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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An affine invariant notion of minimal surface?
The notion of minimal surface (i.e. having vanishing mean curvature) is not "affinely invariant" in the following sense: if $M\subset\Bbb R^n$ is an ($m$-dimensional) minimal surface, then $...
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Definition of uniqueness of tubular neighbourhoods
The tubular neighbourhood theorem states that if $M \subset N$ is an embedding of smooth manifolds without boundary and $\nu: E \to M$ is the normal bundle of $M$ in $N$, then there is a smooth ...
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Missing minus sign in pullback calculation of a $1$ form on $\mathbb{C}$
Consider $S^2$ with a coordinate chart given by the stereographic projection through the north pole. We identify $\mathbb{C}$ with $\mathbb{R}^2$, then for a point $(\theta,\phi)\in S^2$, the ...
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Local cross section in smooth manifolds
I have managed to prove that for every field $X$ of class $C^1$ in $\mathbb R^n$ there is a local cross section at a regular point of $X$. I would like to prove that this fact is also true for ...
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Show that $ \frac{x^2}{(x^2+y^2)^{3/4}} $ is continuous on $\mathbb{R}^2$.
I tried with polar coordinates, I found the limit is infinite (which is not true the graph show it's 0). I tried to majorate with something who has for limit 0 but still impossible.
with polar ...
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Regular homotopy between closed curves
I am stuck at an exercise regarding homotopies between closed curves. Specifically I want to find out whether there is a regular homotopy between the curves
$$
\beta: [0,2 \pi] \to \mathbb{R}^2, \ \...
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Can mathematics distinguish left and right?
Imagine, a mathematician from another galaxy lands on the earth. Is there a way we can explain to him what is "counterclockwise" without showing him a picture?
Things like Green's formula, ...
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Exercise related to vector fields, map degrees and Poincare-Hopf's Theorem
I got stuck with one exercise from Chapter 3.5 in Guillemin and Pollack's book, which I used to study differential topology by myself:
Given a vector field $\overrightarrow{v}$ with isolated zeros in $...
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Are invertibly cobordant manifolds diffeomorphic
Let $M$ and $N$ be oriented, closed, $n-1$ manifolds and $F$ a cobordism from $M$ to $N$ and $G$ a cobordism from $N$ to $M$ such that the composite cobordism $G\circ F\cong M\times I$ and $F\circ G\...
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Convexity in "usual Partition of Unity arguments"
I stumbled upon Problem 13-2 on p.344 in John Lee's Introduction to Smooth Manifolds (2nd Edition) where Lee explains that the proof for the existence of a Riemannian Metric on a manifold is done by a ...
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generalized Poincare conjecture [duplicate]
How to show that the claim that there exists exactly one differentiable structure on $S^4$ iff smooth four-dimensional Poincaré conjecture is true (homotopy equivalent to S4 implies diffeomorphic to ...
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Image of annulus under flow of vector field and differential equations
I'm studying Lee's introduction to smooth manifolds, in chapter 9 he introduces integral curves and vector flows, I'm using the following definitions.
Definition: Let M be a manifold a flow domain for ...
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The derivative of a function from $\mathbb{R}^3$ in $SO(3)$
Let $y=(y_1,y_2,y_3)$ and $\psi:\mathbb{R}^3 \longrightarrow SO(3)$ defined by $$\psi(x)=\prod_{i=1}^{3} R_x(x_i)R_z(y_i)$$
where $R_x,R_z$ are the rotation matrices around $x$-axis and the $z$-axis,...
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Homotopy equivalent to $\mathbb{S}^1$, but not homeomorphic to $\mathbb{R} \times \mathbb{S}^1$ [closed]
What is an intuitive example of a topological object which is homotopy equivalent to $\mathbb{S}^1$, but not homeomorphic to $\mathbb{R} \times \mathbb{S}^1$?
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Is There a Smooth Approximation to Classifying Spaces $BG\,?$
If you look at https://en.wikipedia.org/wiki/Chern%E2%80%93Weil_homomorphism, right above contents the claim is made that we can approximate the classifying space by smooth manifolds. I am aware of at ...
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Lee Smooth Manifolds, why does the Whitney Approximation Theorem fail when the co-domain has non-empty boundary?
I am trying to study chapter 6 of Lee's Introduction to Smooth Manifolds.
In a remark after the Whitney Approximation Theorem, Lee stated that this theorem do not hold because it might not be possible ...
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A question regarding the winding number
My main question is about part B, but I would also be grateful if you can tell me what you think about part A.
Define a smooth vector field
$X$ on $S^1$ as follows:
$X(x,y)=(-y,x)$. For a smooth map $...
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Transverse Foliation to the Flow of a Differential Equation on a Tangent Bundle?
Let $Q^n$ be a closed manifold, $M = TQ$ its tangent bundle, $\xi$ be a differential equation on $M$ that satisfies the "canonical flip on $TTQ$" (a "second-order differential equation ...
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Is the complement of a set with vanishing $(d-2)-$dimensional Hausdorff measure simply connected?
In the same vein as this question, I want to ask whether $\mathbb{R}^d\setminus A$ is simply connected if $A\subseteq \mathbb{R}^d$ has vanishing $(d-2)-$dimensional Hausdorff measure, i.e. ${\cal H}^{...
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Cotangent sheaf of smooth manifold
I was trying to carry out the algebraic geometry construction of the tangent sheaf in the case of smooth manifolds following the ideas outlined in the answer of this question, but I have a couple of ...
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Problem on finding Intersection form of compact,orientable $4-$manifolds .
$\mathbf {The \ Problem \ is}:$ Let $M$ be an $\mathbb{F}$-oriented manifold of dimension $2 n$ for a field $\mathbb{F}$. Consider the non-singular bilinear form $H^{n}(M ; \mathbb{F}) \otimes H^{n}(M ...
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Question about the connected sum of two smooth manifolds
I'm a little confused with the following two questions about connected sum:
(1) Is the covering space$\widehat{M\# N}$ of the connected sum of two smooth closed manifolds $M, N$ the connected sum $\...
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Lee Smooth Manifolds Theorem 6.23's Jacobian Matrix
I am trying to study the tubular neighborhoods and the normal space/bundle section of Lee's Introduction to Smooth Manifolds.
I have a minor question about the proof of Theorem 6.23 which states: If $...
- 511
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Global extension of smooth function
In his book "Topology from the differentiable viewpoint", Milnor defines a function $f:X\to \mathbb{R}^m$ (where $X$ is an arbitrary subset of $\mathbb{R}^n$) to be smooth if it can be ...
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Need help understanding orientation of a manifold as defined in GP
I am reading Guillemin & Pollack's Differential Topology and I love it so far. However, the chapter about orientation of a manifold makes me struggle quite a bit. I have yet fully understood it. ...
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$T\setminus\{p\}$ and $S^1\wedge S^1$ are smoothly homotopically equivalent: rigorous proof
Let $T=S^1\times S^1$. I want to prove that $M:= T\setminus\{p\}$ and $N:=S^1\wedge S^1$ are smoothly homotopically equivalent.
I Just know the following facts:
M and N are $C^0$-homotopically ...
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Lee Smooth Manifolds - Lemma (6.14) for Whitney Embedding Theorem
I am studying Lee's Introduction to Smooth Manifolds but I am stuck by a lemma for Whitney's Embedding Theorem.
This lemma is trying to prove that: If a smooth n-manifold admits a smooth embedding $\...
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Example of a smooth function $f:[0,1)\to \mathbb S^n$ that does not extend to $[0,1]$
I am looking for examples of smooth functions $f:[0,1)\to \mathbb S^n$ that do not extend to
a smooth function $\hat f:[0,1]\to \mathbb S^n$, and
a continuous function $\hat f:[0,1]\to \mathbb S^n$....
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Levi-Civita Connection with Geodesic Spray Containing Flow Lines of Time-Independent Vector Field?
Let $Q^n$ be a closed manifold and $M = TQ$ be its tangent bundle.
In [1], it is worked out in Equation 5.31 that there is a kinetic energy Riemannian metric $g$ on $Q$ with Levi-Civita connection $\...
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What is the lower $n$ such that $\mathbb{HP}^2$ can be embedded in $\mathbb{R}^n$?
Applying basic cohomological calculus it is possible to derive that $\mathbb{HP}^2$ cannot be embedded in a Euclidean space of dimension 11 or less.
From other side, applying spinorial cohomological ...
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Realizing possible accelerations of paths on a sphere
$\newcommand{\al}{\alpha}$
Let $x,v \in \mathbb{S}^n \subseteq \mathbb{R}^{n+1}$, $w \in \mathbb{R}^{n+1}$ satisfy
$\langle x,v \rangle=0, \langle x,w \rangle=-1$.
Does there exist a smooth path $\...
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The way $C^{k_1}$ and $C^{k_2}$ geometries are different.
I want to illustrate the point that $C^{k_2}$ regularity cares more about geometry than $C^{k_1}$ regularity when $k_1 > k_2$. Do you have any example of two $C^1$-submanifolds of $\mathbb{R}^3$ (...
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Characterizing accelerations of paths in a submanifold
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\Hs}{\operatorname{Hess}}$
$\newcommand{\al}{\alpha}$
This is a curious inquiry:
Let $f:\R^N \to \mathbb{R}^n$ be a smooth map,...
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When is the leaf space $M/\mathcal{F}$ of a foliation smooth?
Let $(M,\mathcal{F})$ be a regular smooth foliation on a manifold $M$. In general, the leaf space $M/\mathcal{F}$ is quite pathological. However, when $M$ is a Poisson manifold with compact, 1-...
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Are these two questions asking the same thing?
Question $(I).$
Show that if the $n$-dimensional manifold $M$ is a product of spheres, then there exists an embedding $M \to \mathbb R^{n+1}.$
Question $(2).$
Show that there exists an embedding $S^{...
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showing that a function is smooth [closed]
Here is the question I am trying to understand its answer:
Here is the answer given to me at the back of the book:
Here is the definition of a smooth map:
Still I do not understand the solution ...
- 1,191
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Riesz's representation of a k-current
Reading "Introduction to GMT" by Simon, at page 136 he says that thanks to Riesz's representation theorem we can view k-currents ar Radon measures, or to be more precise he says that given a ...
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The only maximal ideal of the set of all function germs around $p$
Here is the definition we are using for the set of all function germs around p:
Now, I want to show that $m(p) := \{\bar{\phi} \in \mathcal{\varepsilon}(p)| \bar{\phi}(p) = 0\}$ is the only maximal ...
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Is it proper to speak of the inverse image of a set containing elements not in the image of the mapping?
This may be a duplicate of Inverse image of a subset of the codomain with elements without corresponding elements in domain
, but the answer given seems to contradict what is intended by CH Edwards in ...
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The total space of the standard quaternionic Hopf fibration of an $S^3$ fiber bundle over $S^4.$
I am trying to understand the paper "Milnor's Construction of Exotic 7-Spheres" by Rachel McEnroe (link). Here's the abstract:
I do not understand why the total space of this fiber bundle ...
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Chern class of bundle of linear functions on $G(2,4)$
Let $G=G(1,3)$ be the grassmanian of lines in $\mathbb{C}P^3$. Let $E$ be the bundle on $G$ whose fiber over $\ell$ are homogeneous linear functions on $\ell$. I am interested in $c(E)=1+c_1(E)+c_2(E)$...
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Solving a simple partial differential equation.
I'm trying to find every continuously differentiable functions $f:\mathbb{R}^2\to \mathbb{R}$ such that $$\alpha \frac{\partial f}{\partial x}(x,y)+\beta\frac{\partial f}{\partial y}(x,y)=0 $$ where $\...
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Understanding how the local trivializations were calculated.
I am trying to understand the paper "Milnor's Construction of Exotic 7-Spheres" by Rachel McEnroe (link). Here's the abstract:
But when it comes to calculating the local trivialization of ...
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Understanding the transition map in the case of the proof that $S^3$ bundles over $S^4$
Here is the part of the paper of "Rachel Mcenroe" on Milnor's construction of Exotic 7-spheres:
But I do not understand why the transition map is $\frac{1}{z}$ and it does not include any $\...
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Regarding linking number of oriented knots
Let $K,L\subset R^3$ be oriented knots.
Assume that $S\subset R^3$ is a two-dimensional compact connected oriented sub-manifold, $\partial S=K$, and the orientation given on $K$ merges with the ...
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Intuitively understanding why $\pi_1(S^2) = 0$
When we are saying that $\pi_1(S^2) = 0,$ are we speaking about a solid sphere or a hollow sphere? as far as I understand the fundamental group of a topological space is a measurement for the holes in ...
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How to prove that the image is a submanifold?
Here is the question I am trying to tackle:
Show that the map $$f : \mathbb R P^n \to \mathbb R P^{n + 1},$$ defined by $$[p] = [p_0, \dots, p_n] \mapsto [p,0] = [p_0, \dots, p_n, 0]$$
is an embedding....
- 1,191
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Can an higher degree differential equation of the form$ f(x,y)=\prod _{{i=1}}^{{n }}{p-g_{i}(x,y)}$ be solved?
Here is the question I have been given.
$ \qquad\qquad\qquad\qquad px^4 +yx^3+cosec(xy)=0$
I know it can be solved by and then converting it into an exact differential form
$\qquad \qquad \qquad \...
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what about the case $V_x=U_x$ in the proof of paracompactness Theorem (John Lee book)?
Here the proof of "paracompactness Theorem" (Theorem 4.77) of the book "Introduction to topological manifold" - John M. Lee (2ed):
When $\mathcal{U} = \{W_j\}$ then $V_x = U_x \...
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Show that an embedding sends a conjugate transpose to transpose.
Here is the question I am trying to tackle:
Prove that $$\mathbb C \to M_2(\mathbb R),$$ defined by $$x + iy \mapsto \begin{pmatrix} x & -y \\ y & x\end{pmatrix}$$ defines an embedding. Show ...
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