Questions tagged [differential-topology]
Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.
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Prove that each connected component of a surface is also a surface.
QUESTION: Prove that each connected component of a surface is also a surface. (Hint: Connected components of a locally connected space are open.)
Definition: A space $X$ is said to be locally ...
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$\mathbb{R}$ is a smooth manifold
I want to show $M=\mathbb{R}$ with an atlas of chart $U=\mathbb{R}$, $f:U\rightarrow\mathbb{R}$ such that $f(x)=2x$ when $x\geq 0$ and $f(x)=3x$ when $x\leq 0$. Then $M$ is a smooth manifold which is ...
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Proving $D$ is an exterior derivative.
I want to show that following $D$ is an exterior derivative.
$D:\oplus_i\Omega^i(U)\rightarrow \oplus_i\Omega^i(U)$ is a linear map. Also $D^2=0$, $D(\omega\wedge\tau)=D(\omega)\wedge\tau+(-1)^k\omega\...
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Best books for self-studying differential geometry
Next semester (fall 2021) I am planning on taking a grad-student level differential topology course but I have never studied differential geometry which is a pre-requisite for the course. My plan is ...
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29 views
Manifolds with boundaries: Why map boundary points onto boundary points?
The notion of a manifold with boundary has just been introduced in my script. If $\mathbb{R}^n_+$ is defined as $\mathbb{R}^n_+ := \{ (x^1, \ldots, x^n) \in \mathbb{R}^n \; | \; x^n \geq 0 \}$, then a ...
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1answer
38 views
Confusion about lift in the context of tangent bundle
A lift is defined here: https://mathworld.wolfram.com/Lift.html as a tangent vector field $X$ on a manifold, the same way that a section of the tangent bundle gives us $X$ in the context $\dot g = X(g)...
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What part of the Lyapunov spectrum is conserved in a time-delay embedding under Takens' theorem?
My understanding is that time-delay embeddings are often used to estimate the maximum Lyapunov exponent of a chaotic system for which we may not have full state measurements. It seems that you cannot ...
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65 views
Does two different extensions of a map between manifolds have the same derivative in the manifold-domain?
It is well know that the differential calculus is developed with respect open sets but a manifold not necessarily is an open set and actually many times it not. So in Differential Geometry it is usual ...
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25 views
Density of rank $k$ smooth maps
I'm really interested in the questions of density in functional space. I saw in a differential geometry course the notion of rank of smooth maps (the rank of the Jacobian).
Let us consider here the ...
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22 views
divergence theorem: what means continuously differentiable in the presence of countably finite discrete charges
Context
This question is similar, but not identicle, to [3]. Though in [3], it appears the central question revolves around a piece-wise transition between two regions of the real number line.
From [1]...
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47 views
Existence of a smooth homotopy
Suppose I have a contractible neighborhood $B$ inside a manifold $M$. Then I would like to justify that there exists a family of smooth maps $F_t:M\rightarrow M$ such that $F_0=id$ and $F_1(B)=\{p\}$, ...
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131 views
Can we argue that most mappings of $\mathbf{\mathbb{R} P^{\rm 2}}$ into $\bf\mathbb{R}^{\mathrm{3}}$ are “mostly” injective?
From Whitney/Massey/Cohen/etc. we know $\mathbf{\mathbb{R} P^{\mathrm{2}}}$ can be immersed in $\mathbf{\mathbb{R}^{\mathrm{3}}}$ (n.b. the Boy surface). Obviously this is not sufficient to assert ...
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16 views
Example of Inverse Function Theorem for Smooth Manifolds
I'm trying to use the Inverse Function Theorem
Let $f:U\to {R^k}$ be a smooth map with $U$ open in ${R^k}$
The theorem states, If the derivative $df_x:{R^k}\to {R^k}$ is nonsingular, then f maps any ...
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30 views
How differentiate the stereographic projection map
Let $f:\mathbb{S^2} \setminus {(0,0,1)}\to\mathbb{R^2} $ be the stereographic projection of a sphere onto a plane.
Find the derivative of $f$.
Note: You should get a linear map from 2-d space to 2-d ...
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1answer
57 views
Clarification between charts being incompatible and smooth structures being “exotic”
Inspired by this question, $\mathbb{R}$ can be given the atlas $\mathcal{A} = \{f(x) = x : x \in \mathbb{R}\}$ or the atlas $\mathcal{B} = \{g(x) = x^3 : x \in \mathbb{R}\}$.
These are incompatible ...
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144 views
What's the intuition behind the Co-Area formula?
I mainly work in statistics and I know only basic measure theory. I was trying to understand the Co-Area formula by Federer.
If $f:\mathbb{R}^M\to \mathbb{R}^N$ is a Lipschitz function with $M \geq N$...
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1answer
34 views
Helicoid in $\mathbb{R}^4$
I'm considering a parametric equation of helicoid in $\mathbb{R}^4$ is $X(u,v)=(u \cos v, u \sin v,v,v).$
Now I can calculate two unit tangent vectors $t_1=\dfrac{X_u}{\vert X_u\vert}=(\cos u, \sin u, ...
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49 views
Possibility of confusion caused by the use of $\nabla$
This is a question about notation. Of course, as long as the notation is clearly defined, it doesn't matter at all which notation we use, but it's still helpful to ask about a few possible confusions ...
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1answer
34 views
Degree of a polynomial as smooth map - clarification
I would like to a clarification of the following question. I cannot understand the last three lines, where we conclude that the exponents $n_1=n_2=...n_k=1$, which then concludes our proof. Cannot ...
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56 views
Tangent space of a manifold
For a manifold, $M$, at point $p$ we have a set of real-valued smooth function, $C^\infty_p(M)$ passing through $p$. To prove the tangent space at point $p$ we assume that we have map $C^\infty_p(M) =\...
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32 views
Degree of a smooth proper function, smooth homotopy and their pullbacks
Let $M$ and $N$ be compact manifolds, and $F, G: M \rightarrow N$ two smooth maps such that $F \sim G$, where $\sim$ means smooth homotopic. Then a well-known result is that the degree of $F$ is the ...
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1answer
34 views
Is the Whitney topology finer or coincide with uniform topology for codomain is $\mathbb{R}$
Let $C^{r}(U,\mathbb{R})$ be the space of $C^{r}$ functions from open subset $U\subseteq\mathbb{R}^{n}$ to $\mathbb{R}$, $0\leq r<\infty$. This space can equipped with the Whitney topology(or ...
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40 views
How can I prove $\sum\limits_{x \in f^{-1}(y)}\mkern-9mu \operatorname{sgn}(d_xf)=1$?
Definition: If $M \subset \mathbb{R}^k$ and $N \subset \mathbb{R}^d$ are two smooth oriented varieties and if $f: M \to N$ is smooth then we define
$$ \operatorname{sgn}(d_x f)=
\begin{cases}
\phantom{...
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72 views
Let $F:\mathbb{R}^n \to \mathbb{R}^m$ be a smooth surjective submersion. Are the fibers of $F$ necessarily connected?
Let $F:\mathbb{R}^n \to \mathbb{R}^m$ be a smooth surjective submersion. Are the fibers of $F$ necessarily connected? What if we substitute $\mathbb{R}^n$ and $\mathbb{R}^m$ with open neighborhoods of ...
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1answer
38 views
Can an Einstein manifold be scalar flat at some point but not Ricci flat?
Can an Einstein manifold be scalar flat at some point but not Ricci flat? Are the following equations correct, where $k=constant,$ if we start with scalar curvature,
$$scal = k$$
$$g_{\mu\nu}R^{\mu\nu}...
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67 views
Integrating over homotopy classes
We know that $Öæ\pi_2(\mathbb{S}^2)\cong\mathbb{Z}$. Given a fixed degree, say $n\in\mathbb{Z}$, is there a standard, conventional way to parametrize continuous maps $\mathbb{S}^2\to\mathbb{S}^2$ which ...
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1answer
19 views
Local Preservation of Orientation Implies Global Preservation
I am trying to work through some problems in differential topology, and I came across one that I can't figure out, mostly because I'm having trouble understanding some of the definitions.
Let $f \...
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1answer
35 views
Variational Formulation of Yamabe Problem Equivalence.
Suppose we have an n-manifold ($M$, $g_0$) with scalar curvature $S_0$. I'm aware that the Yamabe problem can be classed as the problem of minimising the following Einstein-Hilbert Functional over the ...
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2answers
59 views
Inverse limit without category theory
In Hirsch's book Differential Topology, he by and large does not use any category theory, with the exception of one passage on pg. 52 which I am trying to understand. It is as follows (paraphrased):
...
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1answer
71 views
Computing the Euler characteristic of real projective space $\mathbb{R}P^{n}$
I would like to compute the Euler characteristic of $\mathbb{P}^n(\mathbb{R)}$. I do not know if cohomology could help but I should avoid it because I did not studied it yet. I would like to use only ...
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38 views
Is the product of a differential always equal to the differential of the product?
Let $f \times g:X \times Y \rightarrow Z \times W$ be the product of two smooth functions $f:X \rightarrow Z$ and $g:Y \rightarrow W$ in the category of smooth manifolds.
My question is the following:
...
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1answer
40 views
How can I find a smooth map with arbitrary degree?
Suppose $k \in \mathbb{Z}$. I would like to find a smooth map $f : \mathbb{S}^n \to \mathbb{S}^n$ such that $\deg(f)=k$.
I tried with taking $k$ sets in $\mathbb{S}^n$ diffeomorphic to $\mathbb{D}^n$ ...
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50 views
Proving the Stiefel Manifold is a Manifold
My goal is to prove that the Stiefel manifold, $S(2,n) = \{(v,w) \in \mathbb{R}^n \times \mathbb{R}^n = \mathbb{R}^{2n}: |v|=|w|=1, v \perp w\}$ is a manifold by defining a diffeomorphism from $S(2,n)$...
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61 views
Guillemin and Pollack, Exercise 4.8.7, understanding the hint.
I have copied below exercise 4.8.7 from Guillemin and Pollack's Differential Topology
I'm having trouble following the last part of the hint (Exercise 7 of Section 6 refers to the fact that homotopic ...
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32 views
Last step in Voisin's proof of Ehresmann's lemma
I'm reading Voisin's proof of Ehresmann's lemma, given in the book Hodge Theory and Complex Algebraic Geometry I as theorem 9.3; the proof is essentially reproduced in this answer.
To summarize, there'...
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1answer
48 views
Why a continuous map $f : M \to \mathbb{S}^n$ can be approximated by a smooth function?
Let $M \subset \mathbb{R}^N$ be a smooth manifold of dimension $m$ where $m<n$. Suppose $f : M \to \mathbb{S}^n$ is a continuous map. How can I prove that for every $\varepsilon >0$ I can find a ...
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1answer
73 views
Every embedding gives rise to an embedded submanifold?
I call regular (or embedded) submanifold of a manifold $M$ a subset $i:X \subset M$ endowed with a submanifold structure such that the inclusion map $i$ is an embedding.
I call embedding an immersion ...
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1answer
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A physics approach to the Jordan curve theorem.
The Jordan curve theorem states that if $f:S^1\to \mathbb R^2$ is an injective continuous function then $\mathbb R^2\setminus \text{image}(f)$ has two connected components. I want to discuss an ...
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Why is $ \frac{1}{4\pi} \int \int_S dx \, dy\, \vec{n} \cdot ( \partial_x \vec{n} \times \partial_y \vec{n} ) = 1 $?
I am trying to understand this notion of "topological charge" from a paper in Physics Review Letters. They talk about skyrmions and give this integral (here $\vec{n}$ is a vector field on ...
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1answer
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Question About Maximal Smooth Atlas
Definition of Maximal Smooth Atlas: In the text GTM $218$ by John Lee, the author defines a smooth altas $\mathcal{A}$ to be a maximal smooth atlas if any chart that is smoothly compatible with every ...
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connected components of domain without preimage of critical values
Let $f:M \to N$ be a differentiable map between two connected compact manifolds both of dimension $n\ge 2$ (I think i know the answer to my question if $n=1$). Let us assume that both $M$ and $N$ are ...
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1answer
42 views
Directional derivatives at $P$ are all derivations at $P$
Hi i am reading An introduction to Manifolds by Loring. And have one doubt under section 2.3 Derivations at a point. It is written that
we know that directional derivatives at $p$ are all derivations ...
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1answer
67 views
Open annulus covers any orientable non simply-connected surface without boundary.
$\textbf{Problem:}$ Let $\Sigma$ be an orientable surface without
boundary, possibly non-compact, with the non-trivial fundamental
group. Then there is a covering map $p:\Bbb S^1\times \Bbb R\to
\...
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58 views
Can every cohomology class be represented by an analytic form
Let $M$ be an analytic manifold (you may assume it is equipped with an analytic metric). Must each De Rham cohomology class be representatble by an analytic differential form ?
I think Hodge theory ...
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Almost every vector space $V$ of any fixed dimension $l$ in $\mathbb R^N$ intersects $X$ transversally.
It is a problem from Guillemin and Pollack.
Suppose that $X$ is a submanifold of $\mathbb R^N$. Show that almost every vector space $V$ of any fixed dimension $l$ in $\mathbb R^N$ intersects $X$ ...
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Any neighborhood $\tilde{U}$ of $Y$ in $\mathbb{R}^M$ cointains some $Y^{\epsilon}$
Show that any neighborhood $\tilde{U}$ of $Y$ in $\mathbb{R}^M$ cointains some $Y^{\epsilon}$; moreover, if $Y$ is compact, $\epsilon$ may be taken constant. [HINT: Find covering open sets $U_{\alpha}^...
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1answer
107 views
When does $H^{n-k}(M,\Bbb R)\simeq \Bbb R$ imply $H_{k}(M,\Bbb Z)\simeq \Bbb Z$?
I am not sure that my question is a trivial fact or not or even make sense or not. Anyway I want to know
When does cohomology group $H^{n-k}(M,\color{blue}{\Bbb R})\simeq \Bbb R$ imply homology group ...
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68 views
Homology and mapping spaces?
Suppose that we have an oriented $n$-manifold M and an oriented $i$-manifold $X.$ Consider the mapping space (or function space) $F(X,M)$ equipped with the compact open topology. Now, let $f:M\to M$ ...
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1answer
54 views
Lie algebra of abelian Lie group is compact?
Is it true that the Lie algebra of and abelian group is compact? For a compact Lie algebra I mean a Lie algebra of some compact Lie group.
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43 views
$C^\infty$ Manifolds. Charts.
Let $ (M,\mathcal{O},A_{\infty})$ be a $C^\infty$-Differentiable dimension-d Manifold.
Recall that this setup means : $M$ is a set, $\mathcal{O}$ a topology, $A_{\infty}$ an atlas of charts "$x:M\...
