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Results tagged with differential-topology
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user 121665
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 …
- 28k
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 …
- 28k
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
$$
L(\kappa\circ\gamma)=\int_0^1|D(\kappa\circ\gamma)(t)|\, dt=
\int_0 …
- 28k
9
votes
Accepted
Existence of a certain foliation of $\mathbb R^n$
EDIT: Originally I could prove that there is such a foliation by topological manifolds:
Clearly, if $\mathbb{Q}^n$ is the set if points with all rational coordinates, you can have a foliation by paral …
- 28k
2
votes
Does a $C^1$ perturbation induces diffeomorphic level set?
In general, if we do not assume that $f$ is proper (I missed the word "proper" when I read the question), $c$ need not be a regular value of $g$ for any $\epsilon>0$. For example $0$ is a regular valu …
- 28k
4
votes
Accepted
Is a local diffeomorphism with nice boundary values a diffeomorphism?
This is true and follows from a more general fact. Note that in the dimension $n=2$ the unbounded component of $f(\partial\mathbb{D})$ is simply connected.
Theorem. Let $f: \bar{\mathbb{B}}^n \to \ma …
- 28k
1
vote
Accepted
Approximating continuous functions via diffeomorphisms on compact manifolds
The answer to the last question follows from the following result:
Theorem. If $f:\mathcal{M}\to\mathbb{R}$ is a continuous function on a smooth compact connected manifold without boundary and if
$$
…
- 28k
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 …
- 28k
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 $f|_{S^1\times \{1\}}=\gamma_1$ …
- 28k
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 …
- 28k
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 …
- 28k
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 …
- 28k
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 …
- 28k
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 …
- 28k
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 …
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