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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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Schwartz Space on a Manifold

On $\mathbb{R}^n$, the Schwartz space is an incredibly nice space of functions, and in many ways is more natural than $C_c^\infty (\mathbb{R}^n)$. On a manifold $M$, it of course still makes sense to talk about $C_c^\infty(M)$, but what about…
Jonathan Gleason
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Tangent spaces and $\mathbb{R}^n$

The tangent space of a circle is a line. The tangent space of a sphere (in every point) can be thought of as a plane. Is this a general thing? I mean, having an $n$ dimensional Riemannian manifold, can the tangent space in every point be thought as…
PhoenixPerson
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Quadric in $n$ dimensions is a manifold?

I can show that the sphere $x_1^2+x_2^2+\ldots+x_n^2=1$ is an $(n-1)$-dimensional manifold by considering the map $f(x_1,\ldots,x_n)=x_1^2+\ldots+x_n^2$, and noticing that $1$ is a regular value of $f$, i.e. $Df(p)$ is surjective for all $p\in…
Mika H.
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Why do we consider tangent spaces and what not, when we can just use Whiteney's Embedding Theorem and do calculus in $\mathbb{R^{2m}}$?

Given a real $m$-dimensional smooth manifold why do we consider tangent spaces and what not, when we can just use Whiteney's Embedding Theorem and do calculus in $\mathbb{R^{2m}}$? I assume there is a good reason!
joe ibbs
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Diffeomorphism of tangent spaces

This is my first post on the stackexchange, so I'm sorry if its rambling. I've been working through Lee's Introduction to Smooth Manifolds and I'm having some trouble with one of the exercises. Most of the earlier sections are understandable, and I…
user90242
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Why are 4 dimensional manifolds 'different'?

Bad subject title, but I couldn't quite find the right words. Some years ago I watched the lectures by Frederic Schuller on youtube; one of the things that somehow stuck in my mind was what he said about the number of possible, differential…
j4nd3r53n
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Bijection between $P^2 (\mathbf R)$ and planes in $\mathbf R^3$?

Is it true that there is a bijection between planes through origin in $\mathbf R^3$ and the projective plane $P^2 (\mathbf R)$? It is a remark in a lecture note I am reading. I tried to prove it but it seems to be a mistake. My idea was to intersect…
goobie
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If the pullback of $F:M \to N$ is an isomorphism, is $F$ a diffeomorphism?

Let $M,N$ be smooth manifolds, and let $F:M \to N$ be a smooth map. Then $F$ induces a map $F^*: C^\infty(N) \to C^\infty(M)$, given by $F^*(f)=f \circ F$. Suppose that $F^*$ is an isomorphism. Q1: Is $F$ a bijection? Q2: Is $F$ a homeomorphism?…
awwalker
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Manifold Explained

Is there a good explanation of a manifold on the web somewhere? The wikipedia article isn't really working for me. I was actually hoping for a whiteboard lecture on youtube, but can't find one. My math experience is calculus through differential…
uncle brad
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Linear Subspace of $\mathbb{R}^n$ is a Manifold

How does one prove that a that a linear subspace of $\mathbb{R}^n$ is a manifold? This question arises from Spivak's Calculus on Manifolds, Chapter 5, problem 5-5: Prove that a k-dimensional (vector) subspace of $\mathbb{R}^n$ is a k-dimensional…
dneug
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Lie Bracket and vector fields

Could you please help me solve it? Let X and Y be smooth vector fields on $\mathbb{R}^n$. Suppose that $[X,Y]=0$ (Lie bracket). Show that around each point there exists local diffeomorphism f such that $f_*(X)=∂/∂x$ and $f_*(Y)= ∂/∂y$.
square
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Is a non-orientable surface a kind of manifold?

An example of a non-orientable surface is the Moebius Strip. Iam just curious whether it is indeed a manifold by definition.
nicetyartwork
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Proof that the set is not a manifold

How to show that set: $M = \left\{(x,y,z): x^2+y^2+3z^3 = xy + 6z^{\frac{1}{3}}, z \neq 0\right\} \cup \left\{(0,0,0)\right\}$ is not a manifold? I know the problem is with point $(0,0,0)$. I think i can even show that you cannot define $x$ as $y$…
Prold
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System of two equations form a manifold

Show that the set of solutions of the system of equations $$x_1^2+\ldots+x_n^2=1$$ and $$x_1+\ldots+x_n=0$$ is an $(n-2)$-dimensional submanifold of $\mathbb{R}^n$. I want to take $f(x_1,\ldots,x_n)=(x_1^2+\ldots+x_n^2,x_1+\ldots+x_n)$. Then the…
Mika H.
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Is a topological space with a Minkowsi metric a topological manifold?

The definition of a topological manifold from Wikipedia: tm defines it as a topological space which locally looks like Euclidean space. But what about a topological space that uses the Minkowski metric from special relativity? It doesn't appear to…
user10389
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