Questions tagged [diffeomorphism]
The diffeomorphism tag has no usage guidance, but it has a tag wiki.
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2answers
42 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 ...
5
votes
1answer
51 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 $|\lambda'|<1<|\mu'|$.
...
0
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1answer
41 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 $\...
0
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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 $\...
1
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1answer
49 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 ...
0
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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 ...
1
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0answers
14 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[\...
0
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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):=...
1
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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)|...
1
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1answer
43 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 $T$)
...
1
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1answer
18 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 ...
0
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1answer
50 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 ...
0
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2answers
41 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
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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}\...
0
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0answers
19 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 ...
0
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1answer
27 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-...
0
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1answer
31 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
25 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
47 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 ...
1
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1answer
51 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 ...
0
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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}$$
in finite-dimensional spaces.
Let $A(t)$ be a linear form, i.e. ...
1
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0answers
53 views
How to show that $SO(3)$ and $\mathbb{R}P^3$ are dffeomorphic?
Pardon me for repeating a question, but I am not able to justify one aspect of this proof that $SO(3)$ and $\mathbb{R}P^3$ are diffeomorphic, and I was hoping that someone could clarify it for me.
...
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0answers
37 views
Exponential of a vector field on a manifold and diffeomorphisms
Let $\mathcal{M}$ be a manifold and $f$ be a vector field on $\mathcal{M}$. Let $\exp f = \phi(1)$ where $\phi$ is defined by
\begin{align*}
\phi(0) &= \mathrm{id}_\mathcal{M} \\
\dot{\phi}...
1
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2answers
82 views
Homotopy type of the diffeomorphism group of the sphere.
I've seen in several places the claim that
$$\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}$ ...
0
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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
111 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}...
4
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0answers
53 views
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 ...
0
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1answer
60 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
52 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
179 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
votes
2answers
120 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 ...
4
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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 ...
1
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1answer
49 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
36 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$$ its function graph.
a.) Construct a ...
1
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1answer
30 views
Determining diffeomorphism
I have to determine for all $a,b \in \mathbb{R}$, for which the function $f: \mathbb{R} \rightarrow \mathbb{R}$,$$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
78 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
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1answer
50 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
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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
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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
30 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(...
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0answers
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Explicit construction of Hamilton's counterexample for Implicit Function Theorem in Frechet Space
In his famous paper on Nash-Moser Theory, Hamilton mentioned in Counterexample 5.5.2 that a rotation $f: \theta\mapsto\theta + 2\pi/k$ on a circle can be "pushed a little bit" so that the new ...
0
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1answer
56 views
Diffeomorphism preserves dimension?
I want to apply the inverse function theorem to a general mapping that is a diffeomorphism. The inverse function theorem, as stated in my textbook, requires that the dimension of the range and domain ...
2
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2answers
29 views
How to show that $[c_i, d_i] \subset [a_i- \lambda, b_i + \lambda]$?
In the book of Analysis on Manifolds by Munkres, at page 152, it is given that
However, I cannot see how do we we have that $[c_i, d_i] \subset [a_i- \lambda, b_i + \lambda]$. I mean I have tried ...
0
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0answers
44 views
Level curve and PDE
Suppose you have the following PDE sistem:
$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{\partial^2 u}{\partial y^2}$$ $$\frac{\partial u}{\partial x}+\frac{\partial v}{\...
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0answers
47 views
Definition of hyperbolic set in dynamical systems
Let $M$ be a smooth manifold, $f:M\to M$ a diffeomorphism. An $f$-invariant set $\Lambda\subset M$ is called hyperbolic if there are constants $c>0$ and $0<\lambda<1$ such that:
$$T_pM=E_p^s\...
4
votes
1answer
72 views
Proof that $\text{Diff}(M)$ is a topological group
The set of diffeomorphisms for a compact manifold $\operatorname{Diff}(M)$ forms a group. I was recently discussing with someone that this is a topological group, and gave the usual clever proof ...
1
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2answers
148 views
$F$ is local diffeomorphism $\Leftrightarrow$ $F$ preserves or reverses orientation.
Let $M$ and $N$ manifold oriented with $M$ connected. Show that $F\colon M\rightarrow N$ is local diffeomorphism if and only if $F$ preserves or reverses orientation.
My initial idea is to use ...
0
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0answers
16 views
How to “straighten out” level sets of a function?
I have a function $f: R^2 \rightarrow R$ that is decreasing in both coordinates and symmetric around $y=-x$. The level sets of this function are infinite "bent" lines (bending away from $y=-x$). I ...
