Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [smooth-manifolds]

For questions about smooth manifolds, a topological manifold with a maximal smooth atlas.

A topological manifold of dimension $n$ is a Hausdorff, second countable topological space $X$ such that each $x \in X$ has a neighbourhood homeomorphic to an open subset of $\mathbb{R}^n$.

A smooth atlas on a topological manifold $X$ is a collection of pairs $\left\lbrace(U_{\alpha}, \varphi_{\alpha})\right\rbrace_{\alpha\in A}$ where $U_{\alpha}$ is an open subset of $X$ and $\varphi_{\alpha} : U_{\alpha} \to \varphi_{\alpha}(U_{\alpha}) \subseteq \mathbb{R}^n$ is a homeomorphism such that $\bigcup\limits_{\alpha\in A}U_{\alpha} = X$.

A topological manifold with a maximal smooth atlas is called a smooth manifold.

6282 questions
11
votes
1 answer

Critical points of Torus height function

I'm reading Tu's intro to manifolds. He defines a critical point of a smooth map $F:N\to M$ to be a point where the differential $F_{*,p}:T_pN\to T_{F(p)}M$ fails to be surjective. He then gives illustration of the $2$ torus embedded in three space,…
user124910
  • 3,007
8
votes
1 answer

Is it possible to give the unit square a smooth structure?

At the beginning of Lee's "Introduction to Smooth Manifolds", Lee gives the example of a the square and the circle being homeomorphic as an intuitive motivation for smoothness not being invariant under homeomorphisms. But I read somewhere that the…
rhodonedra
  • 85
7
votes
1 answer

Is the maximal atlas for a topological manifold unique?

I'm reading "An Introduction to Manifolds" by Tu. In this book, the definition of a topological manifold is a Hausdorff, second countable locally Euclidean space and the definition of a smooth manifold is a topological manifold with a maximal…
baby bunny
  • 553
7
votes
2 answers

Is every diffeomorphism an element of a one parameter group of diffeomorphims?

I understand that a smooth vector field on a manifold $M$, generates a "flow"/one parameter group action, lets say $\sigma(t,s): \mathbb{R} \times M \rightarrow M$, and $\sigma_t: M \rightarrow M$ gives a one parameter group of diffeomorphisms. My…
E. Seyma Kutluk
  • 71
7
votes
1 answer

Computation of Pushforward and Pullback

After reading about the pushforward and pullback, I don't really have a concrete grasp of them, so I think these simple questions might clear things up for me; I appreciate any hints or solutions. Let $(s,t)$ be coordinates on $\mathbb{R}^2$ and…
Squirtle
  • 6,698
7
votes
1 answer

Refinement of an Open Cover

This seems rather simple, but just curious about the following definition (pulled from Lee, but definitely standard): Given an open cover $\mathcal{U}$ of $X$, another open cover $\mathcal{V}$ is called a refinement of $\mathcal{U}$ if for each…
Squirtle
  • 6,698
6
votes
1 answer

Why do these unwanted partial derivatives vanish?

Problem Let $(q = 0,\; \dot q=0)$ be an equilibrium point for a dynamical system, that is, a solution of Lagrange's equations $d/dt(\partial L / \partial \dot q^k) = \partial L / \partial q^k$ for which $q$ and $\dot q$ are identically 0. Here $L =…
Buster
  • 235
  • 1
  • 8
5
votes
3 answers

Compute $f_*:T_p\mathbb{R}^m\to T_{f(p)}\mathbb{R}^n$.

I am taking a course on smooth manifolds. We have just defined a tangent space of a manifold $M$ at a point $p\in M$ using paths $\gamma:\mathbb{R}\to M$, $\gamma(0)=p$. So $T_pM:=\{\gamma:\mathbb{R}\to M\;|\;\gamma(0)=p\}/\sim$, where $\sim$ is the…
Joffysloffy
  • 1,334
5
votes
1 answer

On a remark of John Lee concerning smooth manifolds

In John Lee's book Introduction to Smooth Manifolds, on the first page of the first chapter, he writes: ..."But for more sophisticated applications it is an undue restriction to require smooth manifolds to be subsets of some ambient Euclidean…
uniquesolution
  • 18,541
5
votes
0 answers

Why do this kind of tori are not manifold.

A torus is the set of points in $\mathbb{R}^3$ at a distance $b$ from the circle of radius $a$. I want to know why if $a=b$ or $b>a$ then the "torus" given is not a smooth manifold. I can figure it out intuitively speaking, but I want a formal…
EQJ
  • 4,369
5
votes
1 answer

Germs of Smooth Functions vs Germs of Analytic Functions?

Let $M$ be a smooth manifold, $U\subseteq M$ an open subset and $p\in M$. Then $C^\infty(U)$ is naturally an $\mathbb R$-algebra. Define: $$I_p(U):=\{f\in C^\infty(U):\exists W\subseteq U\textrm{ open with }p\in W\textrm{ and }f|_W=0\}.$$ It is easy…
PtF
  • 9,655
4
votes
1 answer

Vector Bundle as a smooth manifold

Definition: It is said that a section $F:M\to E$ of a vector bundle $E$ is smooth if it is smooth as a map between manifolds. Possible Issue: A vector bundle is defined to be, a priori, a smooth manifold, which means that it has some implicit…
Dave
  • 582
4
votes
0 answers

What is the standard smooth structure of a disjoint union of smooth manifolds?

I am following Lee in his introduction to smooth manifolds page 442. He explains that if $\{M_j\}_{j = 1}^\infty$ is a countable collection of smooth $n$-manifolds with or without boundary and let $M = \coprod_{j=1}^\infty M_j$, then there is an…
Mikkel Rev
  • 1,850
4
votes
1 answer

How to prove that the boundary of $Q = [0,1]^3$ isn't smooth manifold?

Given $Q = [0,1]^3$ in $\mathbb{R}^3$, how can we prove that $\partial Q$ is not smooth manifold in $\mathbb{R}^3$? I can understand that its not, because of the connection line between $2$ sides of the cube, but what is a formal explanation for…
Idan Perez
  • 73
4
votes
1 answer

Show that $\mathbb{B}^n $ is a smooth manifold with its boundary diffeomorphic to $S^{n-1}$

Consider a closed unit ball $\mathbb{B}^n = \{ x\in\mathbb{R^n} : \|x\|\le 1\} $ How do I show that $\mathbb{B}^n $ is a smooth manifold with its boundary ($\partial \mathbb{B^n}$) diffeomorphic to $S^{n-1}$ ?
Dark_Knight
  • 2,008
1
2 3
…
10 11