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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
4 views

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 $...
user avatar
  • 28k
3 votes
0 answers
16 views

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 ...
user avatar
  • 1,446
2 votes
1 answer
26 views

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 ...
user avatar
0 votes
0 answers
13 views

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 ...
user avatar
  • 1,463
-2 votes
1 answer
55 views

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 ...
user avatar
  • 59
0 votes
1 answer
40 views

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, \ \...
user avatar
  • 1,129
4 votes
1 answer
282 views

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, ...
user avatar
  • 2,869
4 votes
0 answers
41 views

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 $...
user avatar
3 votes
1 answer
32 views

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\...
user avatar
  • 1,005
4 votes
1 answer
58 views

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 ...
user avatar
  • 2,126
-2 votes
0 answers
53 views

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 ...
user avatar
0 votes
0 answers
22 views

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 ...
user avatar
1 vote
0 answers
44 views

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,...
user avatar
0 votes
1 answer
63 views

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 $\mathbb{R} \times \mathbb{S}^1$?
user avatar
  • 1,155
1 vote
0 answers
38 views

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 ...
user avatar
  • 5,821
1 vote
0 answers
34 views

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 ...
user avatar
  • 511
6 votes
1 answer
118 views
+50

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 $...
user avatar
  • 702
1 vote
0 answers
12 views

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 ...
user avatar
3 votes
1 answer
32 views

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}^{...
user avatar
0 votes
0 answers
46 views

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 ...
user avatar
0 votes
0 answers
44 views

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 ...
user avatar
0 votes
0 answers
39 views

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 $\...
user avatar
2 votes
1 answer
36 views

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 $...
user avatar
  • 511
0 votes
0 answers
15 views

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 ...
user avatar
2 votes
0 answers
21 views

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. ...
user avatar
  • 31
0 votes
0 answers
43 views

$T\setminus\{p\}$ 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 ...
user avatar
  • 179
0 votes
1 answer
37 views

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 $\...
user avatar
  • 511
0 votes
0 answers
40 views

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 $\hat f:[0,1]\to \mathbb S^n$....
user avatar
1 vote
1 answer
52 views

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 $\...
user avatar
2 votes
0 answers
65 views

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 ...
user avatar
  • 2,065
1 vote
2 answers
58 views

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 $\...
user avatar
  • 23.1k
2 votes
0 answers
64 views

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 $\mathbb{R}^3$ (...
user avatar
0 votes
0 answers
51 views

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,...
user avatar
  • 23.1k
1 vote
0 answers
16 views

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-...
user avatar
  • 121
0 votes
1 answer
28 views

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^{...
user avatar
  • 1,191
2 votes
1 answer
69 views

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 ...
user avatar
  • 1,191
2 votes
1 answer
50 views

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 ...
user avatar
  • 2,753
0 votes
0 answers
45 views

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 ...
user avatar
  • 1,191
0 votes
0 answers
39 views

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 ...
user avatar
1 vote
1 answer
52 views

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 ...
user avatar
2 votes
0 answers
66 views

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 $c(E)=1+c_1(E)+c_2(E)$...
user avatar
0 votes
1 answer
46 views

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 $\...
user avatar
0 votes
0 answers
33 views

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 ...
user avatar
0 votes
1 answer
29 views

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 $\...
user avatar
6 votes
0 answers
132 views

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 ...
user avatar
  • 702
4 votes
1 answer
117 views

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 ...
user avatar
  • 47
1 vote
1 answer
160 views

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....
user avatar
  • 1,191
0 votes
0 answers
21 views

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 \...
user avatar
3 votes
1 answer
29 views

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 \...
user avatar
  • 351
0 votes
1 answer
41 views

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 ...
user avatar
  • 1,191

1
2 3 4 5
132