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8 votes
0 answers
238 views
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Maps with small fibers between manifolds of equal dimension

The following question is an attempt to revise this one into what I intended. Important revisions are shown in bold. Are there any known examples of a compact Riemannian manifold $M$ with (possibly ...
Matthew Kvalheim's user avatar
3 votes
1 answer
529 views

Zeros of a function defined on $\mathbb{S}^2 \times \mathbb{S}^2$

Let $u$ be a smooth function on the sphere, and for each $y \in \mathbb{S}^2$, let $R_y$ be the $180^\circ$ rotation about the vector $y$. For each pair $(x, y) \in \mathbb{S}^2 \times \mathbb{S}^2$, ...
MathLearner's user avatar
-4 votes
1 answer
328 views

Does a coarser topology lead to a non-Hausdorff topological manifold? [closed]

Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold? NOTE: PLEASE avoid the ...
Bastam Tajik's user avatar
2 votes
0 answers
136 views

Progess on conjectures of Palis

I came across a "A Global Perspective for Non-Conservative Dynamics" by Palis. He has some conjectures "Global Conjecture: There is a dense set $D$ of dynamics such that any element of ...
NicAG's user avatar
  • 247
0 votes
0 answers
303 views

Proof that a first integral is not a constant function

Let $U$ be an (open) set in $\mathbb{R}^n$. And we are given a set of $m$ basis functions $$B=\{\psi_i(x): U \rightarrow \mathbb{R}\mid i=1,\ldots,m \}$$ such that all of them are differentiable and ...
NicAG's user avatar
  • 247
0 votes
1 answer
525 views

Non-diffeomorphic but homeomorphic (under Lorentzian topology) Lorentzian manifolds

$\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\diff}{\mathrm{diff}}\newcommand{\manifold}{\mathrm{manifold}}$Take a time-oriented Lorentzian ...
Bastam Tajik's user avatar
0 votes
0 answers
70 views

Example of DS with a dense trajectory in the whole state space

Let $U \subset \mathbb{R}^n$ be an open and connected set. We assume there is a vector field $F \in \mathcal{C}^1(\overline{U})$ giving rise to a DS ($\overline{U}$ denotes the closure) $$\dot{\mathbf{...
NicAG's user avatar
  • 247
8 votes
2 answers
489 views

Continuous point map for spherical domains

Consider the space $J$ of Jordan domains on the sphere $\textbf{S}^2$, i.e., continuous injective maps from the unit disk into $\textbf{S}^2$ modulo homeomorphisms of the disk. How can one construct a ...
Mohammad Ghomi's user avatar
29 votes
2 answers
2k views

Contractibility of the space of Jordan curves

Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$. If the curves are ...
Mohammad Ghomi's user avatar
5 votes
1 answer
165 views

Algebraic solutions of polynomial ODEs

Given a polynomial ODE in $n$-dimensions of maximal degree $d$ $$ \dot{x}_j=f_j(x)=\sum_{i_{1},\dots,i_{n}=1}^{d}a_{i_{1},\dots,i_{n}}^{j}x_{1}^{i_{1}}\dots x_{n}^{i_{n}} \quad \forall j=1,...,n ...
NicAG's user avatar
  • 247
1 vote
0 answers
192 views

Simple left earthquakes are dense

i´ve been studying an article from W. P. Thurston about hyperbolic geometry, there, he defines something called left earthquake, whose definition is as follows: Definition. If $\lambda$ is a geodesic ...
Pedro's user avatar
  • 11
2 votes
1 answer
232 views

Existence of diffeomorphism interpolating affine map and identity

$\newcommand{\R}{\mathbb{R}}$Suppose $\Omega$ is a bounded, convex domain in $\R^{m}$. Fix $x_1, x_2\in\Omega$ and an invertible matrix $A\in\mathrm{GL}^{+}(m)$ with positive determinant. Let $U\...
Sven Pistre's user avatar
3 votes
0 answers
109 views

"Practical" references on mapping spaces as infinite-dimensional manifolds

I am studying spaces of the form $C^{k}(\mathcal{M},\mathcal{N})$ between manifolds ($k=\infty$ allowed) and I am looking for extensive references, especially analysing their topology and smooth ...
B.Hueber's user avatar
  • 1,171
2 votes
2 answers
519 views

How to use that the Hessian is negative definite in this proof

Let $X$ be a Riemannian compact manifold on which acts a compact Lie group $H^+$. Let $f^+ : X \rightarrow \mathbb{R} $ be a smooth function on $X$. Consider a Lie subgroup $U$ of $H^+$ and suppose ...
Mira's user avatar
  • 139
6 votes
0 answers
189 views

What is a non-smooth connection?

Let $p : E \to B$ be a map of topological spaces, and $p^I : E^I \to B^I$ the induced map of path spaces. Let $Cocyl(p) = B^I \times_B E$ be the space of paths $\beta$ in $B$ equipped with a lift of $\...
Tim Campion's user avatar
  • 63.9k
2 votes
1 answer
119 views

Density of smooth bi-Lipschitz maps in smooth maps

Setup/Motivation: Let $(M,g)$ and $(N,\rho)$ be complete Riemannian manifolds of respective dimensions $m$ and $n$ and suppose that $m\leq n$. Let $\operatorname{bi-C}^{\infty}(M,N)$ denote the class ...
Carlos_Petterson's user avatar
3 votes
0 answers
126 views

A path with zero increments and positive area

I am studying rough paths from the 2007 St Flour lecture notes and I came across the example at the end of chapter one of the sequence of paths $X(n):[0,2\pi]\to \mathbb R^2$ given by $X_t(n) = \frac{...
Martin Geller's user avatar
2 votes
1 answer
130 views

Gluing isotopic smoothings

Let $M$ be a topological manifold which can be written as $M = U \cup V$ where $U$ and $V$ are open. Suppose both $U$ and $V$ admit smooth structures. Also assume that on the overlap $U \cap V$ the ...
UVIR's user avatar
  • 803
6 votes
0 answers
136 views

A particular case of the general converse to the preimage (submanifold) theorem

I was thinking whether it would be possible to develop a converse to the preimage theorem in differential topology and I found the following post: When is a submanifold of $\mathbf R^n$ given by ...
geooranalysis's user avatar
8 votes
0 answers
198 views

A modified version of the converse to the Sard's Theorem

When I learned Sard's Theorem in differential topology by myself, I was thinking whether it would be possible to prove a converse version of the theorem. That is to say, can we somehow show that each (...
pureorapplied's user avatar
1 vote
1 answer
195 views

Can one explore a surface along ‘piecewise planar’ curves?

Suppose $d\in \{3,4,\dotsc\}$ and $A\subseteq \mathbb{R}^d$ is non-empty, open and connected with its complement $A^c$ connected too and $\text{int}(A^c)\neq \emptyset$. Its boundary $S:=\partial A$ ...
5th decile's user avatar
  • 1,461
6 votes
0 answers
297 views

Regarding homology of fiber bundle

Let $f: X\to Y$ be a smooth map between smooth manifolds, both connected. Let $Y=\cup_{i=1}^k Y_i$ be a finite union of disjoint locally closed submanifold $Y_i$ such that $f^{-1}(Y_i)\to Y_i$ is ...
tota's user avatar
  • 585
1 vote
1 answer
291 views

Isometry and gluing between smooth manifolds - some references

I have a doubt that assails me. The technique of gluing along edges between manifolds is generally considered in the topological context. I don't know if there are other gluing techniques. I was ...
MathDG's user avatar
  • 272
2 votes
0 answers
58 views

Dimension changes from global to local immersion

From Hatcher Corollary A.10. the (global) immersion for an $n-$dimensional CW complex is possible in some $\mathbb{R}^N$. I have started with $M(G,n)$ (Moore space of type $(G,n)$, $G$ is cyclic ...
piper1967's user avatar
  • 1,177
7 votes
2 answers
562 views

Is the union of a compact and the relatively compact components of its complementary in a manifold compact?

I was thinking of a way to prove this and I realised that for my approach the lemma from the title would be useful, and it´s an interesting question on its own. Obviously it is true if the manifold is ...
Saúl RM's user avatar
  • 10.6k
4 votes
1 answer
110 views

Separation of convexity on uniquely geodesic space

A metric $d: X \times X \to [0,\infty)$ is said to be intrinsic provided that the distance between any two points is the infimum of the lengths of paths joining the points. A space is an inner metric ...
Shijie Gu's user avatar
  • 2,083
2 votes
0 answers
74 views

Is the reversibility of inflation of a subset equivalent to its smoothness?

$D_r(x)$ denotes a closed ball of radius $r$ centered at $x$. Definition. Let $M \subset \mathbb{R}^n$. $D_r (M): = \bigcup\limits_{x \in M} D_r (x)$ $Int_r (M): = \{x ~|~ D_r(x) \subset M\}$ ...
Arshak Aivazian's user avatar
5 votes
2 answers
252 views

Hermitian vector bundles and Hilbert $C^*$-modules

Let $X$ be a compact Hausdorff space and $C(X)$ its algebra of continuous complex valued functions. The Gelfand-Naimark theorem tells us that we have a duality between commutative $C^*$-algebras and ...
Jake Wetlock's user avatar
  • 1,144
13 votes
0 answers
364 views

What is known about differentiable and analytic structures on the long line (and half-line)?

When reading about this question which recently became active for some reason, I wanted to make a comment, as a warning regarding non-metrizable manifolds, to the effect that the every $C^\infty$ ...
Gro-Tsen's user avatar
  • 32.5k
1 vote
1 answer
649 views

Local diffeomorphisms, covering maps and smooth path lifting

Let $f: M\to N$ be a surjective local diffeomorphism of noncompact smooth manifolds. Suppose that every smooth path is liftable, that is, for any smooth path $\gamma: [0,1]\to N$ and any point $p\in f^...
Dmitrii Korshunov's user avatar
4 votes
2 answers
335 views

If $\Omega$ is locally Lipschitz, then $\Omega = \bigcup_{k = 1}^N \Omega_k$ for $\Omega_k$ star shaped with respect to an open ball $B_k$

I am reading Galdi's Introduction to the mathematical theory of Navier Stokes equations and there is an argument which comes up quite often that I really don't understand. In many theorems of Chapter $...
Falcon's user avatar
  • 452
3 votes
1 answer
102 views

Parametrised proper map

I recently tried to wrap my head around the following problem: Let $f\colon \mathbb{R} \times K \rightarrow \mathbb{R}$ be a smooth map, where $K$ is a compact manifold. Assume that for each $k\in K$, ...
Alexander Schmeding's user avatar
5 votes
0 answers
154 views

Sheaf-like reconstruction of a continuous function

Let $X$ and $Y$ be topological manifolds and let $\{(\phi_x,U_x)\}_{x \in X}$ and $\{(\psi_y,Y_y)\}_{y \in Y}$ be respective atlases of $X$ and $Y$; with each $\phi_x:U_x\rightarrow \mathbb{R}^n,\...
ABIM's user avatar
  • 5,405
2 votes
1 answer
560 views

Collar neighborhood theorem for manifold with corners

I was reading this wonderful sequence of posts: nlab: manifold with boundary and nlab: collar neighbourhood theorem and I couldn't help but wonder. Is there an extension of the Collar neighborhood ...
ABIM's user avatar
  • 5,405
1 vote
1 answer
283 views

Poincare duality-differential geometry

Let $ M $ be a smooth and compact manifold with boundary $\partial M = X \times F $ on which the structure of a smooth locally trivial bundle $$ \pi: \partial M \longrightarrow X $$ where the $ X $ ...
Ady Fall's user avatar
2 votes
2 answers
447 views

Reconciling some result about the exponential map, the Chow-Rashevskii theorem, and $\mathrm{Diff}_0(M)$

Let $M$ be a $C^{\infty}$ manifold $C^{\infty}$-diffeomorphic to $\mathbb{R}^d$. I've recently come across some results which I'm trying to reconcile. Let $\mathfrak{X}(M)$ denote the set of ...
ABIM's user avatar
  • 5,405
5 votes
1 answer
380 views

Non-density of continuous functions to interior in set of all continuous functions

Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed with the ...
ABIM's user avatar
  • 5,405
2 votes
1 answer
301 views

Density of continuous functions to interior in set of all continuous functions

Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold with boundary. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed ...
ABIM's user avatar
  • 5,405
1 vote
0 answers
61 views

Minimal radius of a ball admitting a trivialization of a vector bundle

Let $X$ be a compact Hausdorff space and $p : V \to X$ a complex vector bundle of rank $n$. For $r > 0$ let $B(r,x)$ denote the open ball of radius $r$ around $x$. Does there exist an $r$ such that,...
Francine Laporte's user avatar
5 votes
2 answers
308 views

Is this subset of matrices contractible inside the space of non-conformal matrices?

Set $\mathcal{F}:=\{ A \in \text{SL}_2(\mathbb{R}) \, | \, Ae_1 \in \operatorname{span}(e_1) \, \, \text{ and } \, \, A \, \text{ is not conformal} \,\}$, and $\mathcal{NC}:=\{ A \in M_2(\mathbb{R}) \...
Asaf Shachar's user avatar
  • 6,741
4 votes
1 answer
648 views

Essential simple closed curves on a punctured torus vs those in the torus

Let $T$ be a compact oriented torus, let $p\in T$ be a point, and let $T^*$ be $T - \{p\}$. In Farb-Margalit's Primer on mapping class groups, in the discussion after Proposition 1.5 they say that "...
Will Chen's user avatar
  • 10.7k
2 votes
0 answers
263 views

Are these two definitions of smooth $k$-manifold as a Euclidean subset equivalent?

I am struggling to reconcile the two definitions of smooth k-manifold in $R^n$ from M.Spivaks Calculus on Manifolds (pg 109) and J.W Minor's Topology from differential point of view (pg 01). Milnor's ...
Senthan Sara's user avatar
36 votes
1 answer
3k views

Is there a general theory of "compactification"?

In various branches of mathematics one finds diverse notions of compactification, used for diverse purposes. Certainly one does not expect all instances of "compactification" to be specializations of ...
Tim Campion's user avatar
  • 63.9k
2 votes
1 answer
152 views

How to define "interior" for the unit arc? [closed]

Let the unit arc be, $$\{x \in \mathbb{R}^2| x_1^2 + x_2^2 =1, x_1 \geq 0, x_2 \geq 0\}$$ There is something I found curious about the unit arc which is that, It has an empty interior viewed as a ...
Olórin's user avatar
  • 179
1 vote
0 answers
92 views

Topological space modeled by special topological structures

Let $X$ be a topological space. Suppose it is "modeled by" topological spaces of the form $\text{Spec}(A)$ for some commutative ring $A$, then, (along with some other conditions/structure), we call $...
Praphulla Koushik's user avatar
6 votes
0 answers
132 views

Generalization of pseudogroups

Pseudogroups are defined here: https://ncatlab.org/nlab/show/pseudogroup One of the problems with defining manifolds in terms of pseudogroups is that it gives no notion of a morphism between manifolds,...
Joshua Meyers's user avatar
10 votes
1 answer
1k views

Normal bundle of Whitney embedding

Let $X$ be a real $n$ dimensional manifold. One knows that it can be embedded into $\mathbb{R}^{2n}$ by the Whitney embedding theorem. The normal bundle for such an embedding will be a rank $n$ real ...
Arkadij's user avatar
  • 988
4 votes
0 answers
105 views

How is the instanton Floer homology of Seifert fibrations related to that of a trivial fibration

My question centers around the relationship of the Chern-Simons theories of a Seifert fibration and the trivial product space $\Sigma_g \times S^1$, and its implication for instanton Floer homology. ...
Mtheorist's user avatar
  • 1,155
4 votes
1 answer
757 views

Homotopy groups of fiber products

Let $X, Y, B$ be three smooth manifolds, and $f : X\to B$, $g : Y\to B$ submersions. Then $X\times_BY$ exists. (1) If $X, Y, B$ have the homotopy type of a finite CW complex, does $X\times_BY$? (2) ...
John P.'s user avatar
  • 180
6 votes
0 answers
376 views

Topological Singularities in Affine Varieties

Let $X$ be an affine variety over $\mathbb{C}$. Let $x\in X$. If $x$ is non-singular, then $x$ is locally holomorphic (in the Euclidean topology). See here for a relevant MO post. By results of ...
Sean Lawton's user avatar
  • 8,529