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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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Complexified tangent space

Let $M$ be a complex manifold of dimension $n$ and $p\in M$. So $M$ can be viewed as a real manifold of dimension $2n$ and we can consider the usual real tangent space at $p$, $T_{\mathbb{R},p}(M)$, that is the space of $\mathbb{R}$ linear…
gradstudent
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How to show $\omega^n$ is a volume form in a symplectic manifold $(M, \omega)$?

I have the following problem: Let $(M, \omega)$ be a symplectic manifold. How can I show $$\omega^n=\underbrace{\omega\wedge \ldots\wedge \omega}_{n-times},$$ satisfies $\omega^n(p)\neq 0$ for all $p\in M$. I believe that is not too dificult but I'm…
PtF
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the equivalence between paracompactness and second countablity in a locally Euclidean and $T_2$ space

suppose $M$ is a locally Euclidean Hausdorff space, show that $M$ is second countable if and only if it is paracompact and has countably many components. This is Problem 2-15 p.59 (or 1-5 p.30 in the new version) in: Introduction to smooth…
Alex
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The graph of a smooth real function is a submanifold

Given a function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ which is smooth, show that $$\operatorname{graph}(f) = \{(x,f(x)) \in \mathbb{R}^{n+m} : x \in \mathbb{R}^n\}$$ is a smooth submanifold of $\mathbb{R}^{n+m}$. I'm honestly completely…
ackshooairy
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Is the inclusion map always smooth?

This maybe a very silly question. But I am quite confused. If $M$ is a smooth manifold, and $A\subset M$ is a submanifold, then is the inclusion map $i:A\longrightarrow M$ smooth? My guess is that it might not be smooth. Because the manifold…
gradstudent
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Showing $[0,1] \times [0,1]$ is a manifold with boundary

I'm familiarizing myself with manifolds. I tried to show $[0,1]\times[0,1]$ is a manifold with a boundary. Can you please tell me if my proof is correct: The definition for manifold with boundary: A manifold with boundary $M$ is a second countable…
goobie
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What exactly is a tangent vector?

I'm a physics student with a very loose understanding of the mathematics I use. I'm trying to learn a little more about very basic topology, manifolds, and Riemannian geometry. I'm using Nakahara's Geometry, Topology, and Physics for self-study. I'm…
Dargscisyhp
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Transition functions in manifold

Following is given as the definition of a manifold in a book which I am reading. Let $\{U_{\alpha}:\alpha\in I\}$ be an open covering of a Hausdorff topological space $X$ and $\phi_{\alpha}$ be homeomorphisms from $U_{\alpha}$ onto open subsets of…
Kumara
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Boundary of the boundary of a manifold is empty

As mentioned in the title, it's well know that boundary of the boundary of a manifold is empty. That is, if $M$ is the boundary of a manifold $N$, i.e. $M=\partial N$, then $M$ is a manifold without boundary, i.e. $\partial M=\varnothing$. For…
Paul
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is triangle a manifold?

Is a triangle (its sides and the region enclosed by its sides) in a 2D Euclidean space $\mathbb{E}^2$ a manifold? I was thinking to use the identity mapping as its charts, but for each point on the sides of the triangle, there is no neighborhood of…
Tim
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Manifold and maximal atlas

1) I didn't understand really what is a maximal atlas. Is it as set of compatible chart maximal in the sens that adding one more chart will yield the atlas not compatible ? 2) Let two atlas $\mathcal A$ and $\mathcal A'$. So if they are compatible,…
MSE
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How many charts are needed to cover a 2-torus?

Can you please answer this question with explanation ? I just learned about the charts needed to cover 1 and 2 spheres but got confused for the case of torus. It would be great if you guys could help.
Mush
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Finding a diffeomorphism between two smooth structures of $\Bbb R$

This is taking from Tu's Introduction to Manifolds book. We have defined $\mathbb{R}$ as the real line with the differentiable structure given by the maximal atlas of the chart $(\mathbb{R},\phi=\operatorname{Id}_\mathbb{R}:\mathbb{R}\to\mathbb{R})$…
ThatOneMathGuy
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What is the relationship between Grassmann Manifolds with different dimensions?

I'm an EE student and I'm just beginning to learn about the Grassmann Manifold. As is known that the Grassmann Manifold is a space treating each linear subspace with a specific dimension in the vector space $V$ as a single point, for example we can…
Helmholz
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Looking for an atlas with 1 chart

Can we provide the set $\{(x,y,z)\in\mathbb{R^3}|x^2+y^2=1\}$ with a 2-dimensional manifold structure involving only 1 chart? I can see it with 2 charts with cylindrical coordinates, but not with only one...
Lisa
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