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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.

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What is this $\mathfrak{D} =\mathfrak{D}(\mathfrak{U})$ in differentiable topology

From the textbook: Introduction to Differential Topology by TH. Brocker and K. Janich. Defintion 1.3: An atlas of a manifold is called differentiable if all its chart transformations are differentiable. Then a few lines below: If $\mathfrak{U}$…
Cedric Martens
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$f : \mathbb{R}^n\rightarrow\mathbb{R}^n$ is smooth. Prove: if its tangent maps are orthogonal transformations, then f is orthogonal transformation...

This is a problem that I meet in manifold class. At the beginning, I want to show $\left $, $\forall x \in\mathbb{R}^n$, and then use Taylor's theorem with remainder, but I found that it is possibly incorrect since I meet the…
Dieck-W
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Alter a map to be orientation preserving

I'm reading the h-cobordism by Milnor and he claims that we can alter a map to make it orientation preserving(second paragraph in page 58). I'll give a detailed description in the following: Suppose $f:\mathbb{R}^n\to \mathbb{R}^n$ an orientation…
taiat
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What is the "standard" way of proving that the transition map is $C^{\infty}$?

Sorry if this is a vague question. I was handed a map between two manifolds, say, $M\to \mathbb{R^n}$. I was indeed able to find an open cover of $M$ and show that there is a homeomorphism between each open set and some open subset of…
able20
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Is every embedded smooth submanifold either open or closed?

One often wants to show that a smooth embedded submanifold is open or closed. Is it sufficient to show such a submanifold is not open to imply it is closed and vice versa?
Jeff
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When is it true that $F: N \rightarrow M_1 \times \cdots \times M_k$ is smooth iff each component map is smooth?

I'm reading Introduction to Smooth Manifolds by John M. Lee, and I have a question about Proposition 2.12 (=Problem 2-2) on P.36. Proposition 2.12. Suppose $M_1, \cdots, M_k$ and $N$ are smooth manifolds with or without boundary, such that at…
hidenoris
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How upper sheet of hyperboloid can be covered by a single coordinate system?

How upper sheet of hyperboloid can be covered by a single coordinate system? First let consider $f(x,y,z)=z^2-x^2-y^2-a^2$. Now i visualize from here that if we consider any open ball ($x^2+y^2
RAM_3R
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Smooth manifold and singular point

I came across a statement that looks like this: Let $f(x): x\in U\subseteq\mathbb{R}^d\rightarrow\mathbb{R}$ be an analytic function. Then, $\left\{x\in U:f(x)=y\right\}$ is a smooth manifold if $\nabla f(x)\neq0$ for all $x$ such that…
user15988
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Constructing a Differential Structure

I am currently undergoing a first course on Differentiable Manifolds. An an example I was trying to construct a differential structure for a 2D unit sphere $\mathbb{S}^2=\{(x,y,z):x^2+y^2+z^2=1\}$. By definition, the coordinate charts are as…
creative
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Proving that the Grassmanian is a smooth manifold

I am trying to find a differentiable structure on the Grassmannian, which is the set of all $k$-planes in $\mathbb{R}^{n} $. To do this, I have to show that for any given $\alpha$, $\beta$, the set $$\left\{…
Eigenfield
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Finding an explicit integral manifold

Let $\mathscr{D}$ be the $2$-dimensional distribution on $M:=\{(x,y,z)\in\mathbb{R}^3:x,y,z>0\}$ generated by $\{X,Y\}$ where $X=x\partial_x - 2y\partial_y, Y=xy\partial_y - xz\partial _z$. By a direct computation, I showed that $\mathscr{D}$ is…
Rubertos
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If $TM$ and $TN$ are diffeomorphic, then $M$ and $N$ are diffeomorphic.

I want to prove following statement: If $TM$ and $TN$ are diffeomorphic, then $M$, and $N$ are diffeomorphic. First my trial was using a natural projection between tangent bundle and manifold \begin{align} &\pi_M : TM \rightarrow M \\ …
phy_math
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if $f:M\to N$ is local diffeomorphism implies $f_{\star p}:T_pM\to T_{f(p)}N$ is an isomorphism

Let $M$ and $N$ be two finite dimensional and real smooth manifolds and $f:M\to N$ a local diffeomorphism at some point $p\in M$. Is the induced map $f_{\star p}:T_pM\to T_{f(p)}N$ an isomorphism?
FUUNK1000
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Change of coordinates in manifolds

Let $M$ be a smooth manifold of dimension $m$. Let's consider $(U,\varphi)$ a coordinate neighbourhood, such that $\varphi =(x^1,...,x^m)$. And let $(y^1,...,y^m)$ smooth maps between an open neighbourhood $V$ of a point $p \in U$ of $M$ and…
J. Salieri
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$M^n$ compact, $ f:M^n\to \mathbb{R}^n$ smooth $\Rightarrow f$ is not a submersion

Let $M^n$ be a smooth, compact manifold. Show that if $f:M\to\mathbb{R}^n$ is smooth, then $f$ is not a submersion. Let $n=1$, $M=(0,1)$ and $f:x\mapsto x$, then $f_{*_{x}}=1\neq 0$ for all $x\in(0,1)$, so $f$ is a submersion. Isn't this a counter…
rmdmc89
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