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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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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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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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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• 319
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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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1 vote
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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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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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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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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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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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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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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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