All Questions
Tagged with gn.general-topology dg.differential-geometry
127 questions
8
votes
0
answers
229
views
+300
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 ...
- 491
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 $(x, y) \in \mathbb{S}^2 \times \mathbb{S}^2$, ...
- 434
-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 ...
- 612
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 ...
- 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 ...
- 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 ...
- 612
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)
$$\dot{\mathbf{...
- 247
8
votes
2
answers
489
views
Continuous point map for spherical domains
Consider the space $J$ of Jordan domains on the sphere $\textbf{S}^2$, i.e., continuous injective maps from the unit disk into $\textbf{S}^2$ modulo homeomorphisms of the disk. How can one construct a ...
- 7,149
29
votes
2
answers
2k
views
Contractibility of the space of Jordan curves
Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.
If the curves are ...
- 7,149
5
votes
1
answer
165
views
Algebraic solutions of polynomial ODEs
Given a polynomial ODE in $n$-dimensions of maximal degree $d$
$$
\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
...
- 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 ...
- 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\...
- 193
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 ...
- 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 ...
- 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 $\...
- 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 ...
- 195
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{...
- 177
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 ...
- 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 ...
- 197
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 (...
- 349
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 $S:=\partial A$ ...
- 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 ...
- 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 ...
- 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 ...
- 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 ...
- 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 ...
- 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 $M \subset \mathbb{R}^n$.
$D_r (M): = \bigcup\limits_{x \in M} D_r (x)$
$Int_r (M): = \{x ~|~ D_r(x) \subset M\}$
...
- 6,962
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 ...
- 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 $C^\infty$ ...
- 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^...
- 2,934
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 $...
- 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 $k\in K$, ...
- 2,388
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,\...
- 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 ...
- 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 $ ...
- 27
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 ...
- 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 ...
- 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 ...
- 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,...
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}) \...
- 6,741
4
votes
1
answer
647
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 "...
- 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 ...
- 21
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 ...
- 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 ...
- 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 $...
- 6,023
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,...
- 181
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 ...
- 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. ...
- 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) ...
- 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 ...
- 8,529