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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”.
4
votes
Gluing two diffeomorphisms together
The answer to the question the way it is formulated is no, because if $\varepsilon<\delta$ and $f(x)=ax$ for some small $a$ we would have that $f$ equals identity on $B(\delta')\setminus B(\varepsilon …
- 28k
5
votes
Accepted
Homogeneous regular (= polynomial component) maps with odd degree and their being global hom...
The answer is yes.
If a homogeneous polynomial map $F:\mathbb{R}^m\to\mathbb{R}^m$ is a local homeomorphism, then it is a global homeomorphism.
We say that a map is proper if preimages of compact s …
- 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
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
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 …
- 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
3
votes
Topologies in the vicinity of Euclidean space
Under reasonable assumptions about $\Sigma$ the answer is yes. For example if $\Sigma$ is smooth and compact $(n-m)$-dimensional submanifold of $\mathbb{R}^n$ and it has trivial normal bundle*, that f …
- 12.9k
10
votes
Books in advanced differential topology
I highly recommend an amazing and highly underestimated trilogy Modern Geometry.
It covers not only differential geometry, but also differential and algebraic topology of manifolds.
Dubrovin, B. A.; F …
- 28k
5
votes
Accepted
Smoothness of distance function to a compact set
If a domain $\Omega$ has boundary of class $C^k$, $k\geq 2$, then in fact the distance function $d$ to the boundary of $\Omega$ is of class $C^k$ in a neighborhood of the boundary. This is exactly wha …
- 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
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
$$
…
- 28k
6
votes
Accepted
Is $L^1$ strong convergence of Jacobians valid for maps between manifolds?
You actually do not need to assume that the mappings are Lipschitz as it is true for general $W^{1,n}$ mappings
Theorem. If $\mathcal{M}$ and $\mathcal{N}$ are smooth compact and oriented manifolds, …
- 28k
2
votes
Is a smooth transformation of a plane domain onto a plane domain with everywhere nonzero Jac...
As Alexandre Eremenko pointed out, in general the answer is in the negative. However, in a comment the OP asked a modified question:
What if we assume that both $U$ and $V$ are simply connected?
The …
- 28k
6
votes
Global diffeomorphisms of $\mathbb R^n$
$\kappa$ must be an affine isometry. If $\gamma:[0,1]\to\mathbb{R}^n$ is a smooth curve and $L(\gamma)$ denotes its length, then
$$
L(\kappa\circ\gamma)=\int_0^1|D(\kappa\circ\gamma)(t)|\, dt=
\int_0 …
- 28k
24
votes
Accepted
Examples of odd-dimensional manifolds that do not admit contact structure
According to a well known result of Martinet, every compact orientable $3$-dimensional manifold has a contact structure [2], see also [1] for various proofs. On the other hand we have
Theorem. For …
- 28k