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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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$C^\infty$ structure subordinate to a $C^k$ structure of a manifold

In Ludwig Arnold's Random Dynamical Systems, he says On a paracompact manifold $M$ there is a $C^\infty$ structure subordinate to a $C^k$ structure of $M$ for $1\le k<\infty$ (Whitney). I don't quite understand his meaning (like, what does…
Fan
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Parametrize $\{(x,y,z)\in\mathbb{R}^3\mid x+y^2+z^2=1, x\geq0\}$

Let $M =\{(x,y,z)\in\mathbb{R}^3\mid x+y^2+z^2=1,x\geq0\}$. It seems to me that this manifold is a "cone" since we have $y^2+z^2=1-x$ for $x\in[0,1]$ which, geometrically, is a circle in the $yz$ plane starting at radius $1$ at $x=0$ and shrinking…
trystero
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Why is the dimension of a real submanifold uniqely determined?

Let $X$ denote a finite dimensional normed space. A non empty set $M \subset X$ is called $d$-dimensional differentiable submanifold of $X$, if for all $a \in M$ there exists an open neighborhood $U$ of $a$ and a diffeomorphism $\varphi \colon U…
el_tenedor
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Given a one-t0-one function f that maps M onto an arbitrary set A, prove there is a unique way to make A a manifold s.t. f becomes a diffeomorphism.

I'm really unsure of how to proceed, I've drawn a picture and can understand the general setting but don't know how to actually prove it.
Daniel Ward
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Derivation of $f\in \mathcal C^1(M,N)$ where $M,N$ are smooth manifold.

I have a question about derivation of fonction $f:M\longrightarrow N$ where $M$ and $N$ are smooth manifold of dimension $n$. In my course, we try to compute $$\mathrm d_p f\left(\frac{\partial }{\partial x_i}\right)$$ where $\frac{\partial…
user301068
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Why the function $f$ is $\mathcal C^\infty (M)$?

Let $M$ a manifold of dimension $n$. Let $g_\varepsilon:\mathbb R^n\longrightarrow \mathbb R$ s.t. $g_\varepsilon\in\mathcal C^\infty (\mathbb R^n)$ and $supp(g_\varepsilon)=[-\varepsilon,\varepsilon]^n.$ Let $\varphi: U\longrightarrow V$ an…
MSE
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Let $W$ a neighborhood of $p\in M$. Why is there $V\subset M$ s.t. $V\subset \overline{V}\subset W$?

Let $M$ be a manifold and let $W$ a neighborhood of $p$. Why is there an open $V$ s.t. $p\in V\subset \overline{V}\subset W\subset M$ where $V$ is compact ? It's in a proof of a theorem of my course, but I don't understant why.
MSE
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Classification of manifold of dimension 1

By my course, all manifold of dimension 1 is isomorphic to $(0,1), (0,1],[0,1)$ or $\mathbb S^1=\{x^2+y^2=1\mid x,y\in\mathbb R\}$. I was thinking of a curve in the plan, with a knot. (See picture) I agree that the gluing (in pink) is possible in…
Rick
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Show $\varphi(x_1,...,x_{n+1})=\frac{1}{1-x_{n+1}}(x_1,...,x_n)$ is surjective.

We consider in fact the stereographic projection. Let $$\mathbb S^{n}=\{x_1^2+...+x_{n+1}^2=1\mid (x_1,...,x_{n+1})\in\mathbb R^{n+1}\}$$ and $E=\text{span}(e_{n+1})^\perp$ where $e_{n+1}=(0,...,0,1)$. Let $$\varphi:\mathbb S^{n+1}\backslash…
Rick
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Partial derivatives on manifolds in terms of local charts

Let $\phi=(u^1,\cdots, u^n)$ be a coordinate system in manifold $M$ at $p$. If $f \in c^{\infty}(M)$, we define $$\frac{\partial f}{\partial u^i} (p) = \frac{\partial(f \circ \phi ^{-1})}{\partial x^i} \qquad 1 \le i \le n.$$ Where $x^1, \cdots ,…
maxim
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Tangent Bundle of the open positive orthant

What is the expression for the tangent bundle of the open positive orthant $\mathbb R_+^n$? I think I know the answer, but just to be sure. Thanks
M.A
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Manifolds with boundaries and vertices

I heard about manifolds with boundaries, but I never heard about manifolds with boundaries and vertices except perhaps in Spivak's book. Take a solid cube. It's a 3-dimensional manifold with a boundary and 8 vertices. So I think manifolds with…
Makoto Kato
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What kind of manifold is a configuration manifold?

I have recently been learning about the basic properties of topological, smooth, and Riemannian manifolds. But I frequently hear the term configuration manifold referenced in relation to Lagrangian mechanics. What kind of manifold is a…
Stan Shunpike
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Is $x:\emptyset\to\emptyset$ a chart?

In the definition of a manifold, one defines, in particular, a chart as a homeomorphism $x:U\to O$ where $U\subseteq M$ is an open set of the topological space $M$ and $O\subseteq \mathbb{R}^n$. Question : Is the empty function…
Guest
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Connected manifold with disconnected boundary?

Is there any simple example of a connected manifold with disconnected boundary?
Lucas Colucci
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