Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [diffeomorphism]

The tag has no usage guidance, but it has a tag wiki.

0
votes
0answers
13 views

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 \...
0
votes
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^...
0
votes
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 ...
4
votes
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
votes
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
votes
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 $f$ ...
1
vote
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 $F_s(x)=F(x,s)$ ...
0
votes
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 ...
1
vote
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 ...
5
votes
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 $|\lambda'|<1<|\mu'|$. ...
0
votes
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 $\...
0
votes
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
vote
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 ...
0
votes
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
vote
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[\...
0
votes
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
vote
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
vote
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 $T$) ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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}\...
0
votes
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 ...
1
vote
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-...
0
votes
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
votes
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?
0
votes
0answers
13 views

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 ...
1
vote
0answers
40 views

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
votes
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 ...
1
vote
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 ...
0
votes
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
vote
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. ...
2
votes
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}...
2
votes
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 $C^{\infty}$ ...
0
votes
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
vote
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}...
4
votes
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
votes
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\...
1
vote
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
votes
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
votes
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 ...
4
votes
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
vote
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 $p \in \partial M$. ...
1
vote
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 ...
1
vote
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
votes
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
votes
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(...