U$ is a parallelogram

Let $U\subseteq\mathbb R^2$ be a open parallelogram with vertices $(2,3), (6,4), (8,6), (4,5)$. Find a diffeomorphism $\varphi:(0,1)\times (0,1)\rightarrow U$. How can I find such an Diffeomorphism? I tried to do it with polar coordinates but I…
Moritz
  • 151
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Showing that $f$ is $C^{\infty}$ using the equation $\frac{d^2f}{dx^2} = f\frac{df}{dx}$.

I'm trying to answer Tu's Introduction to Manifolds exercise 1.3, which says in the first item: Let $U \subset \mathbb{R}^n$ and $V \subset \mathbb{R}^n$ be open subsets. A $C^\infty$ map $F \colon U \to V$ is called a diffeomorphism is it is…
Lucas Giraldi
  • 526
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Differentiability of a composite map

Suppose $U,V$ are open subsets of $R^2$ and $S\subseteq R^3$ . Suppose $f:U\to S$ and $g:V\to S$ are bijective differentiable maps whose Jacobians have rank 2. Is the composite map $g^{-1}f$ from $U$ to $V$ a diffeomorphism?
Drooga
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Find some diffeomorphism $A$ on $B$

Find some diffeomorphism $A$ on $B$: $$ A=\left\{(x,y)\in \mathbb R^2: x>0, y>0, xy<1 \right\}$$ $$ B=\left\{(x,y)\in \mathbb R^2: 10 \right\}$$ I completely do not know how to go about such tasks, because I do not understand…
William8649
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Trying to study about homeomorphism, local diffeomorphism, diffeomorphism.

If I am given a function, how can I tell if it is a homeomorphism, local diffeomorphism, or diffeomorphism. Does one imply the other(s)? (I just started studying this topic and having a hard time understanding it. It's quite confusing. I think some…
Seungchan Lee
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Most general diffeomorphisms of a sphere.

Given a n dimensional vector $V$, such that $|V|=1$, how can one write a general diffeomorphism which preserves it's length as an orthogonal matrix $M(V)$ which acts on V?
zooby
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Find a diffeomorphism $F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $F(x)=y$

Suppose $x,y \in \mathbb{R}^n $. Find a diffeomorphism $F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ and $r>0$ such that $F(x)=y$ and $F(z)=z$ for any $z\in \mathbb{R}^n-B(x,r)$ .
Edward Grammer
  • 67