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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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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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Hamiltonian diffeomorphisms on a Poisson manifold

Let $(M,\{-,-\})$ be a Poisson manifold. An Hamiltonian isotopy is a smooth family of diffeomorphisms $\{\varphi^t:M\to M\}_{t\in [0,1]}$ such that $\varphi^0=\text{id}_M$ there exists a smooth ...
Kandinskij's user avatar
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Is a diffeomorphism between two manifolds just a diffeomorphism between the sets of the manifolds

I'm reading Intro to GR by Sean Carroll, and in that book he defines a $C^\infty$ n-dimensional manifold as a set M along with a maximal atlas. In the same chapter, he also defines a diffeomorphism ...
Chidi 's user avatar
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Proving $\phi$ is a smooth map and constructing an explicit isometry

Consider a Lorentzian manifold $(\zeta,g)$ with metric: $$g=\frac{dudv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2}.$$ For $u,v,w,r,>0$. Suppose we take a Cauchy foliation of $\zeta,$ called $\mathcal F$,...
John Zimmerman's user avatar
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Why are local submersions and local immersions not studied in Lee's Introduction to Smooth Manifolds?

I'm reading through Lee's Introduction to Smooth Manifolds and wondered why local submersions or local immersions are not studied. Let $M$ and $N$ be smooth manifolds (with or without boundary) and ...
Sam's user avatar
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Is it possible $S^n\cong \mathbb{RP}^n$? [closed]

I found that $S^1\cong \mathbb{RP}^1$. I wonder if this can happen for any dimension other than 1. I have inquired about it but have had no response....
3435's user avatar
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Isotopic diffeomorphisms of surface are diffeotopic?

Due to Epstain, if two homeomorphisms $f_0, f_1$ of a closed two-dimensional manifold $S$ are homotopic then they are isotopic. I've heard, that if $f_0, f_1$ are diffeomorphisms then the isotopy can ...
Elena's user avatar
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If $x:U\subset\mathbb{R}^2 \to S$ is a local map, then $U$ and $x(U)$ are diffeomorphic.

First, this is the definition of regular surface I'm using. A subset $S\subset\mathbb{R}^3$ is a regular surface if, for each $p\in S$, there exist a neighborhood $V\subset\mathbb{R}^3$ and a map $x:...
Fabrizio Gambelín's user avatar
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Diffeomorphic to $\mathbb{R}^2\times S^1$ [duplicate]

Let $M_c=\{(x_1, x_2, x_3, x_4)\in \mathbb{R}^4 \ |\ x_1x_2+x_2x_3+x_3x_4=c\}.$ Prove that $M_c$ is diffeomorphic to $\mathbb{R}^2\times S^1$ if $c\neq 0$ . I tried to get a diffeomorphic mapping, but ...
Rosalina's user avatar
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Definition of uniformly $C^1$-regular boundary: Why last coordinate of diffeomorphism $\mathbb{e_d} \cdot \Phi(y) >0 \iff y \in \Omega $

In my partial differential equations class, we introduce Sobolev spaces. We define that for an open, connected set $\Omega$ to have a uniformly $C^1-$regular boundary, it means that there are positive ...
Len's user avatar
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Why $f(x)=x^3$ is not diffeomorphism?

Currently I am reading Tu's book 'An introduction to Manifolds', and I have some problem about the Proposition 6.10 in the book. I have already known that $f(x): x \mapsto x^3$ is not diffeomorphism ...
Tanblade's user avatar
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Smooth map is diffeomorphism iff its pullback is an algebra isomorphism?

newbie here in Diff Geo/Man! I am given topological spaces $M$ and $N$, their algebras of continuous functions, $C(M), C(N)$, respectively, and a map $F^* = — \circ F : C(N) \to C(M)$, which is easily ...
Jos van Nieuwman's user avatar
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Boundary case of adapted orthonormal frame exercise in Professor Lee's Introduction to Riemannian Manifolds (IRM)

In the post Adapted Orthonormal frame on Riemann manifold, the poster offered the outline of a proof for IRM Proposition 2.14. I can fill in the details to make that proof work when the embedded ...
Jeff Rubin's user avatar
6 votes
1 answer
135 views

Uncertain about the statement "equivalent condition for an isometry of Riemannian manifolds" in Lee's Intro to Riemannian Manifolds

In Professor Lee's Introduction to Riemannian Manifolds, second edition on page 12, the first paragraph on Isometries reads Suppose $(M,g)$ and $(\tilde{M},\tilde{g})$ are Riemannian manifolds with ...
Jeff Rubin's user avatar
4 votes
2 answers
58 views

Diffeomorphisms of $(-1,1)^n$ sending fixed point to origin

Consider $I^n\equiv(-1,1)^n:=\{(x_1,\dots,x_n)\in \mathbb{R}^n: |x_i|<1 \ \text{for each} \ 1\leq i\leq n\}.$ Let's fix a point $a=(a_1,\dots,a_n)\in I^n$. Question: How to construct a $C^1$-...
RFZ's user avatar
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A complete closed uni-leaf foliation on $\Bbb B^2$

In foliations on the open 3-ball by complete surfaces, Inaba and Masuda give an example for $\Bbb B^2$ at the end of page $2.$ I would like to understand this example first, before I keep progressing ...
John Zimmerman's user avatar
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Are diffeomorphisms induced by non-autonomous flows the same as autonomous flows on a compact subset?

Short version (as suggested by Mariano, and as in title): For every diffeomorphism $f: \Omega \rightarrow \Omega$, (where $\Omega$ is a compact subset of $\mathbb{R}^n$) that is the time-1 map of a &...
Rohit Jena's user avatar
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2 answers
82 views

Diffeomorphism from $\{(x, y) \in \mathbb R^2 : 0 < x^2 + y^2 < 1\}$ to $\{(x, y) \in \mathbb R^2 : x^2 + y^2 > 1\}$

Give an example of a diffeomorphism of the class $C^1$ that takes the set $\{(x, y) \in \mathbb R^2 : 0 < x^2 + y^2 < 1\}$ on $\{(x, y) \in \mathbb R^2 : x^2 + y^2 > 1\}$. I did this task ...
qerty149's user avatar
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2-modulus of curves over a trapezoid

Definition. For a real number $p \geq 1$, the $p$-modulus of a curve family $\Gamma$ in a metric measure space $(X,d,\mu)$ is $$\operatorname{mod}_p(\Gamma) := \inf\left\{\int_X \rho^p \mathrm{d}\mu : ...
Philippe Knecht's user avatar
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Prove that $\varphi\circ X_1$ is also a parametrization if $\varphi$ is a diffeomorphism and $X_1$ is a parametrization

Suppose that $X_1:U_1\subset R^2 \to S_1$ and $\varphi:S_1 \to S_2$ is a diffeomorphism, I want to prove that $\varphi \circ X_1:U_1 \to S_2$ is a parametrization of $S_2$ Here is my attempt: Suppose ...
Gang men's user avatar
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Open neighborhoods for diffeomorphism.

I have a question regarding the following task: Examine whether the mapping $g: \mathbb{R}^3 \rightarrow \mathbb{R}^3$, defined by $$(r,\phi,\theta) \mapsto (r \sin(\theta) \cos(\phi), r \sin(\phi) \...
alex.'s user avatar
  • 47
2 votes
2 answers
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Proving a certain map on $S^n$ is a diffeomorphism

Define $f:S^n\to S^n$ by $x\to -x$. Prove this is a diffeomorphism. My attempt: My definition of a smooth map between two manifolds $M,\,N$ is that a map $g:M\to N$ is smooth iff the maps: $$\psi_i\...
Gal Barkol's user avatar
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1 answer
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Can a diffeomorphism of the circle with irrational rotation number have vanishing derivative?

I'm reading about Denjoy's theorem right now and noticed a few different forms. If $f$ is a diffeomorphism of $\mathbb{S}^1$ with bounded derivative and irrational rotation number, it is sometimes ...
Gecko1111's user avatar
3 votes
0 answers
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Question about the existence of a diffeomorphism between nonsmooth domains in $\mathbb{R}^2$

Suppose $\Omega_1 \subset \mathbb{R}^2$ is an $n$-sided convex polygon while $\Omega_2 \subset \mathbb{R}^2$ is simply connected with a piecewise smooth Lipschitz boundary comprised of $n$ smooth ...
Yamamoto's user avatar
2 votes
1 answer
223 views

Give an example of a diffeomorphism between $\{ (x, y) \in \mathbb R^2: 1 < x^2 +y^2 < 2, x < y < 2x \}$ and a square

Give an example of a diffeomorphism of the class $C^1$ that takes the set $\left\{ (x, y) \in \mathbb R^2: 1 < x^2 +y^2 < 2, x < y < 2x \right\}$ on square $\left\{ (x, y) \in \mathbb R^2: ...
qerty149's user avatar
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Diffeomorphisms (fixing $p,q$) preserving the smooth structure on the class of all closed embedded surfaces $S \subset [0,1]^3$

Consider the space of diffeomorphisms between $X$ and $Y$: $$\mathrm{Diff}(X,Y)$$ For $X,Y:=[0,1]^3,$ such that $\mathrm{Diff}(X,Y)$ are maps preserving the smooth structure on the class of all closed ...
John Zimmerman's user avatar
1 vote
0 answers
30 views

Do the orbits of the flow of a vector field change when multiplied by a function? [duplicate]

Assume we have a vector field $X$ on open subset $U\subseteq\mathbb{R}^n$ containing the point $0 \in \mathbb{R}^n$, and let $f: U\rightarrow\mathbb{R}$ be a smooth function. I want to know whether ...
Khashayar's user avatar
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0 answers
46 views

Local diffeomorphisms and orientability

Given a differentiable map $f \colon S_1 \to S_2$ between regular surfaces such that $df_p$ is an isomorphism for each $p \in S_1$, I want to study whether the following statements are true or false: ...
David's user avatar
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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's user avatar
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2 votes
1 answer
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Flowing a vector field along a diffeomorphism

For simplicity I am happy to work on Euclidean space $\mathbb{R}^d$. Let $X$ be a vector field on $\mathbb{R}^d$, which I think of as a first-order differential operator (that is, I would write $Xf$ ...
felipeh's user avatar
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2 votes
1 answer
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Can non-homeomorphic topological manifolds be built over the same set?

Let $(M,\mathfrak{X})$ be a smooth manifold, where $M$ is a topological manifold and $\mathfrak{X}$ a smooth structure on it. It is commonly mentioned (in introductory resources on smooth manifolds) ...
qubitsandwich's user avatar
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Limit of diffeomorphism group of n torus

I am taking a second course in topology (fundamental group and covering spaces). I had a small discussion with the teacher, and in passing he said: "The limit of diffeomorphism groups of the n ...
DevVorb's user avatar
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What is meant by the cross section of an orbit?

In Folland's quantum field theory book he says: Let $\{\sigma_t : t \in \mathbb{R}\}$ be a one-parameter group of measure-preserving diffeomorphisms of $\mathbb{R}^n$ whose orbits are (generically) ...
CBBAM's user avatar
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Prove that a transformation is a diffeomorphism and a flow. Solution verification

Let $F: \mathbb{R}^n \to \mathbb{R}$ be an integrable function, and $\phi: \mathbb{R}^n \to \mathbb{R}^n$ be a smooth function with compact support (i.e., vanishing outside a certain ball). Justify ...
thefool's user avatar
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3 votes
2 answers
181 views

Uniqueness of smooth structures on submanifolds with boundary

In Professor Lee's Introduction to Smooth Manifolds (Second Edition), he states and proves Theorem 5.31, which guarantees that the smooth structure on an embedded or immersed submanifold of a smooth ...
Jeff Rubin's user avatar

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