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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6
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0answers
33 views

Is Gauss map a diffeomorphism onto its image in this case?

Let $D$ be a properly embedded free boundary disk in the closed unit ball $\mathbb{B}^3$ of $\mathbb{R}^3$. This means that $D$ is a smooth disk contained in this ball, $D \cap \partial \mathbb{B}^3 = ...
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0answers
26 views

Is the inverse of an interior chart of a submanifold an immersion?

Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$, $\Omega$ be an open subset of $M$, $\phi$ be a $C^1$-diffeomorphism$^1$ from $\Omega$...
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1answer
20 views

Need the result of composing an infinite number of smooth functions be smooth?

$f$ is a smooth function from a manifold to itself. So is $f\circ f$, and $f\circ f\circ f$ and so on... If this sequence is extended forever, and supposing that it converges to some function, need ...
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53 views

Classification of diffeomorphisms by association of differentials with Lie groups

Suppose we are given an oriented Riemannian manifold $S \subset \mathbb{R}^3$ (which I'll refer to as a surface) and a diffeomorphism on $S$, $\Psi: S \rightarrow S$ where $d\Psi\vert_{\bf q}:T_{{\bf ...
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1answer
20 views

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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1answer
43 views

Show that these two diffeomorphisms cannot exist simultaneously

Let $d\in\mathbb N$, $x\in M\subseteq\mathbb R^d$ and $\psi^{(i)}:\Omega_i\to\psi^{(i)}(\Omega_i)$ be a diffeomorphism with $x\in\Omega_i$, $$\psi^{(1)}(M\cap\Omega_1)=\psi^{(1)}(\Omega_1)\cap(\mathbb ...
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2answers
91 views

Intuitive meaning of Diffeomorphism

Let $U\subset\mathbb{R}^n$, $V\subset\mathbb{R}^m$ and a bijection $f:U\to V$ is a diffeomorphism if $f$ and $f^{-1}$ are differentiable. I would like to know the intuitive meaning of two open sets ...
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1answer
42 views

Pushforward of a Vector Field by a Diffeomorphism

A question concerning when the pushforward of a vector field is well-defined. if $ F:N \rightarrow M $ is a smooth map between manifolds, the pushforward of a tangent vector $ X_p \in T_PN $ is given ...
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1answer
36 views

prove that any Jordan measurable set in $\mathbb{R}^n$ under Diffeomorphism stays Jordan measurable set

prove that any Jordan measurable set in $\mathbb{R}^n$ under Diffeomorphism stays Jordan measurable set So what I thought is to split the set to bounded sets and show that for each the boundary is ...
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1answer
15 views

Composition of local diffeomorphisms is a local diffeomorphism

Let $F: M\rightarrow N$ , $G:N\rightarrow P$ be local diffeomorphisms, where $M,N,P$ are smooth manifolds. I would like to show that $G\circ F: M\rightarrow P$ is a local diffeomorphism. My attempt: ...
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26 views

Diffeomorpishm preserves dimension

Theorem:Suppose $U \subset \mathbb{R}^n, V \subset \mathbb{R}^m$ are open and $f: U \to V$ is a diffeomprhism with inverse $g : V \to U$. Then (i) $n=m$, and (ii) at each $x \in U$ the differentials ...
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1answer
74 views

Diffeomorphism from $\mathbb{R}^m\to\mathbb{R}^n$

I have a question about diffeomorphism between $\mathbb{R}^m$ and $\mathbb{R}^n$. From this page of the internet we have the following definition: Let $U\subseteq\mathbb{R}^m$ and $V\subseteq\mathbb{...
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1answer
21 views

Dimension of an open subset of a submanifold?

Suppose that $S$ is an embedded/regular submanifold of $M$ with $\mathrm{dim}\ S = s < \mathrm{dim}\ M$. If $U$ is an open subset of M, then $S' = U \cap S$ is an open subset of $S$ in the subspace ...
1
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1answer
32 views

$\mathbb{Z}$-invariant function on $\mathbb{R}^3$ manifold.

Let $\phi_k : k\in \mathbb{Z}, \phi: \mathbb{R}^3\rightarrow\mathbb{R}^3$ be a family on mappings given by: $$\phi_k (x,y,z)=(x+k,e^k\cdot y, e^{-k}\cdot z).$$ I've proven the following statements: $\...
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1answer
31 views

Checking that the obvious map $SO(n, \mathbb{R}) \times \mathbb{R}^{n(n+1)/2 - 1} \to SL(n, \mathbb{R})$ is a diffeomorphism.

It's well-known that $SL(n, \mathbb{R})$ is diffeomorphic to $SO(n,\mathbb{R}) \times \mathbb{R}^{n(n+1)/2-1}$. The argument goes like this: Let $T^+$ be the subset of $SL(n,\mathbb{R})$ consisting of ...
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1answer
15 views

is a diffeomorphism regular?

I have learned the inverse function theorem which ensures that a regular mapping (which has its inverse) is a (local) diffeomorphism. But I wonder whether a diffeomorphism is regular. I guess the ...
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1answer
32 views

Tangent space of a group of diffeomorphisms

In a paper I was reading the following result was used: Let $\Gamma= Diff^{+}([0,1]^2)$ be the set of all boundary preserving diffeomorphisms on $[0,1]^2$, then the Tangent space $\mathcal{T}_{\gamma_{...
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0answers
30 views

Is the parameterization of a regular surface a local diffeomorphism?

Let $\left\{ U_{\alpha },\varphi _{\alpha }\right\}$ be a local chart of a regular surface S $\subset \mathbb{R}^{3}$. That is: $\varphi_{\alpha}:U_{\alpha}\subseteq\mathbb{R}^2\rightarrow \varphi_{\...
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1answer
17 views

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 ...
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0answers
19 views

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 $U$ ...
7
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1answer
120 views

Approximation of smooth diffeomorphisms by polynomial diffeomorphisms?

Is it possible to (locally) approximate an arbitrary smooth diffeomorphism by a polynomial diffeomorphism? More precisely: Let $f:\mathbb{R}^d\rightarrow\mathbb{R}^d$ be a smooth diffeomorphism. For ...
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0answers
28 views

Smooth image of a zero measure set is of measure zero

I'm trying to proof that given a diffeomorphism $f:\mathbb{R}^n\to\mathbb{R}^n$, the image of a set of measure zero, is also a set of measure zero. I've seen several answers to similar questions here ...
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1answer
30 views

Does any oriented atlas of a manifold have maps diffeomorphic to either $\mathbb{R}^n$ or the upper half plane in $\mathbb{R}^n$?

I came across this claim: let $M$ be an oriented manifold of $\dim M=n$ and $\mathcal{A}$ an atlas for $M$. Then any $U \in \mathcal{A}$ is diffeomorphic to either $\mathbb{R}^n$ or $\mathbb{H}^n:= \...
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0answers
10 views

How to prove this aplication between surfaces is bijective

I have two different surfaces: $$ S=\{(x,y,z)\in\mathbb{R}:z=x^2+y^2\} $$ $$ C=\{(x,y,z)\in\mathbb{R}:y=x^3\} $$ A candidate to diffeomorphism between them is $F:S\longrightarrow C$, defined by $F(x,...
2
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1answer
31 views

Flows are Stable Under Diffeomorphism

In chapter 2 of this book (entitled: The Simplicity Of Diffeomorphism Groups) the author says that given any compactly supported smooth vector field $V$ on a simply connected and connected (finite-...
3
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0answers
24 views

Jacobian Matrix and Exponential Map?

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a diffeomorphism. Can $f$ always be represented as $$ f(x)=\exp(F(x))x, $$ where $F:\mathbb{R}^n\to Mat_{n\times n}$? Intuition/Direction Somehow it seems ...
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1answer
39 views

diffeomorphism which is not a translation of the integral curve for some vector field.

For some smooth path-connected manifold $M$, is there any diffeomorphism which could not be represented by the translation of the integral curve for some vector field $X$ on $M$? Also, is there an '...
3
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1answer
48 views

If $f$ is a $C^1$-diffeomorphism between open subsets of Banach spaces, is ${\rm D}f$ surjective at each point?

Let $E_i$ be a $\mathbb R$-Banach space, $\Omega_i\subseteq E_i$ be open and $f:\Omega_1\to\Omega_2$ be bijective. If $f$ is Fréchet differentiable at $\omega_1\in\Omega_1$ and $f^{-1}$ is Fréchet ...
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0answers
19 views

If $f\colon X \to Y$ is one-to-one and a local diffeomorphism, then $f$ is diffeomorphic to an open subset of $Y$.

This question is taken from the book "Differential Topology" by Allan Pollack. It is question 5 from section 1.3 and $X$ and $Y$ are manifolds. It is not for an assignment or a homework. I am just ...
0
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1answer
32 views

Set of fixed point for the self-diffeomorphism in manifold

For smooth manifold M, does the set of fixed point for the element in $\sigma \in Diff(M)$ in $M$ become a manifold? How about a sub-manifold of M?
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0answers
27 views

Lipschitz Constant of Exponential Map on Finite-Dimensional Subgroup of Diffeomorphism Group

Consider the setup of this question. I recapitulate (+small modifications) the setup of that question's OP: Let $M$ be a connected, simply connected, (finite-dimensional) non-compact manifold (...
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0answers
27 views

Transforming differentiable, non-injective functions that are not compactly supported (part 2)

Let $m,n\geq 2$, and $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ be a $C^k$ function that is not compactly supported and not injective. Is it always possible to find a diffeomorphism $\phi$, such that $...
0
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1answer
8 views

Transforming differentiable functions that are not compactly supported

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ be a $C^k$ function that is not compactly supported. Is it always possible to find a diffeomorphism $\phi$, such that $\phi\circ f$ is compactly ...
3
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1answer
33 views

Reference Requiest: Representation Theory of Diffeomorphism Groups

This wonderful and insightful Wikipedia page contains a lot of interesting facts about representations of diffeomorphism groups. However, there are no references. In general, I'm curious if $...
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2answers
56 views

Understanding diffeomorphism concept. [closed]

I understood why homeomorphism is isomorphism in category of topological spaces. Because the structure we are interested about topological space is open set and the way homeomorphism is defined i.e. ...
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1answer
52 views

Diffeomorphism of $\Bbb R^2-\{0\}$ with a subspace of $\Bbb R^2\times \Bbb RP^1$

Consider the subspace $Y$ of $\Bbb R^2\times \Bbb RP^1$ defined by $$ Y=\{(x_1,x_2,[y_1:y_2])\in\Bbb R^2\times \Bbb RP^1 : x_1y_2=x_2y_1\} $$ This is clearly a well-defined subspace of $\Bbb R^2\...
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1answer
66 views

Application of Implicit Function Theorem to a function $\psi:U\subset\Bbb{R}^{2}\rightarrow \Bbb{R}^{4}$

Let $U$ be an open subset of $\Bbb{R}^{2}$ and \begin{align*}\psi:U&\rightarrow\Bbb{R}^{4}\\ x\mapsto &(\psi_1(x),\psi_2(x),\psi_3(x),\psi_4(x)) \end{align*} a $\mathcal C^1$ function. ...
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0answers
54 views

Intuitive notion of diffeomorphism

Until now, I had thought (intuitively) that two objects were diffeomorphic if and only if they had the same number of peaks, but I think that I have found a counter example for this intuition. ...
0
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1answer
16 views

Is $f$ a diffeomorphism

$f:\Bbb{R}^{2}\to \Bbb{R}^{2}$ be the function $f(x,y)=(e^{x}cosy,e^{x}siny)$. If $f$ is diffemorphism, then i need to show that $f$ is bijective and $f$, $f^{-1}$ are smooth. I know that the Jacobian ...
2
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0answers
67 views

$f:U\rightarrow V$ a $C^2$ diffeo, $\forall\, a\in U, \exists r>0$ such that $f\left(B_{a}(\epsilon)\right)$ is convex $\forall\epsilon\leq r$

Let $U,V$ open sets in $\Bbb{R}^n$, $f:U\rightarrow V$ a diffeomorphism of class $C^2$. I need to prove that, for all $a\in U$, exists $r>0$ such that the image of the open ball centered at $a$ ...
2
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1answer
53 views

Why is this mapping a diffeomorphism?

Let $\varphi:\mathbb{R}^2 \rightarrow \mathbb{R}$ satisfy $$\varphi_{xx}\varphi_{yy} - \varphi_{xy}^2 = 1$$. Then define the mapping $(x,y) \mapsto (\xi, \eta)$ by: $$ \xi(x,y) = x+\varphi_x(x,y)\:, \...
2
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1answer
40 views

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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1answer
40 views

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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0answers
22 views

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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0answers
39 views

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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1answer
94 views

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 ...
2
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1answer
46 views

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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0answers
12 views

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 ...
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0answers
31 views

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 $\alpha$...
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0answers
27 views

On the statement of the characteristic property of surjective smooth submersion in John Lee's Smooth Manifolds

In John Lee's Introduction to Smooth Manifolds, he states a characteristic property of surjective smooth submersions as for quotient maps. He refers to this "characteristic" property in Problem 4-7, ...

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