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Questions tagged [manifolds]

For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

In mathematics, a manifold of dimension $n$ is a topological space that near each point resembles $n$-dimensional Euclidean space. More precisely, each point of an $n$-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension $n$. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot.

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Why $M = \{(x,y)\in \mathbb{R}^2 \vert x^3+y^3 = 0\}$ seems not to be a manifold for $(x,y) = (0,0)$?

Mathematically this can't be counted as manifold because $\textbf{rank}D_f(0,0) = (3\cdot 0^2\quad 3\cdot0^2) = 0$. However, I see no problem at this point the reason being the manifold is a straight line $(y = -x)$
Leon
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For Which Additional constant a Manifold is guaranteed?

Set is given by $M = (x,y) \in \mathbb{R}^2 \vert \underbrace{e^{x^2+2\,y^2+2}}_{f} = c$. Instantly I noticed $M$ can be counted as Manifold if $c \in(e^2,\infty)$ because $f \geq e^2$. Now apparently this is not complete, so I wonder which…
Leon
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On definition of charts of tangent manifold

Let $\{\phi_i:U_i\to V_i\}$ be an atlas of manifold $M^m$. Tangent vector is a an equivalence class $[x,i,u]$ where, $x\in U_i, u\in\mathbb R^m$. Two classes $[x,i,u]$ and $[x',j',v]$ are equal if $x=x'$ and…
Madara Uchiha
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Example of $C^k$ manifold but not $C^{k+1}$ manifold

I try to construct an example which is not smooth but is $C^k$ manifold, but fail. In past, I always think (without any think) the graph of $y=|x|$ is not $C^1$ manifold, but just consider $(x, |x|) \rightarrow x$, I think the graph of $y=|x|$ has…
Enhao Lan
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Function from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$ - Higher dimensions

If we have a function such as: $\mathbb{R}$ to $\mathbb{R}$, $f(x)=x$, we have a one varible function, that is living in a two dimensional space, because n+n = 1+1 = 2, so, if we have the following function: $f(x,y)= (xy, xy^{2})$, it must be a two…
Astrum
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When does $f_{a,b}$ become immersion or embedding?

For $a, b \in \mathbb{Z}$, we define $f_{a, b}: S^1 = \mathbb{R}/\mathbb{Z} \rightarrow T^2 = \mathbb{R}^2/\mathbb{Z}^2; x \bmod \mathbb{Z} \mapsto (ax, bx) \bmod \mathbb{Z}^2$. When does $f_{a,b}$ become immersion or embedding? [My consideration so…
user765710
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What is meant by "Find an explicit trivialization of $\mathbb{S}^{3}$?

The problem is "Find an explicit trivialization of the tangent bundle of $\mathbb{S}^{3}$", but I am confused because aren't all trivialization's of a tangent bundle just $\Phi(v^{i}\frac{\partial}{\partial u^{i}}|_{p})=(p,v^{1},...,v^{n})$? How…
SihOASHoihd
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why hypersurface is called hypersurface and not Hyposurface

from wikipedia: "Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface" Now, hyper- is the prefix of "over" hypo- is the prefix of "under" If we move form n to n-1 soundn't it be called…
iddober
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A closed subset of a manifold is precompact (Lemma 1.10 from Lee)

I can't understand the end of the proof of the following lemma from "Introduction to smooth manifolds" by Lee Lemma 1.10: Every topological manifold has a countable basis of precompact coordinate balls In order to prove it he takes a manifold…
Phi_24
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Injective Map between Manifolds is Surjective

Suppose $f : M \rightarrow N$ is a smooth map of compact connected manifolds of the same dimension which is an injection. How do I show that $f$ must be a surjection? Any help/hints would be appreciated.
user100101212
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Does this make S into a one-manifold?

I have the following problem : Let $S =(0,1) \times (0,1) \subset \mathbb{R} ^{2} $ and for each $s,0\leq s \leq 1$ let $V_{s} = {s}\times (0,1)$ and $f_{s} : V_{s} \rightarrow \mathbb{R} ,(s,t) \rightarrow t.$Does this make S into a…
Kevin
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Any open covering is contractible?

Let $M$ be a smooth manifold. Then, is any open covering $\{U_i\}$ of $M$ is contractible?, i.e. each $U_i$ is contractible?, where $\{(U_i, \phi_a)\}$ is an atlas of $M$, i.e. $M = \bigcup_i U_i$ and $\phi_a:U_a \mapsto \phi_a(U_a)$ is a…
math student
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There is an open contractible covering anytime?

Let $M$ be a simply-connected smooth manifold with dimenison $n$. Then, is there an open covering $\{U_i\}$ of $M$ such that each $U_i$ is contractible?
math student
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Construction of a diffeomorphism

Let $M$ be an arbitrary smooth manifold with dimenison $n$, and $\{U_i\}$ be an open covering of $M$. Can we construct a diffeomorphism $f_i:U_i \to B_i$?, where $B_i$ is the open unit ball in $\mathbb{R}^n$.
math student
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Any manifolds have an open covering such that each is constractible?

Let M be an arbitrary smooth manifold. Then, is there an open covering $\{U_i\}$ of $M$ such that each $U_i$ is constractible anytime?
math student
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