# Questions tagged [diffeomorphism]

This tag is for questions regarding to "diffeomorphisms", a map between manifolds which is differentiable and has a differentiable inverse.

479 questions
Filter by
Sorted by
Tagged with
56 views

### Condition for section of the frame bundle of $M$ to be the 1-jet of a local diffeomorphism $\phi: M \rightarrow \mathbb{R}^n$?

We're given an n-dimensional Riemannian manifold $M$ and its frame bundle $FM$. The tangent bundle can be locally regarded as an invertible map $\phi:M\rightarrow\mathbb{R}^{n}$. Let's confine $\phi$ ...
1 vote
55 views

### Closed curve in a torus diffeomorphic to a circle?

In Example 15.9 of Tu's "An Introduction to Manifolds: Second Edition", it is written Let G be the torus $R^2/Z^2$ and L a line through the origin in R2. The torus can also be represented ...
44 views

### Isomorphism and local diffeomophism

If $S_1,S_2$ are regular surfaces and $\varphi:U\subset S_1\rightarrow S_2$ is a differentiable map such that the differential $d\varphi_p$ of $\varphi$ at $p\in U$ is an isomorphism, then $\varphi$ ...
47 views

### A question related to the action of a group on the real projective space

I am trying to make sense of the following proof showing that $\mathbb{P}\simeq \text{O}(n+1)/(\text{O}(1)\times \text{O}(n)),$ where $\mathbb{P}:=\mathbb{P}^n(\mathbb{R})$ is the $n$-dimensional real ...
34 views

### Is there an ill-embedded ball in the 4-sphere?

In https://arxiv.org/pdf/2102.04391.pdf, there is an explanation of how one could theoretically use a pair of knots $K$ and $K'$ (one slice and the other not) with the same 0-surgery to generate a ...
49 views

96 views

27 views

### Diffeomorphisms (fixing $p,q$) preserving the smooth structure on the class of all closed embedded surfaces $S \subset [0,1]^3$

Consider the space of diffeomorphisms between $X$ and $Y$: $$\mathrm{Diff}(X,Y)$$ For $X,Y:=[0,1]^3,$ such that $\mathrm{Diff}(X,Y)$ are maps preserving the smooth structure on the class of all closed ...
1 vote
30 views

### Do the orbits of the flow of a vector field change when multiplied by a function? [duplicate]

Assume we have a vector field $X$ on open subset $U\subseteq\mathbb{R}^n$ containing the point $0 \in \mathbb{R}^n$, and let $f: U\rightarrow\mathbb{R}$ be a smooth function. I want to know whether ...
46 views

### Local diffeomorphisms and orientability

Given a differentiable map $f \colon S_1 \to S_2$ between regular surfaces such that $df_p$ is an isomorphism for each $p \in S_1$, I want to study whether the following statements are true or false: ...
21 views

### Can we extend a diffeomoprhis of surface to symplectomorphism?

I think surface diffeomorphism can be extend to a symplectomorphism but i can't describe this. Is there some reference? If this is not turue, please tell me.
88 views

### Flowing a vector field along a diffeomorphism

For simplicity I am happy to work on Euclidean space $\mathbb{R}^d$. Let $X$ be a vector field on $\mathbb{R}^d$, which I think of as a first-order differential operator (that is, I would write $Xf$ ...
110 views

### Can non-homeomorphic topological manifolds be built over the same set?

Let $(M,\mathfrak{X})$ be a smooth manifold, where $M$ is a topological manifold and $\mathfrak{X}$ a smooth structure on it. It is commonly mentioned (in introductory resources on smooth manifolds) ...
34 views

### Limit of diffeomorphism group of n torus

I am taking a second course in topology (fundamental group and covering spaces). I had a small discussion with the teacher, and in passing he said: "The limit of diffeomorphism groups of the n ...
46 views

### What is meant by the cross section of an orbit?

In Folland's quantum field theory book he says: Let $\{\sigma_t : t \in \mathbb{R}\}$ be a one-parameter group of measure-preserving diffeomorphisms of $\mathbb{R}^n$ whose orbits are (generically) ...
Let $F: \mathbb{R}^n \to \mathbb{R}$ be an integrable function, and $\phi: \mathbb{R}^n \to \mathbb{R}^n$ be a smooth function with compact support (i.e., vanishing outside a certain ball). Justify ...