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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.

8723 questions
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Manifold notes in more informal way

When defining the properties of scalar functions that live in manifold $M$ in a less formal way, the following is said: "We no longer refer to a covering by coordinate patches. Instead we conceive of the manifold as a set whose points may be…
PhilosophicalPhysics
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Manifold that is Hausdorff and second countable

Why are we usually assume that a manifold $M$ has to be a Hausdorff space and Second countable ? Is it really hard to study smooth manifolds without making these assumptions?
SamC
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Prove that an atlas is $C^{\infty}$

The sphere $S^2$ can be covered by the following $6$ subsets (hemispheres) $$ O_i = \{(x^1, x^2, x^3) \in \mathbb{R}^3 | x^i > 0, i = 1, 2, 3\}$$ Each of these subsets can be mapped by the unit open disk $D$ or $\mathbb{R}^2$ via the projection: $$…
mesllo
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Non-compact manifold with compact boundary

What is an example of a non-compact manifold with compact boundary?
Lucas Colucci
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Infinite Genus Riemann Surfaces

I want to show that every infinite genus Riemann surface $M$ has a proper closed subset such that, $M^*\setminus E$ ($M^*$ is the one-point compactification of $M$) is connected and locally connected, and $E$ is not of essentially of a finite…
user191988
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Can the Differential be Considered as a Covariant Functor?

First, I apologize if this question is poorly-worded or otherwise vague, I'll try to be as clear as possible. If $F:N\rightarrow M$ is a smooth map between smooth manifolds $N$ and $M$, then at each point $p \in N$ the map $F$ induces the derivation…
ItsNotObvious
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Definition of Submanifold of Topological Manifold

Analogy to smooth manifold, I want to define the submanifold of topological manifold. There are two ways. Let $M$ be a topological manifold, and $N\subset M$. If $N$ is a topological manifold, then we call $N$ is a submanifold of $M$. For any $p\in…
gaoxinge
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A question about metrizable manifolds

Let $(M,d)$ be a metric space and $B(x, \epsilon) = \{ y \in M \mid d(x,y) < \epsilon \}$ and $\bar{B}(x, \epsilon) = \{ y \in M \mid d(x,y) \leqslant \epsilon \}$. In general, it is not true that $\bar{B}(x, \epsilon) = \overline{B(x, \epsilon)}$…
onebengaltiger
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Atlas/chart for a Hyperboloid

given is the following hyperboloid: $$H = \{(x,y,z) \mid \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\},$$ where a,b,c are free parameters. I have to find an $C^\infty$-atlas for H. In order to do this, I have to find one (or more)…
Paul85
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Trying to prove M is a manifold

Let $M$ be the set of all points $(x, y, z) \in \mathbb{R^3}$ satisfying both of the equations $x^3 + y^3 + z^3 = 1$ and $x + y + z = 1$. Prove that M is a manifold, except perhaps near the points $(x; y; z) = (-1; 1; 1) ; (1;-1; 1) ; (1; 1;-1)$.…
Cian
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Charts on a Manifold

Let $f^{1},\cdots ,f^r,\Phi ^1,\cdots , \Phi ^{n-r}$ be functions of class $C^{(1)}$ on an open set $D$. suppose that $F=(f^1\mid S,\cdots ,f^r\mid S)$ is a coordinate system for $S$, that $S=\lbrace x\in D:\Phi (x)=0\rbrace $, and that $D\Phi (x)$…
sami
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Immersed submanifold

I need help finding an example that shows an immersed submanifold might have more than one topology and smooth structure with respect to which it is an immersed submanifold. This is problem 5-15 from Lee's book on manifolds. I thought I could use…
kumhmb
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Prime Manifold in Different Dimensions

Definition Let $M$ be an $n$ dimensional manifold. If $M$ = $M_1$ # $M_2$, we have $M_1=\mathbb S^n$ or $M_2=\mathbb S^n$. Then $M$ is called a prime manifold. $\mathbb T^2$ and $\mathbb RP^2$ are prime from the classification of surface and…
gaoxinge
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Is the cube with boundary and corners a manifold with boundary?

The definition of a n-manifold with boundary as I understand it, is that the manifold without boundary is an n-manifold, and the boundary is an (n-1)-manifold. Thus because the boundary of cube has edges and corners, which are of a lower dimension…
NateZ
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$SO(2)$ is a manifold?

I am failing to see why $SO(2)$ is a manifold. I see that $SO(2)$ is isomorphic as a group to $([0,2\pi),+_{\mod 2\pi})$. But then when we look for a manifold we need a collection of charts. And since we are dealing with $[0,2\pi)$ this would…
joe ibbs
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