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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.

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Change of coordinates in a regular surface $S$.

I'm quite confused in the following situation: Suppose that $S \subset \mathbb{R}^3$ is a regular surface. Let $U_1$ and $U_2$ be open sets in $\mathbb{R}^2$, and consider the pair of ...
Joel Marques's user avatar
2 votes
1 answer
37 views

Why are quasiconformal maps orientation preserving under the analytic definition?

Quasiconformal maps admit a lot of equivalent definitions. One of the geometric definitions states that a homeomorphism $f$ is quasiconformal, iff it preserves orientation and changes the modulie of ...
Beno Učakar's user avatar
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1 answer
48 views

Question about Lemma 19.1 in Munkres' Analysis on Manifolds

In Munkres' Analysis on Manifolds, page 162 Lemma 19.1 Step 2 it states: Third, we check the local finiteness condition. Let $\mathbf{x}$ be a point of $A$. The point $\mathbf{y}=g(\mathbf{x})$ has a ...
studyhard's user avatar
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9 votes
1 answer
177 views

Contradiction in Computation of Homology Groups of the Mapping Class Group of a Surface?

One of the two main results of a paper by Nathalie Wahl on homological stability of the mapping class group of a surface is the following: Theorem 1.2 The map $H_*(\delta_g) : H_*(\Gamma_{g,1};\mathbb{...
jasnee's user avatar
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1 vote
1 answer
50 views

Inverse of smooth family of diffeomorphisms [closed]

Let $M$ be a smooth manifold, and let $I$ be an open interval. We say a map $\varphi:I \rightarrow \text{Diff}(M)$ is a smooth family of diffeomorphisms if the map $I \times M \rightarrow M$ given by $...
Joseph Kwong's user avatar
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0 answers
36 views

Question 5, section 19 of Munkres Analysis on Manifolds

Let $A$ be an open rectifiable set in $\mathbb{R}^{n-1}$. Given the point $p$ in $\mathbb{R}^n$ with $ p_n > 0 $, let $ S $ be the subset of $\mathbb{R}^n$ defined by the equation $$ S = \{ x \mid ...
Emanuelly Santos Lima's user avatar
2 votes
0 answers
48 views

Do diffeomorphisms map increasingly smaller sets to increasingly "smaller" sets?

Let $\Omega \subset \mathbb R^n$ be an open set. Let $\phi: \Omega \rightarrow \Omega$ be a diffeomorphism. Suppose $y \in \Omega$ is such that for each $n \in \mathbb N$, $\overline{B(y, 1/n)} \...
rosecabbage's user avatar
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52 views

Diffeomorphism between half plane and plane

Let $H$ be the half plane of $\mathbb{R}^n$, that is the set of $x\in \mathbb{R}^n$ such that $x_n\geq 0$. My aim is to prove that there are no $C^1$-diffeomorphism from $\mathbb{R}^n$ to $H$. EDIT: ...
Laurent Claessens's user avatar
1 vote
1 answer
49 views

Extending Special Automorphisms of Lie Manifolds

I call a smooth manifold $M$ special if it is diffeomorphic to a connected Lie group $G_M$ of automorphisms of $M$. Let $M$ be a special smooth manifold and $N$ a special smooth submanifold of $M$. $...
Jannis's user avatar
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Extending Diffeomorphisms of Lie Manifolds

Let $M$ be a connected smooth manifold that is diffeomorphic to a Lie group $G$ and $N$ be a connected smooth submanifold of $M$ that is diffeomorphic to a Lie group $H$. Can every diffeomorphism of $...
Jannis's user avatar
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1 answer
38 views

Why is this local chart on a fibre bundle $(E, M, \pi, L)$ compatible with the given smooth structure on $E$?

In these lecture notes, at Remark 16.5, Merry states: ''Suppose $(W, \varepsilon)$ is a bundle chart on $E$. Let $(U, x)$ and $(V, y)$ be (manifold) charts on $M$ and $L$ respectively with $W \subset ...
Dave's user avatar
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4 votes
1 answer
62 views

Property of Lipschitz Domains

I have been working on a research problem of mine and came across the concept of Lipschitz domains. I am curious about whether it is possible to show that there always exists a bi-Lipschitz map from $\...
sudeep5221's user avatar
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An example of a fixed point of a diffeomorphism which is not a hyperbolic point

Find an example of a $C^1$ diffeomorphism with a non-hyperbolic fixed point which is an accumulation of other fixed points. A fixed point which is not an hyperbolic point must satisfy that $$f^n(p)=1,...
Superunknown's user avatar
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3 votes
1 answer
76 views

A differentiable homeomorphism that isn't a diffeomorphism?

From the book Introduction to Differentiable Topology by TH Brocker and K Janich. 1.7 Definition: A diffeomorphism is an invertible differentiable map Below: A differentiable homeomorphism need not ...
Cedric Martens's user avatar
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8 views

Symmetry of the partial derivative of a diffeomorphic function

I am wondering if there is symmetry among the single partial derivative of a multivariate diffeomorphic function. Let $f: \mathcal{X} \rightarrow \mathcal{X}$ be a diffeomorphism defined over $\...
ajl123's user avatar
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2 answers
56 views

Whether $\mathbb{T}\times [0,1]$ is diffeomorphic to $\mathbb{D}^2$?

Here $\mathbb{T}$ denotes the torus $\mathbb{R}\backslash \mathbb{Z}$, and $\mathbb{D}^2$ the closed unit ball in $\mathbb{R}^2$. Since $\mathbb{T}\times [0,1]$ and $\mathbb{D}^2$ are both smooth ...
Jiawen Zhang's user avatar
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0 answers
29 views

on the definition of $H^\infty$-diffeomorphisms

The group of orientation-preserving $H^\infty$-diffeomorphisms on $\mathbb{R}^n$ is defined as $$\operatorname{Diff}(\mathbb{R}^n):=\{\operatorname{id}+f\,|\,f\in H^\infty(\mathbb{R}^n,\mathbb{R}^n)\...
Stuck's user avatar
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Differential of a family of diffeomorphisms

Suppose one has a smooth curve $\gamma:\mathbb{R}\to M$ on a manifold $M$ and a smooth one-parameter family $f_t:M\to M$ of diffeomorphisms (not necessarily a one-parameter subgroup). Then, the curve $...
Miguel Wazowski's user avatar
2 votes
2 answers
270 views

Tangent bundle of a sphere $T\mathbb S^n$ is diffeomorphic to $\mathbb S^n \times \mathbb S^n - \Delta$

Let $\mathbb S^n$ denote the $n$-sphere, which is the smooth manifold consisting of all points in $\mathbb R^{n+1}$ with Euclidean norm one. Recall that the tangent bundle of $\mathbb S^n$, denoted $T ...
Joseph Kwong's user avatar
0 votes
0 answers
142 views

Why is the diffeomorphism group a manifold?

Let $M$ be a differentiable manifold. The diffeomorphism group of $M$ is the group of all $C^{\infty}$ diffeomorphisms of $M$ to itself, denoted by $\text{Diff}(M)$. This space of diffeomorphism $\...
Swakshar Deb's user avatar
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1 answer
76 views

Example of a homeomorphism that is not a diffeomorphism between two diffeomorphic surfaces in $\mathbb{R}^3$

Give an example of a homeomorphism that is not a diffeomorphism between two diffeomorphic surfaces in $\mathbb{R}^3$. The only function that came in my mind is $(x,y) \mapsto (x^3, y^3, \sqrt{1-x^6-y^...
Andreadel1988's user avatar
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0 answers
25 views

Are submersions and immersions sometimes, always, or never classifiable as diffeomorphisms?

From Wikipedia: Submersions: In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential ...
Nate's user avatar
  • 894
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38 views

Diffeomorphic Regular surfaces

I came across the following question whilst preparing for an exam. Consider the map $f:\mathbb{R}^{3} \to \mathbb{R}$ given by the polynomial $f(x,y,z)=x^{2}+y^{2}+z^{2}$. Let $s,t \neq 0$. Show that ...
hizerain's user avatar
4 votes
1 answer
52 views

Constructing a Diffeomorphism between a Cone like and cylinder surface

I am told very specifically that a cone has the following definition: $$C_1 = \{(x,y,z) \in \mathbb{R^3} | z^2 = x^2 + y^2 , z > 0\}$$ Please note the $z>0$ part. I want to map it to a cylinder ...
Dr. Ernesto Chinchilla's user avatar
1 vote
3 answers
111 views

Show that $S^2$ is not diffeomorphic to $T^2$ [duplicate]

I know this question have already been answered many times. However I want to prove this using the De Rham cohomology of them, showing that they are not isomorphic and thus $S^2$ and $T^2$ are not ...
NiJuice's user avatar
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1 vote
1 answer
78 views

Diffeomorphism between open surfaces

Is it true that for every pair of surfaces $S_1, S_2$ in $\mathbb{R}^3$, there exist two non-empty open sets $W_1 \subseteq S_1$ and $W_2 \subseteq S_2$ that are diffeomorphic? Here is my attempt: Let ...
Andreadel1988's user avatar
0 votes
2 answers
75 views

Diffeomorphism between intervals implies derivative is either positive everywhere or negative everywhere

Let $\varphi: [c,d] \rightarrow [a,b]$ be a diffeomorphism between two closed intervals in $\mathbb{R}$. I am following a proof that claims that since $\varphi$ is a diffeomorphism either $\varphi' &...
CBBAM's user avatar
  • 6,285
0 votes
0 answers
29 views

Existence of a diffeomorphism matching a finite point cloud transform.

Given a differentiable manifold $M$, a compact subset $U ⊆ M$, a finite subset $C ⊆ U$ and an injective function $f_C: C → U$, when does there exist a diffeomorphism $f: M → M$ that is "supported&...
Lydia Marie Williamson's user avatar
4 votes
0 answers
108 views

Exponential map of an $SO(n)$-invariant metric on $\mathbb R^n$ is diffeomorphism

Let $g$ be a complete $SO(n)$-invariant metric on $\mathbb R^n$. That is, for each $A \in SO(n)$, the map $\varphi_A:\mathbb R^n \rightarrow \mathbb R^n$ given by $x \mapsto Ax$ is a Riemannian ...
Joseph Kwong's user avatar
0 votes
0 answers
54 views

isometric isomorphism implies diffeomorphism

Suppose $E$ be a $\mathbb R-$vector space that is isometrically isomorphic to $\mathbb R^n$ via the map $f:\, E\to\mathbb R^n$, which means $f$ is bijective and \begin{align} d_E(x,y)&=d_{\mathbb ...
PermQi's user avatar
  • 579
1 vote
0 answers
48 views

simply connected Lie groups are isomorphic iff locally isomorphic

I am reading a lecture note on Lie groups, and in the note it states that upon "standard topological considerations", two simply connected Lie groups $G,H$ are locally isomorphic if and only ...
Kai's user avatar
  • 11
2 votes
0 answers
59 views

Some properties of Lipschitz diffeomorphisms [closed]

Suppose I have got two bounded Lipschitz domains $A$ and $B$, both subsets of $\mathbb{R}^n$. Suppose I have a Lipschitz diffeomorphism $F\colon \bar A \to \bar B$. So $F$ is a Lipschitz function ...
math_guy's user avatar
  • 465
0 votes
0 answers
50 views

difeomorphism l<k or $l\geq k$

I have just learned from Sikorski Book diferentiation of several variables this fact: there are no diffeomorphisms of $G\subseteq E^k$ to $G'\subseteq E^l$ for $l<k$ but they do exist for $l\geq k$...
user122424's user avatar
  • 3,978
0 votes
0 answers
65 views

Valid interpretation of Higher order frame bundles and their jet groups?

I've been trying to develop an intuition for higher order frame bundles to help me understand them and this is what I've come up with. Criticisms welcome, as I'm not sure it's valid? NOTE: Always I ...
R. Rankin's user avatar
  • 340
0 votes
1 answer
85 views

Condition for section of the frame bundle of $M$ to be the 1-jet of a local diffeomorphism $\phi: M \rightarrow \mathbb{R}^n$?

We're given an n-dimensional Riemannian manifold $M$ and its frame bundle $FM$. The tangent bundle can be locally regarded as an invertible map $\phi:M\rightarrow\mathbb{R}^{n}$. Let's confine $\phi$ ...
R. Rankin's user avatar
  • 340
1 vote
0 answers
80 views

Closed curve in a torus diffeomorphic to a circle?

In Example 15.9 of Tu's "An Introduction to Manifolds: Second Edition", it is written Let G be the torus $R^2/Z^2$ and L a line through the origin in R2. The torus can also be represented ...
Foo's user avatar
  • 11
0 votes
1 answer
56 views

Isomorphism and local diffeomophism

If $S_1,S_2$ are regular surfaces and $\varphi:U\subset S_1\rightarrow S_2$ is a differentiable map such that the differential $d\varphi_p$ of $\varphi$ at $p\in U$ is an isomorphism, then $\varphi$ ...
F.Liu's user avatar
  • 3
2 votes
1 answer
87 views

A question related to the action of a group on the real projective space

I am trying to make sense of the following proof showing that $\mathbb{P}\simeq \text{O}(n+1)/(\text{O}(1)\times \text{O}(n)),$ where $\mathbb{P}:=\mathbb{P}^n(\mathbb{R})$ is the $n$-dimensional real ...
neophyte's user avatar
  • 520
0 votes
0 answers
43 views

Is there an ill-embedded ball in the 4-sphere?

In https://arxiv.org/pdf/2102.04391.pdf, there is an explanation of how one could theoretically use a pair of knots $K$ and $K'$ (one slice and the other not) with the same 0-surgery to generate a ...
horned-sphere's user avatar
3 votes
1 answer
76 views

Existence of an isomorphism between $X$ and $Y$ when $g(X)=f(Y)$

Assume we have regular real random variables $X$ and $Y$ and real-valued functions $g,f \in \mathcal{G}$. Assume that $g(X)=f(Y)$ almost surely. I would like to know under which conditions on $\...
Julien's user avatar
  • 153
3 votes
0 answers
51 views

Representations of a Function of a random variable

Assume we have a real random variable $Y\in \mathbb R^p$, such that $Y= g(X)$ where $X \in\mathbb R^n $, $n< p$, is some random variable with continuous distribution $P$ and $g:\mathbb R^n \...
Julien's user avatar
  • 153
0 votes
0 answers
131 views

Prove $f(x) = x/(1+|x|)$ is a diffeomorphism.

Let $|{}\cdot{}|$ be the Euclidean norm on $\mathbb{R}^{n}$ and $f(x) = x/(1+|x|)$, $x \in \mathbb{R}^n$. Prove $f$ is a $C^{1}$ diffeomorphism on $\mathbb{R}^n - \{0\}$. Here is my sketch of proof: $...
Irbin B.'s user avatar
  • 172
3 votes
1 answer
82 views

Diffeomorphism from the unit disc to the $n$-dimensional Euclidean space

Let $f: B^n \to \Bbb R^n$ be a map from the unit disc given by $x \mapsto \dfrac{x}{\sqrt{1-\|x\|^2}}$. Show that this map is a diffeomorphism. So I'm trying to find a way to derive the inverse for ...
dejinka's user avatar
  • 71
1 vote
0 answers
34 views

embedding continuous function as diffeomorphism

For any $\varepsilon>0$, any continues function $f$ from $[0,1]^2\times \{0\}$ to $\mathbb{R}^2 \times \{0\}$, can we find a diffeomophism $\Phi$ of $\mathbb{R}^3$ such that $$ \|f(x) - \Phi(x)\| &...
Lightmann's user avatar
  • 313
2 votes
0 answers
69 views

Function on "figure eight" which is not a diffeomorphism due to its topology

In my lecture, we've considered $G(x)=F(g(x))$ the "figure eight", where $F(x) = (2\cos(x-\frac{1}{2}\pi), \sin(2(x-\frac{1}{2}\pi))$ and $g(x)=\pi+2\arctan(x)$. I understand that this is ...
notimportant's user avatar
1 vote
1 answer
138 views

Explaining the tangent bundles of $S^n$ for $n=1,3,7$

I have seen several posts on the tangent bundles of $S^1$, and $S^3$. Basically, it seems that the idea is to find $n$ linearly independent smooth vector fields. In particular, for $S^1$, we pick $x\...
Ook's user avatar
  • 211
3 votes
1 answer
72 views

Extension of Homeomorphism of boundaries

Let $A,B \subset \mathbb{R}^2$ be two open bounded sets with smooth boundary (boundary is a smooth closed curve). Let us assume that we know $$\Phi \colon \partial A \to \partial B$$ is a ...
Rooibos's user avatar
  • 179
0 votes
0 answers
16 views

Locally conjugate diffeomorphisms, but does not take orbits to orbits.

I'm working on the Palis's ''Geometrical Theory of Dynamical Systems''. I have the next problem: Let $X$ and $Y$ be $C^1$ vector fields on $\mathbb{R}^m$. Suppose that $0$ is an attracting hyperbolic ...
Nestor Bravo's user avatar
1 vote
0 answers
158 views

Show that the lens space is a smooth 3-manifold

Suppose we view $S^3\subset C^2$. Then for coprime integers $p,q$ we define the lens space by $M_{p,q}=S^3/\sim$ where $(z_1,z_2)\sim(z_1e^\frac{2\pi i}{p},z_2e^\frac{2\pi iq}{p})$. I want to show ...
Ook's user avatar
  • 211
0 votes
0 answers
77 views

Why a vector field on a manifold defines a 1-1 mapping on the manifold?

I am reading paragraph 3.1 (Introduction: how a vector field maps a manifold into itself) of chapter 3 (LIE DERIVATIVES AND LIE GROUPS) of the book Geometrical Methods of Mathematical Physics, by B. ...
green's user avatar
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