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Results tagged with differential-topology
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user 46290
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”.
5
votes
1
answer
309
views
Is every map of rank smaller than r dominated by a constant rank map?
Let $\, f:M \to N$ be a smooth map, with rank $df \le r$ everywhere.
Does there exist a smooth map $\tilde f:M \to N$ of constant rank $r$, such that each level set of $\tilde f$ is contained in some …
- 6,741
10
votes
1
answer
404
views
is there a diffeomorphism with only finite orbits but of infinite order?
I asked this in stackexchange, but got no answer, so I am trying here.
Is it possible for a diffeomorphism $\phi$ (of a smooth manifold $M$) to have the following properties:
All its orbits are fi …
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21
votes
1
answer
2k
views
Differential Topology over $\mathbb{Q}$
I have a specific question in mind, but it requires some explanation and context before it can be formally stated. To summarize it in a sentence, this is it:
Are every two rational manifolds of the …
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14
votes
3
answers
3k
views
When a homeomorphism is a diffeomorphism w.r.t to a suitable smooth structure?
Assume we have a homeomoprhism $\phi:M\rightarrow M$, where $M$ is a topological manifold which admits at least one smooth structure.
Is it always possible to construct a smooth structure on $M$ w.r. …
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6
votes
2
answers
1k
views
Riemannian metrics preserved by diffeomorphisms
Let $f \neq Id$ be a diffeomorphism (of a smooth manifold $M$) which admits some Riemannain metric on $M$ making it an isometry. How many different metrics are preserved by $f$?
Note that $Met(f)=\{g …
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0
votes
Riemannian metrics preserved by diffeomorphisms
I am just adding a few details to Vladimir's answer:
Lemma: Assume there exists a point $x$ such that the orbit w.r.t. the iterations of the $\phi$ (i.e., the set $\{x,ϕ(x),ϕ(ϕ(x)),...\}$ is dense in …
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4
votes
2
answers
415
views
Do trivial homotopy groups imply existence of boundary preserving homotopies?
This is a cross-post from MSE.
Let $N$ be a smooth $d$-dimensional connected orientable manifold which have the following property:
For every smooth $d$-dimensional manifold $M$ with non-empty bounda …
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28
votes
3
answers
2k
views
Does isometric immersion map boundary to boundary?
Let $M$ be a compact, connected, oriented, smooth Riemannian manifold with non-empty boundary. Let $f:M \to M$ be a smooth orientation preserving isometric immersion.
Is it true that $f(\partial M) \ …
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0
votes
0
answers
320
views
Unit sphere of a norm is a submanifold implies the norm is smooth?
Let us call a norm on $\mathbb{R}^n$ smooth if its restriction $\| \cdot \|:\mathbb{R}^n\setminus \{ 0 \} \to \mathbb{R}$ is a smooth map.
Suppose the unit sphere of a norm $\| \cdot \|$ is an embedd …
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0
votes
Smoothness of the closest point on a submanifold
$\newcommand{\til}{\tilde}$
This is an attempt to prove rigorously that there exists an open subset $\Omega$ of the normal bundle to $S$, such that $exp:\Omega \to M$ is a bijection. This proof seems …
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7
votes
1
answer
208
views
Is a Sobolev map with smooth minors smooth on the whole domain?
Let $d\ge 3$ and $2 \le k \le d-1$ be integers, where at least one of $k,d$ is odd. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for some $p \ge 1$.
Q …
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3
votes
0
answers
308
views
Is the image of the map $A \to \bigwedge^{k}A $ a weakly embedded submanifold?
$\newcommand{\End}{\operatorname{End}}$
$\newcommand{\GL}{\operatorname{GL}}$
Let $V$ be a $d$-dimensional real vector space. ($d \ge 4$). Fix an odd $2 \le k \le d-2$. Define
$H_{>k}=\{ A \in \End(V) …
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3
votes
0
answers
81
views
Is there a transitive Lie group action on the space of matrices with rank bigger than $k$?
$\newcommand{\GL}{\operatorname{GL}}$
Let $H_{>k}$ be the space of real $d \times d$ matrices of rank bigger than $k$, for some fixed $k$. $H_{>k}$ is an open connected submanifold of $ \mathbb{R}^{d …
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4
votes
0
answers
168
views
Can the rank of harmonic maps decrease far from the boundary?
Let $\mathbb D^n$ be the closed unit ball in $\mathbb R^n$. Let $f:\mathbb D^n \to \mathbb{R}^n$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^n \to \mathbb{R}^n$ be t …
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6
votes
1
answer
368
views
Can we choose smoothly the singular vectors of a matrix?
$\newcommand{\GLm}{\text{GL}_n^-}$Let $A$ be a real $n \times n$ matrix with non-positive determinant. Suppose that the smallest singular value of $A$ is strictly smaller than all the others (it has m …
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