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The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.

2 votes

Is a smooth transformation of a plane domain onto a plane domain with everywhere nonzero Jac...

As Alexandre Eremenko pointed out, in general the answer is in the negative. However, in a comment the OP asked a modified question: What if we assume that both $U$ and $V$ are simply connected? The …
Piotr Hajlasz's user avatar
6 votes
Accepted

Smooth functions on subsets of $\mathbb{R}^n$

The answer is yes for functions defined on closed sets $X\subset\mathbb{R}^n$. In Section 1.5.5 in [1] we have a necessary and a sufficient condition of the existence of an extension to a $C^m$ functi …
Piotr Hajlasz's user avatar
6 votes

Global diffeomorphisms of $\mathbb R^n$

$\kappa$ must be an affine isometry. If $\gamma:[0,1]\to\mathbb{R}^n$ is a smooth curve and $L(\gamma)$ denotes its length, then $$ L(\kappa\circ\gamma)=\int_0^1|D(\kappa\circ\gamma)(t)|\, dt= \int_0 …
Piotr Hajlasz's user avatar
9 votes
Accepted

Existence of a certain foliation of $\mathbb R^n$

EDIT: Originally I could prove that there is such a foliation by topological manifolds: Clearly, if $\mathbb{Q}^n$ is the set if points with all rational coordinates, you can have a foliation by paral …
Piotr Hajlasz's user avatar
2 votes

Does a $C^1$ perturbation induces diffeomorphic level set?

In general, if we do not assume that $f$ is proper (I missed the word "proper" when I read the question), $c$ need not be a regular value of $g$ for any $\epsilon>0$. For example $0$ is a regular valu …
Piotr Hajlasz's user avatar
4 votes
Accepted

Is a local diffeomorphism with nice boundary values a diffeomorphism?

This is true and follows from a more general fact. Note that in the dimension $n=2$ the unbounded component of $f(\partial\mathbb{D})$ is simply connected. Theorem. Let $f: \bar{\mathbb{B}}^n \to \ma …
Piotr Hajlasz's user avatar
1 vote
Accepted

Approximating continuous functions via diffeomorphisms on compact manifolds

The answer to the last question follows from the following result: Theorem. If $f:\mathcal{M}\to\mathbb{R}$ is a continuous function on a smooth compact connected manifold without boundary and if $$ …
Piotr Hajlasz's user avatar
7 votes
Accepted

Map between manifolds and open dense subsets

Yes. I think you do not even have to assume that the preimages are finite. $f$ can be any $C^1$ mapping from $X$ onto $Y$. Let $y\in Y$ be a regular value of the mapping $f$. Then for some $\epsilon>0 …
Piotr Hajlasz's user avatar
1 vote

Existence of isotopy preserving the action

This is not an answer but an idea. If you can show that there is an Lagrangian embedding of $f:S^1\times [0,1]\to\mathbb{R}^4$ such $f|_{S^1\times \{0\}}=\gamma_0$ and $f|_{S^1\times \{1\}}=\gamma_1$ …
Piotr Hajlasz's user avatar
6 votes
Accepted

$C^1$ perturbation of diffeomorphism is diffeomorphism?

Assuming that $M$ is a compact manifold, the answer is yes. Indeed, $\det Df(x)\neq 0$ for $x\in M$ and if $|Df(x)-Df_\epsilon(x)|$ is small, then $\det Df_\epsilon(x)\neq 0$, because the set of inver …
Piotr Hajlasz's user avatar
9 votes

Whitney embedding theorem for Hölder manifolds

Every $C^1$ manifold admits a compatible $C^\infty$ structure. You can find a proof in Hirsch's "Differential topology". It is actually quite easy and based on a fact that smoothing a $C^1$ diffeomorp …
Piotr Hajlasz's user avatar
5 votes

Thom's gradient conjecture and analyticity

The result quoted by the author of the questions is not correct. Suppose we have an analytic function $f: U \to {\mathbb R}$, where $U\subset {\mathbb R}^n$ is an open subset and $0 \in U$ is a c …
Piotr Hajlasz's user avatar
13 votes
Accepted

Is there a sensible notion of a winding number of a closed curve in $\mathbb{R}^n$, $n\geq 3...

No, you cannot define a winding number if $n\geq 3$ since, as pointed out in a comment by Anthony Carapetis, any two curves in $\mathbb{R}^n\setminus\{ p\}$ are homotopic, and a winding number should …
Piotr Hajlasz's user avatar
3 votes
Accepted

Nice decomposition of surface diffeomorphisms

If the Jacobian of $f$ is not equal $1$ on any set of positive measure, then the mapping will not be measure preserving on any set of positive measure so a decomposition is not possible. A classical …
Piotr Hajlasz's user avatar
13 votes

Checking that the image of a curve is not contained in a hyperplane

A curve $\alpha$ in $\mathbb{R}^3$ is called non-degenerate if $\alpha'$ and $\alpha''$ are linearly independent at every point. A curve parametrized by arc-length is a Frenet curve if $\alpha''\neq 0 …
Piotr Hajlasz's user avatar

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