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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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What branch of mathematics studies manifolds?

Hi: I am considering to study a book called General Relativity for Mathematicians. In the book the word manifold appears many times. Consider manifolds in the context of the general relativity. What could be the branch of mathematics studying them?…
stf91
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Let $M$ a manifold and $\Sigma\subset M$ a curve. How can we choose a coordinate system s.t. $\Sigma=\{(x,y)\mid x=0\}$?

I'm quite new with manifold. Let $M$ a manifold of dimension $2$ and let $\Sigma\subset M$ a curve (i.e. a sub-manifold of dimension $1$). They say in my lecture : We choose a coordinate system s.t. $\Sigma=\{(x,y)\in \mathbb R^2\mid x=0 \}$. Q1)…
John
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How is a tensor defined on the tangent space in the point associated with each point of a differentiable manifold?

Differential structure allows one to define the globally differentiable tangent space, differentiable functions, and differentiable tensor and vector fields. Differentiable manifolds are very important in physics Yes, but I don't understand which…
Kann
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Creating manifold from subsets?

In the definition of Manifold, we are given countable atlases $(U_i, \psi_i)$, such that every point $p$ is contained in one such $U_i$ and $\psi_i(U_i)$ is homeomorphic to some subset in $\mathbb{R}^n$. Now I want to reverse this process. Suppose I…
henceproved
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The circle group acts on complex plane by complex multiplication.

My question arises from Chapter 21 (page 541-542, example (c)) of Lee's book, Introduction to Smooth Manifolds, 2nd edition. I find in Lee's book ''Introduction to Smooth Manifolds'' this example. He says that the quotient map is f : $\Bbb C$…
Ioannis Vousnakis
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Intersecting open sets in manifolds

Given a smooth manifold, say a two-sphere, and given two disjoint open sets on it, containing two different points of the manifold (Hausdorff property) can I find a third open set that intersects them both? If yes; in what property of the manifold…
EEEB
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Invariant manifold — tangent space

Suppose we have a manifold W and on it a vector field. Is the manifold invariant under the vector field exactly if the vector field lies in the tangent space of W at w? The tangent vectors are the vectors which are invariant? Disclaimer: I know this…
Salamo
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Does a manifold of dimension one has curvature?

Recently I have seen an interesting answer to an "obvious" question. That is "why can we pull a curve back into a line"? And the answer is "because a manifold of dimension one has no curvature". So I was wandering whether that answer is correct or…
Yuyi Zhang
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For a locally euclidean space X, what is the domain of the coordinate map for a chart?

Given a locally Euclidean space X of dimension n and a point $p \in X$, by definition there exists a neighborhood $U \subset X$ and a homeomorphism $\phi$ such that $\phi (U)$ is an open subset in $\mathbb{R}^n$ I wanted to show that every locally…
eager2learn
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Determining whether a subset is a manifold given two different graphs, one of which is not everywhere differentiable

A smooth manifold in $\mathbb{R}^2$ is locally the graph of a $C^1$ function. Consider the graph of $f(x)=x^{1/3}$. Since $f$ is not differentiable at zero, we are in trouble. However, this subset is the graph of $x=y^3$, which IS differentiable at…
Wouter Zeldenthuis
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$[0,1]\times [0,1]$ is a manifold with boundary

Possible Duplicate: Showing $[0,1] \times [0,1]$ is a manifold with boundary Definition: A manifold with boundary $M$ is a second countable Hausdorff space so that for a $p \in M$ there is an open set $U \subseteq M$ so that there is a…
goobie
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Every point of a manifold M has a neighborhood homeomorphic to an open subset of $R^n$

"Every point of M has a neighborhood homeomorphic to an open subset of $R^n$." I would like to understand this definition a bit better. With a homeomorphism, I understand it to be a continuous map with a continuous inverse. But why just continuous…
Higgsino
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Is it true that the wedge of two manifolds is not a manifold?

The wedge sum of two circles is not a manifold since it contains a cross point. Can we generalize this property? In other words, is it true that the wedge sum of two $n$-manifolds, $n \geq 1$, is not a $n$-manifold?
Nicolas Boutry
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the tangent bundel TM of a manifold is also manifold of dimension twice the dimension of M

I want to prove that the tangent bundel TM of a manifold is also manifold of dimension twice the dimension of M? could you help me? Thanks!
mary
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Constructing a non-vanishing vector field on $M \times S^1$, $M$ manifold $S^1$ the circle.

The title pretty much says it all. How do I go about constructing a non-vanishing vector field on $M\times S^1$ where $M$ is a manifold and $S^1$ the circle? Thank you for your time.
BoogieMonster314
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