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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Prove that $\Phi_{(f, h)}(s)=\left[h(s), h(f^{\tau}(s)), \ldots, h\left(f^{2\tau d}(s)\right) \right]^T$ is smooth and has a smooth inverse.

Let $f: \mathcal M \rightarrow \mathcal M$ be a smooth diffeomorphism and $h: \mathcal{M}$ $\rightarrow \mathbb{R}$ be a smooth function, where $\mathcal M$ is a $d$-dimensional manifold (which we ...
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Differential geometry and surfaces [closed]

Are the cone $\{(x, y, z) \in\mathbb{R}^3 | z^2 = x^2 + y^2,z > 0\}$ and the cylinder $\{(x, y, z)\in\mathbb{R}^3 | x^2 + y^2 = 1\}$ diffeomorphic?
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Diffeomorphisms imply bijections between the set of smooth functions?

This interesting answer raised a question in my mind: Suppose that $M$ and $N$ are smooth manifolds and $f:$ $M \rightarrow N$ is a diffeomorphism. Can we conclude that $C^{\infty} (M)$ and $C^{\infty}...
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Diffeomorphism $\varphi:(0,1)^2\rightarrow U$. $U$ is a parallelogram

Let $U\subseteq\mathbb R^2$ be a open parallelogram with vertices $(2,3), (6,4), (8,6), (4,5)$. Find a diffeomorphism $\varphi:(0,1)\times (0,1)\rightarrow U$. How can I find such an Diffeomorphism? ...
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Is the operation of inverting a function continuous wrt. this bounded norm?

Let $C:=\big(C^{1,1}(\mathbb{R}^d;\mathbb{R}^d), \|\cdot\|_\infty\big)$ be the space of diffeomorphisms on $\mathbb{R}^d$ with the classical pseudonorm $\|f\|:=\min\!\big(\!\sup_{x\in[0,1]}(|f(x)| + |...
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Diffeomorphism between star-space and sphere-space

I am a robotics student who has very poor knowledge of topology, thus I hope my question is not ill-posed. Studying the classical textbook [1], I found an interesting diffeomorphism from stars* to ...
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2 answers
222 views

How to see that "two manifolds are diffeomorphic when you can give them each a coordinate atlas with the same transition maps"

This question is about the diffeomorphism of $\mathbb{C}P^1$ and $S^2$. At the end of youler's answer, we read "the general fact that two manifolds are diffeomorphic when you can give them each ...
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2 answers
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Is the surface of a 3d cube homeomorphic to the 2-sphere

Is the surface of a three dimensional cube i.e. $[0,1]^3$ (surface is like a hardboard box, I don't know if it has a symbol to represent) homeomorphic to $S^2$. If yes, is it diffeomorphic to $S^2$ ...
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Reducing stiffness with coordinate transformation (diffeomorphism)

Brief question: Is there a body of works about reducing stiffness using coordinate transformation? From my little bit of work on linear systems (see below), it seems like the stiffness (defined by the ...
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Are these two questions asking the same thing?

Question $(I).$ Show that if the $n$-dimensional manifold $M$ is a product of spheres, then there exists an embedding $M \to \mathbb R^{n+1}.$ Question $(2).$ Show that there exists an embedding $S^{...
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finding the inverse of a function from $M_2(R).$

I am trying to prove that the given map below is a diffeomorphism and it is pretty clear to me that it is a bijection but I do not know how to show that the inverse of the given map is smooth? in fact ...
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Construction of a special diffeomorphism with some special properties

Let $x,y\in(a,b)$ be real numbers. I am trying to find a diffeomorphism $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying that $f(x)=y$ and $f(t)=t$ for all $t\notin(a,b)$. Here is my attempt. Let $g\in ...
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What is the constant of hyperbolicity?

I am studying dynamical systems of discrete time, and I am having some trouble in understanding what is the constant of hyperbolicity for a closed hyperbolic set $\Lambda \in M$ of a diffeomorphism $f:...
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How to prove that the image is a submanifold?

Here is the question I am trying to tackle: Show that the map $$f : \mathbb R P^n \to \mathbb R P^{n + 1},$$ defined by $$[p] = [p_0, \dots, p_n] \mapsto [p,0] = [p_0, \dots, p_n, 0]$$ is an embedding....
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Show that an embedding sends a conjugate transpose to transpose.

Here is the question I am trying to tackle: Prove that $$\mathbb C \to M_2(\mathbb R),$$ defined by $$x + iy \mapsto \begin{pmatrix} x & -y \\ y & x\end{pmatrix}$$ defines an embedding. Show ...
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Why is this called the conformal tensor?

Suppose you have a Riemannian manifold $(M, g)$ and a diffeomorphism $f: M \longrightarrow M$. Define $E^f:= f^*g - g$. I can understand that $E^f$ defines a $(0, 2)$-tensor which in a sense keeps ...
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1 answer
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How to prove that two set are diffeomorphic?

Let $f:\mathbb{R}^{n-1}\rightarrow\mathbb{R}$ be a smooth map, and $F:\mathbb{R}^n\rightarrow\mathbb{R}$ be defined as $F(x_1,\cdots,x_n)=f(x_1,\cdots,x_{n-1})-x_n$. How to prove that $F^{-1}(0)$ is ...
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Existence of non-volume preserving automorphisms of normal distribution

Let $X=\mathbb{R}^d$, let $p$ the standard normal distribution on $X$, with zero mean and identity covariance, and let $f:X \to X$ be a diffeomorphism that preserves this normal distribution. For ...
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Does there exist a fibre preserving diffeomorphism from $T S^1$ onto $S^1 \times \mathbb R\ $?

Does there exist a fibre preserving diffeomorphism $\phi : TS^1 \longrightarrow S^1 \times \mathbb R\ $? (Here $TS^1$ denotes the tangent bundle of $S^1$) Any help would be warmly appreciated. Thanks ...
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2 votes
1 answer
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Question regarding integration and diffeomorphisms on $\mathbb{R}$

Suppose $I, J, K$ are three (possibly infinite) intervals in $\mathbb{R}$. Suppose that $\phi: K \times I \to J$ is such that $\phi(x, \cdot) \colon I \to J$ is a $C^\infty$-diffeomorphism for all $x \...
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Normal vector under diffeomorphism

I have a smooth diffeomorphism $\Phi\colon M \to N$ between two 2D hypersurfaces in $\mathbb{R}^3$, eg. the sphere and some deformed version of the sphere. If I have a unit (outward) normal vector $n$ ...
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How many connected components are there in $\text{Diff}^+(S^3 \times \mathbb R)$?

I am reading an article about overtwisted contact structure and I am stuck at some point. I will not add all the context because it is quite long but I can summarize my question as follows: I would ...
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2 votes
1 answer
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Regarding diffeomorphism on manifolds

I am trying to show these claims: If $M,N$ are smooth manifolds without boundary. Prove that: $T(M\times N)$ is diffeomorphic to $TM\times TN$. Prove that $TT^n$ is diffeomorphic to $T^n\times R^n$....
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Finding and showing that the inverse of a transition map is smooth

I'm trying to show that stereographic projection from the north and south pole determine a smooth atlas, and I just showed that the transition map $$ \varphi_N\circ\varphi_S^{-1}(x_1, \ldots, x_n) = \...
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1 vote
1 answer
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Proving a vector field without explicit inverse surjective

Consider the vector field given by \begin{align} F: \mathbb{R}^3 \mapsto \mathbb{R}^3, (x,y,z)\mapsto (y+e^z,z+e^x,x+e^y). \end{align} I want to show that $F$ is surjective/onto. After some trying, I ...
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1 answer
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Visual and conceptual intuition for diffeomorphisms

I have recently learned the notion of diffeomorphism in the context of defining regular surfaces as those that are locally diffeomorphic to a plane. I have read the answers on this great question, but ...
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Diffeomorphisms of disks (not rel boundary)

There is a lot known about $\pi_0(Diff(D^n), \partial)$, diffeomorphisms of the closed n-disk that act trivially on the boundary. I was wondering what is known about $\pi_0(Diff(D^n))$, where ...
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Construct a $C^\infty$ atlas $\mathcal{A}$ so that $(M, \overline{\mathcal{A}})$ becomes a differentiable $2$-manifold.

Let $M=\{(x,y,|x|) \in \Bbb R^3 \mid (x,y) \in \Bbb R^2 \}$. Construct a $C^\infty$ atlas $\mathcal{A}$ so that $(M, \overline{\mathcal{A}})$ (where $\overline{\mathcal{A}}$ is the maximal atlas) ...
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Let $M$ be a k-dim. $C^1$-Manifold of $\mathbb{R}^n$ and $\phi:\mathbb{R}^n \rightarrow \mathbb{R}^n$ a diffeomorphism, proofe $\phi(M)$ is a manifold

That task is, that given a k-dimensional $C^1$-submanifold of $\mathbb{R}^n$ and $\phi \colon \mathbb{R}^n \rightarrow\mathbb{R}^n$ a diffeomorphism. Show, that $M' := \phi(M)$ also is a $k$-...
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An explicit expression for a diffeomorphism

I know that the open unitary ball $B^n$ of $\mathbb R^n$ are diffeomorphic and I also know some explicit forms of a diffeomorphism, many can be found on this very website. Now I would like to find a ...
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1 vote
1 answer
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On proving that Constant Rank Theorem $\implies$ Inverse Function Theorem

I am referring to the discussion on presented here: Rank theorem implies inverse function theorem. We known that the composition of diffeomorphisms is a diffeomorphism and that we have our ...
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3 votes
1 answer
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Condition on local flows that imply completeness of vector field

Given any smooth vector field $X$ on a manifold $M$ we can cover $M$ by open sets $\{U_\alpha\}_{\alpha\in A}$ such that for each $\alpha$ there is an $\epsilon_\alpha>0$ such that if $p \in U_\...
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Prove that $\mathbb R\mathrm P^1$ is diffeomorphic to certain submanifold of $\mathbb R^3$.

I am trying to solve the following exercise: Prove that the Real Projective Line $\mathbb R\mathrm P^1$ is diffemorphic to certain submanifold $M$ of $\mathbb R^3$ via the map $f:\mathbb R\mathrm P^1\...
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Are transition maps diffeomorphisms?

For two charts $(U, \phi)$ and $(V,\psi)$ on a topological manifold that are $C^\infty$-compatible, the transition maps $$\phi \circ \psi^{-1}:\psi(U \cap V)\rightarrow \phi(U \cap V)$$ $$\psi\circ\...
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3 votes
1 answer
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Restriction of vector bundle to norm 1 is a covering map

Given a mainfold $M$, the vector bundle \begin{equation*} \pi:\wedge^k T^*M = \sqcup_{p \in M} \wedge^k T_p^*M \rightarrow M \end{equation*} has the property that its section are exactly the $k$-forms ...
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0 votes
2 answers
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Prove that a map $f:B(0,1)\subset \mathbb R^n\to \mathbb R^n$ is a diffeomorphism

Let $B(0,1)$ be the open ball given by $x\in \mathbb R^n$ such that $\|x\|<1$ and consider $f:B(0,1)\to\mathbb R^n$ given by: $$f(x)=\frac{x}{\sqrt{1-\|x\|^2}}$$ In order to prove that $f$ is a ...
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Diffeomorphism between $SO(3)$ and $STS^2$ [duplicate]

I am studying differential geometry. I'm having a problem I don't know, so here is the question. Problem) Show that $SO(3)$ is diffeomorphic to $STS^2=\{v\in TS^2 : \Vert v \Vert_g=1\}$, where $g$ is ...
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2 votes
0 answers
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If M is a non-compact manifold, then Diff(M) is not a manifold. Yet it still has manifold-like properties. What is it? [closed]

If $M$ is a non-compact smooth manifold, then $\text{Diff}(M)$ is not locally compact and hence not a manifold. Yet it still has manifold-like properties. In fact it has Lie group-like properties. It'...
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1 vote
3 answers
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Smoothness of a manifold implies all maps of its altas' charts are diffeomorphism?

I need to show that : On any smooth manifold $(M,A)$ all chart maps are $C^{\infty}$-diffeomorphisms. Definitions : Let $M$ be a Hausdorff second countable topological space. Then a pair $(U, \phi)$...
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3 votes
0 answers
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Prove that $f:\mathbb{R}^n \rightarrow \mathbb{R}^n $ is a $C^1$ diffeomorphism [duplicate]

Suppose that $f:\mathbb{R}^n \rightarrow \mathbb{R}^n,f\in C^1.$ If there exists a constant $a>0$,$\forall x,y\in \mathbb{R}^n,|f(x)-f(y)|\geqslant a|x-y|,$ prove that $f:\mathbb{R}^n \rightarrow \...
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The inverse of the differential of f inverse

If I have a diffeomorphism $f:S_1 \rightarrow S_2$, where $S_1, S_2$ are regular surfaces, then is $(df)^{-1}=df^{-1}$? How would I show this? I saw a hint that said to consider $f^{-1}\circ f=id$ at ...
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2 votes
2 answers
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Diffeomorphism between $\mathbf S^2 \times \mathbf S^2$ and complex projective hypersurface

I have a problem while studying differential geometry and I have a question. I want to show that $\{[{z_0:z_1:z_2:z_3]:z_0^2+z_1^2+z_2^2+z_3^2=0}\}$ and $\mathbf S^2 \times \mathbf S^2$ are ...
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1 vote
0 answers
38 views

How to show if two regular surfaces are diffeomorphic

How can I show if regular surfaces $x^2+y^2-z^2=1$ and $x^2+y^2-z^2=-1$ are diffeomorphic? I know that they are not, but how can I show it?
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Differentiable manifold homeomorphic to the 2-torus

I think of the 2-torus as $\mathbb T^2=\mathbb R^2/\mathbb Z^2$. Suppose that I have constructed some new charts on it (with $C^{1+\alpha}$ transition maps) that are not smooth differentiable with ...
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4 votes
0 answers
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Are a (co)tangent vector's "coordinate components" equivalent to the "coordinate components of its pushfoward to R^n"?

[All equation numbers reference Wald, Robert M., General relativity, Chicago-London: The University of Chicago Press. XIII, 491 p. 34.50 (1984).] Consider a subset $O$ of an $n$-dimensional, $C^{\...
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7 votes
1 answer
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Diffeomorphisms of sphere and homotopies and Smale's conjecture in $n\geq 4$ dimensions

Short version of question: Does $\operatorname{dif}(S^n)$ have more than two connected components? Reading this article on Smale's conjecture and the resolution in higher dimensions, I had a ...
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Restriction of a diffeomorphism as composition of functions

Let $h\colon\mathbb R^2\to\mathbb R^2$ be a $C^1 $ diffeomorphism. How do we show that any $a \in\mathbb R^2$ has a neighborhood U such that $h|_U = g_1 ◦ g_2 ◦ · · · ◦ g_m$, where each $g_i$ is a $C^...
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Prove a point $x\in \Bbb{R^n}$ is a zero-dimensional manifold

Let $x\in \Bbb{R^n}$. We may claim that there is an open $U$ containing $x$. For the proof, is it enough to define a diffeomorphism $h:U\to\emptyset$? This would mean that $$h(U\cap \{x\})=V\cap(\Bbb{...
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1 vote
1 answer
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Regarding the Definition of an Orientable Foliation on a Manifold in terms of Transition Maps

Introduction to the Geometry of Foliations, Part $A$, Authors Gilbert Hector and Ulrich Hirsch, Page $15$. Let $\mathcal{F}$ be a Foliation on a Manifold $M$ defined by the atlas $\{(U_i,\phi_i)\}$. ...
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Rectifiable set under diffeomorphism

We need to show that image $f(A)$ of a rectifiable set $A$, under a diffeomorphism $f$, is also rectifiable. Definition: A rectifiable set is a set which is closed, bounded & has boundary of ...
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