Questions tagged [diffeomorphism]
The diffeomorphism tag has no usage guidance, but it has a tag wiki.
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Does there exist a diffeomorphism satisfying this condtion?
Let $a_1,...,a_k,b_1,...,b_k \in \mathbb{R}^n$ and $p_1,...,p_k \in (0,\infty)$. ($a_1,...,a_k$ are distinct and $b_1,...,b_k$ are distinct)
Then, does there exist a diffeomorphism $f:\mathbb{R}^n \...
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0answers
29 views
Requirements on Coordinate Change
Let $\Omega$ be a differentiable $n$-manifold, let $f:\Omega\to\mathbb R$ be Riemann integrable, let $u:\Omega\to\Gamma\subset\mathbb R^n$ be a chart, and let $v:\Lambda\subset\mathbb R^n\to\mathbb R^...
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2answers
29 views
Prove that if $f$ is a diffeomorphism than its differential $D_{f}$ is an isomorphism
I want to prove that if $f : U \to V$ , when $U,V \subset R^n$, is a diffeomorphism than its differential $D_{f}$ is an isomorphism.
Since $D_{f}$ is a linear transformation, isomorphism means that ...
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0answers
36 views
Diffeotopy group, Mapping Class group, Isometry group
There are several closely related concepts on the symmetries or symmetry groups of the space.
My apology, but some vague imprecise definitions may be as:
Mapping class group (MCG) is an important ...
3
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3answers
100 views
Diffeomorphism and homeomorphism between open ball and polydisc in $\mathbb{R}^n$
Recently I came to know that for $n\geq 2$, the open ball $B(0,1) = \{z\in \mathbb{C}^n: |z|<1\}$ and the polydisc $\Delta(0;1) = \{z\in \mathbb{C}^n: |z_j|<1, j=1,\ldots, n\}$ are not ...
3
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0answers
55 views
Nonzero Jacobian implies diffeomorphism? [duplicate]
Suppose $f: U\to V$ is a $C^1$ surjection from a $k$ dimensional manifold $U$ onto a $k$ dimensional simply connected manifold $V$ with everywhere nonzero Jacobian or nonsingular differential. Is $f$ ...
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0answers
46 views
Show that embeddings, diffeomorphisms, etc. are stable classes of maps
This is part of Problem 16 in Chapter 6 of Lee's Smooth Manifolds.
Let $N,M,S$ be smooth manifolds. A smooth family of maps is a collection $\{F_s:N\to M \;|\; s\in S\}$ such that $F_s(x)=F(x,s)$ ...
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0answers
54 views
Diffeomorphism Between Surfaces Preserves Orientability
From Do Carmo (Exercise 2.6.2).
Let $S_2$ be an orientable regular surface and $\varphi:S_1\rightarrow S_2$ be a local diffeomorphism at every $p\in S_1$. Prove $S_1$ is orientable.
Up until this ...
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2answers
50 views
Showing that there is not a global diffeomorphism between unit quaternions and $\mathrm{SO}(3)$
I am new to differential geometry. I have the following question:
Let $\mathbf{Q}$ denote the set of unit quaternions. I already have proved using the implicit function theorem that $\mathbf{Q}$ is a ...
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1answer
63 views
Relation between the eigenvalues of matrices conjugated by a homeomorphism.
Let $A, B$ be $2\times 2 $ matrices satisfying:
The eigenvalues $\lambda,\mu$ of $A$ satisfy $|\lambda|<1<|\mu|$.
The eigenvalues $\lambda',\mu'$ of $B$ satisfy $|\lambda'|<1<|\mu'|$.
...
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1answer
59 views
Diffeomorphism Preserves Tangency (Do Carmo 2.4.25)
Suppose $C_1$ and $C_2$ are regular curves on a regular surface $S$. Suppose $p$ is a point in $S$ where $C_1$ and $C_2$ are tangent, then if $\varphi:S\rightarrow S$ is a diffeomorphism, prove that $\...
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0answers
22 views
Change in unit normal of moving surface
Suppose I have a smooth open region $W\subseteq \mathbb{R}^3$ that is being continuously acted upon by a family of diffeomorphisms $\{\phi_t:\mathbb{R}^3\to\mathbb{R}^3\ : \ t\in\mathbb{R}\}$, where $\...
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1answer
50 views
Show that if $\left|T\left(x\right)-T\left(y\right)\right|\geq c\left|x-y\right|$ then $v\left(T\left(E\right)\right)\geq c^{n}v\left(E\right)$
Let $c>0$ be a real positive number and let $T\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$
be a diffeomorphism such that $\left|T\left(x\right)-T\left(y\right)\right|\geq c\left|x-y\right|$.
I wish to ...
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1answer
35 views
For all matrices A exists unique matrix B with $B^2=A$ in a sufficient neighborhood.
Show existence of $\epsilon > 0, \delta > 0$ such that for all $A\in U_\epsilon(Id_n)$ exists unique $B \in U_\delta(Id_n)$ with $B^2=A$.
$U_\epsilon(Id_n):=\{X \in M_{n\times n}| \Vert X-Id_n ...
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0answers
15 views
How do I determine which $V$ to exclude to obtain a Diffeomorphism
In order to use the Lebesgue Transformation Formula, I need to find a $V$ to exclude, but I am unsure of how exactly to do it.
Example:
Spherical Coordinates
$\Phi: \mathbb R_{>0 }\times]0,\pi[\...
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0answers
19 views
Trouble with Spherical Coordinates and the Transformation Formula
I know for a function $f: \mathbb R^{3} \to \mathbb R$ that is integrable
Then we have a coordinate map:
$\Phi: \mathbb R_{\geq 0}\times[0,\pi]\times[0,2\pi] \to \mathbb R^{3}, \Phi(r,\theta, \phi):=...
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1answer
12 views
Meaning of $h_*$ complex map.
I have come across the following:
Let $\Omega$ be a convex domain in $\mathbb{C}^n$ and let $h: \Omega \rightarrow \mathbb{C}^n$ (or $\mathbb{R}^n$) be a smooth mapping with $||h_*||= sup_x || h_*(x)|...
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1answer
48 views
Diffeomorphism between the unit ball to itself
Let $T:B\to B$ be a diffeomorphism , where $B\subset\mathbb{R}^n$ is the unit ball.
I want to show there exists $x\in B$ such that $|J_T(x)|=1$ ($|J_T|$ is the absolute value of the Jacobian of $T$)
...
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1answer
21 views
$C^1$-diffeomorphism $\implies \parallel (D\varphi)^{-1} \parallel$ is bounded
Let $U,V$ be open subsets of $\mathbb{R}^n$ and $\varphi:U \rightarrow V$ a $C^1$-diffeomorphism. We know that $\varphi, D \varphi$ and $(D\varphi)^{-1}$ are defined on compact set $K$ with $U \subset ...
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1answer
52 views
Prove $\Phi$ is a diffeomorphism
Let $d \in \mathbb N$.
Define $\Phi:]0, \infty[ \times \{ x \in \mathbb R^{d-1}: |x|<1\}\to \{y \in \mathbb R^d:y_{1}>0\}, (r,x) \mapsto r(\sqrt{1-|x|^2},x)$
Show that $\Phi$ is a ...
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2answers
43 views
Are the manifolds $N=(\mathbb{R},\text{Id})$ and $M=(\mathbb{R},x\mapsto x^{\frac{1}{3}})$ diffeomorphic?
Are the manifolds $N=(\mathbb{R},\text{Id})$ and $M=(\mathbb{R},x\mapsto x^{\frac{1}{3}})$ diffeomorphic?
I have already shown that $\text{Id}: N \rightarrow M$ is a homeomorphism but not a ...
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0answers
14 views
Id $f,g$ orientation preserving cricle-diffeomorphisms, then $\rho(g^{-1}\circ f \circ g) = \rho(f)$.
Let $F, G : \mathbb{R} \rightarrow \mathbb{R}$ be a lift of $f$ and $G$ of $g$. That is, $ \pi \circ F = f \circ \pi$ with $\pi(x) = e^{2\pi i}$. We define $$\rho_{0}(F) = \lim_{n\rightarrow \infty}\...
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0answers
23 views
Point with dense orbit in locally maximal hyperbolic set is recurrent
Let $f$ a diffeomorphism, $\Lambda$ a locally maximal hyperbolic set and the restriction of $f|_\Lambda$ a transitive map. Let $x\in\Lambda$ be the point with dense orbit. I would like to show that ...
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1answer
36 views
Question about the diffeomorphism group of a Lie group.
I was reading about the diffeomorphism group of varying Lie groups. Wikipedia states that:
When M = G is a Lie group, there is a natural inclusion of G in its
own diffeomorphism group via left-...
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1answer
40 views
integral curve starting at a zero of a vector field
This is a question from Loring Tu's book "Introduction to manifolds" (Page-161 14.6(b))
Show that if X is the zero vector field on a manifold M, and ct(p) is the maximal integral curve of X starting ...
0
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1answer
32 views
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?
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0answers
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coordinate systems produce submanifolds
In his book "Semiriemannian geometry with applications to relativity", Barrett O'Neil says on page 16 under definition 26 that "coordinate systems produce submanifolds.If T:U->R^n is a coordinate ...
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0answers
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Space of orientation-preserving diffeomorphisms on the circle. [duplicate]
Diff$_+(\mathbb{T})$ is the space of orientation-preserving diffeomorphisms on the circle. Then is it true that it is connected?
I have looked through the similar questions on here, but seem to find ...
3
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1answer
51 views
Given that derivative of a function is bounded. Prove surjectivity
Given a differentiable function $f:\mathbf{R} \to \mathbf{R},$ such that $|f'(x)| < c < 1$. Consider a function $g:\mathbf{R}^2 \to \mathbf{R}^2$, such that $g(x,y) = (x+f(y),y+f(x))$. Prove ...
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1answer
57 views
How does a Lie algebra transform under a diffeomorphism?
I'm pretty new to the mathematicians notation in this area so apologies in advance. Suppose we have some manifold M equipped with a left invariant metric $g(u,v)$ and a full set of $n$ left-invariant ...
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0answers
42 views
(Nonlinear after Linear)-Differential Equations
I know the theory needed to solve nonhomogeneous linear differential equations looking like
$$\dot{x}(t) = A(t)x + b(t)\tag{1}$$
in finite-dimensional spaces.
Let $A(t)$ be a linear form, i.e. ...
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0answers
56 views
How to show that $SO(3)$ and $\mathbb{R}P^3$ are dffeomorphic?
Pardon me for repeating a question, but I am not able to justify one aspect of this proof that $SO(3)$ and $\mathbb{R}P^3$ are diffeomorphic, and I was hoping that someone could clarify it for me.
...
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0answers
38 views
Exponential of a vector field on a manifold and diffeomorphisms
Let $\mathcal{M}$ be a manifold and $f$ be a vector field on $\mathcal{M}$. Let $\exp f = \phi(1)$ where $\phi$ is defined by
\begin{align*}
\phi(0) &= \mathrm{id}_\mathcal{M} \\
\dot{\phi}...
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2answers
111 views
Homotopy type of the diffeomorphism group of the sphere.
I've seen in several places the claim that
$$\mathrm{Diff}(S^n) \approx O(n+1) \times \mathrm{Diff}(D^n\,\text{rel}\,\partial D^n),$$
where:
$\mathrm{Diff}(S^n)$ is the group of $C^{\infty}$ ...
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0answers
23 views
Regular parametrisations of surfaces and diffeomorphisms
I am stuck at the following exercise:
Let $f:D \subset \mathbb{R}^k \rightarrow \mathbb{R}^n$ with $k \le n$ be a regular parametrisation and let $\sigma: \mathbb{R}^k \rightarrow D$ be a ...
1
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1answer
143 views
Existence of a smooth map from the Circle to the Square
I know that there is no diffeomorphism from the unit circle, $S^1$ to the square of side length 2 centered at 0.
However, can we construct a bijective map from $f : \mathbb{R}^2 \rightarrow \mathbb{R}...
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0answers
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Toy example of deformed diffeomorphism group
Consider a toy example of a diffeomorphism group – the group of diffeomorphisms of a 1-dimensional manifold with a disconnected boundary (2 points).
The group is a group of monotonically increasing ...
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1answer
72 views
Is the total derivative of a diffeomorphism a bijection?
I'm looking to confirm some properties of the total derivative of a diffeomorphism. Suppose $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ is a diffeomorphism. Then its total derivative at a point $p\in\...
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0answers
80 views
Diffeomorphism Implies Equality of Dimensions
I'm trying to prove the following.
If $f:M\to N$ is a diffeomorphism from an $m$-dimensional (non-empty) manifold to an $n$-dimensional manifold, then $m=n$.
My attempt: Fix a point $p\in M$ and ...
2
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2answers
193 views
Diffeomorphism mapping interior point to boundary point of manifolds with boundary
Why can a diffeomorphism $F: M \to N$ between two smooth manifolds with boundary not take an interior point of $M$ to a boundary point of $N$?
Let $(U, \varphi)$ be a smooth chart for $M$, $(V, \psi)$...
3
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2answers
132 views
If a composition of functions is smooth and one of them is smooth, then the other is smooth
Show that a map $\xi$ between smooth manifolds $M$ and $N$ is
smooth if and only if $f ◦\xi$ is a smooth function on $M$ whenever $f$ is a smooth function on $N$.
One implication is clear because I ...
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0answers
60 views
What being diffeomorphic means (except that there is a diffeomorphism)?
I was wondering what exactly "$A$ and $B$ are diffeomorphic" means (instead of "having a diffeomorphism $f:A\to B$). I know the definition of a diffeomorphism. For example, the set $A$ and $B$ are ...
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1answer
64 views
Is a map with invertible differential a diffeomorphism onto its image near a boundary point?
Let $M,N$ be smooth manifolds of the same dimension, and suppose $M$ has a non-empty boundary. Let $f:M \to N$ be a smooth map, and suppose that $df_p$ is invertible for some $p \in \partial M$.
...
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0answers
37 views
Constructing a diffeomorphism which maps $\Gamma$ to a parabola
$f : \mathbb{R} \to \mathbb{R}$ be a differentiable function and
$$\Gamma = \left \{ (x,y) \in \mathbb{R}^2 | y=f(x) \right \} \subset \mathbb{R}^2$$ its function graph.
a.) Construct a ...
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1answer
31 views
Determining diffeomorphism
I have to determine for all $a,b \in \mathbb{R}$, for which the function $f: \mathbb{R} \rightarrow \mathbb{R}$,$$f(x) = x^3+ax^2 +bx$$ is a diffeomorphism.
This is how far I got:
I know that det $...
2
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1answer
87 views
Are Riemannian manifolds only referred to as diffeomorphic if the diffeomorphism is an isometry?
Let $M$ and $N$ be Riemannian manifolds. My understanding is that strictly speaking, a diffeomorphism $\phi:M \to N$ only acts on the smooth manifold structure, not the metric tensor. But there is a ...
3
votes
1answer
52 views
Star domain are regular
Are all closed star-shaped domains in $\mathbb{R}^d$ with d-dimensional interior, regular closed sets? I know all closed star-shaped domains have a star-shaped interior and all d-dimensional open ...
0
votes
1answer
25 views
What transformations of the sphere keep triple product fixed?
Given three unity vectors from the origin to the surface of a sphere centred on the origin we can get a quantity $(A\times B).C$
If we rotate the sphere by rigid rotations this quantity remains the ...
6
votes
1answer
114 views
$\mathbb CP^n$ with a quadric removed is homeomorphic to $T(\mathbb RP^n)$.
Let $V_f$ be the zero set of a quadratic $z_1^2+\dots +z_{n}^2$ in $\mathbb CP^{n}$. I would like to show that
$P^{n}(\mathbb C) \setminus V_f$ is diffeomorphic to the total space of the tangent ...
0
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0answers
32 views
Express the volume bounded below by the surface $z=x^2 + y^2$, and above by the plane $z = 2x + 6y + 1$ as the integral of a function over unit ball
I was trying to solve this question, and calculate the volume of that region, and before reading the answer, I have come up with the following solution, but is my solution correct ?
Solution:
Let $g(...
