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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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Manifold with boundary : Definition using locally ringed space

Suppose we define a manifold with boundary, using the locally ringed space definition, with the local model being either open subsets of Euclidean spaces, or open subsets of the half-spaces in $\mathbb R^n$, together with the sheaf of differentiable…
Severus Snape
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Poincaré lemma for star shaped domain

I would like to know if someone can help me out to prove the Poincaré lemma for a star shaped domain without using the Stokes theorem.
Lommel
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Show that $f^{-1}(0)$ is a manifold on $U\times V$

Consider the following set in $\mathbb R^3$: $$ M=\{(x,y_1,y_2):x^3y_1+x^2y_1y_2+x+y_1^2y_2=0\}. $$ a) Show that there are open neighborhoods $U$ of $1\in\mathbb R$ and $V$ of $(-1,1)\in\mathbb R^2$, and a $C^1$-function $\phi\colon V\to U$,…
Sha Vuklia
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Is every manifold sigma-compact?

Thanks for reading my post. Here is my question. I want to know if every surface is hemicompact, i.e., there is a compact exhaustion. I think that question could be asked for every manifold. I know that every locally compact and sigma compact space…
juliho
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$S^1$ is the only compact connected 1-manifold.

I want to find a proof of $S^1$ is the only compact connected 1-manifold. (In here, manifold means Hausdorff, locally euclidean space) Is there any reference? Or is it easy and can be proved simply?
Gobi
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why can't a topological manifold have a finite underlying set?

According to a course notes I am following, a topological space cannot have a finite underlying set, because it is locally homeomorphic to $R^d$, and $R^d$ is an uncountable set. However, why can't I define the set $M=$ { $1$ } together with the…
user56834
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Diffeomorphism between two manifolds and a differential function in a neighborhood of $\varphi(p)$

Let $M$ be a manifold in $\mathbb R^n$. Observe that if $\varphi : M \to M$ is a diffeomorphism, $v \in T_p M$ and $f$ is a differentiable function in a neighborhood of $\varphi(p)$, we have $$ (d\varphi(v)f)\varphi(p) = v(f \circ…
New day rising
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Smooth Manifolds Exercise

There's a naive exercise that I'm having trouble to finish. Suppose $M$ is a smooth manifold and $f:$ $M$ $\rightarrow$ $\mathbb{R}^{k}$ is a smooth function. Show that: $$f\circ\varphi^{-1}:\varphi(U)\rightarrow\mathbb{R}^{k}$$ is smooth for every…
Br09
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Is this a $c^\infty$ Atlas?

I'm watching Frederic Schuller's "Lectures on the Geometric Anatomy of Theoretical Physics," and came across the following in this video: We want to define an atlas on the real line with the standard topology. We choose the atlas to be $\mathscr{A}…
Dargscisyhp
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If $M$ is a manifold, then $\partial(\partial M)) = \emptyset.$

If $M$ is a manifold, then $\partial(\partial M)) = \emptyset.$ I've searched this question here and I did not find any solution. I know that this problem is equivalent to show that $\partial(\partial \mathbb{H}^n)) = \emptyset,$ but I still have no…
L.F. Cavenaghi
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Constructing a Tangent Space Functor

This is a follow up of this question I asked some time ago regarding the tangent space functor. I am wondering though if there is a simpler way that this situation can be characterized without formally invoking all of the machinery of vector…
ItsNotObvious
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Manifold and derivative: change of coordinate.

I'm stuck on the following problem. Let $M$ a manifold of dimension $n$ and let $\varphi_1:U_1\longrightarrow W_2$ and $\varphi_2:U_2\longrightarrow W_2$ two charts at the neighborhood of $p\in U_1\cap U_2$. Let $$h=\varphi_2\circ…
MSE
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What's "ancient time"?

I've found a reference to "ancient time" from Google. It's mentioned in e.g. the book Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincare Conjecture, by Qi S. Zhang. E.g. The second case is when the manifold $M$ is compact and…
mavavilj
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Does there exist "the implicit function existence theorem" in the differential manifold theory?

Does there exist "the implicit function (existence) theorem" in the differential manifold theory? We have known that there exist inverse function theorem and implicit function theorem in the Euclid space. And I have known that in the differential…
David Chan
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Questions on topological manifolds

Does the dimension of a manifold depend on the topology? That is, can I endow a set with a topology $T$ and get an $n$-dimensional manifold, and endow the exact same set with another toplogy $T'$ and get an $m$-dimensional manifold, with $n\neq…
kein
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