Mathematics Stack Exchange
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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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Is $F$ a differentiable function between manifolds?

I want to know if this statement is true. Let $F: M \to N$ a function between manifolds, such that the restriction of $F$ to each element of an open cover $\mathcal{O}$ of $M$ is differentiable. Is $F$ a differentiable function?
user95747
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Is this curve an embedding?

The map $\varphi : M \to N$ is an embedding if $\varphi$ is a homeomorphism, an immersion, and $\varphi(M) \subset N$, where $M$ and $N$ are manifolds. The curve $\alpha(t)=(t^3-4t,t^2-4)$ has a self-intersection for $t=2$ and $t=-2$ (because…
New day rising
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Showing $[0,\infty)^2$ is not a differentiable manifold

I have to show that $[0,\infty)^2$ is not a differentiable manifold. The problem is $(0,0)$ (because there doesn't exist a diffeomorphism between $[0,\infty)^2$ and $R^2$) but I don't know how to show that. $[0,\infty)^2$ = $[0,\infty) \times…
user401588
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Stereographic projection of $S^1$ onto $\mathbb R$

Can you tell me if this is correct: I define $f_{(0,1)}: S^{1+} \setminus \{(0,1)\} \to \mathbb R$ as $ f((x,y)) =\frac{x}{1-y} $ where $S^{1+} = \{ (x,y) \in S^1 \mid y \geq 0 \}$. Since I removed the north pole this map is continuous and it is…
Rudy the Reindeer
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Is $\mathbb R^2\backslash \{0\}$ a manifold?

Is $\mathbb R\backslash \{0\}$ a manifold ? Is $\mathbb R^2\backslash \{0\}$ a manifold ? I would say yes, but in the doubt, I prefer to ask.
user330587
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number of roots on SO(3)

Suppose we have a smooth map$ f:SO(3) → SO(3)$ of manifolds s.t.$ f(X)=X^2$. $I$ though since I is a regular value of this map and f is orientation preserving, to calculate degree of it, it is enough to check the number of roots of $X^2-I$ in SO(3).…
CruiousBeginer
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What is concretely a vector field?

Let $M$ be a manifold and $TM=\coprod_{x\in M}T_xM$ be the tangent bundle. By definition, a vector field is an application \begin{align*} X: M&\longrightarrow TM\\ m&\longmapsto X_m\ni T_mM \end{align*} but I don't see concretely how it looks like.…
MSE
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Quotient space of a linear space space is also linear?

Suppose, $C$ is a linear manifold (i.e., manifold which is closed under addition and multiplication) and $\Gamma$ is a Lie group. Can we say in general $C/\Gamma$ is also a linear manifold? Can we the same about the following case: Let $C$ is space…
Janak
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Show that $M$ is a 2-dimensional submanifold.

Let $f:\mathbb{C}\to\mathbb{C}$ be a complex polynomial $f=a_{0}+a_{1}z+...+a_{n}z^{n}$ without double zeroes. Consider for every natural number $k\geq 2$ the set $$M=\{(z,w)\in\mathbb{C}^{2}: w^{k}-f(z)=0\}$$ Prove that $M$ is a 2-dimensional…
asdasd asd
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How to embed 3-manifold being homology sphere to $S^4$?

Can someone explain for me following sentence "Mike Freedman has proven all homology 3-spheres admit tame topological embeddings into $S^4$.", which I found on "open problem garden"…
mmm
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Is it possible to define a differentiable manifold structure on a cone?

A cone is a topologic manofold but can we define a differentiable manifold structure on it?
user297564
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Using the pullback to compute $f^*\mathrm d x$ where $f(r,\theta)=(x,y)=(r\cos\theta,r\sin\theta)$

My formula of the pullback is given by If $f:M\longrightarrow N$, then \begin{align*} f^*:\Omega ^k(N)&\longrightarrow \Omega ^k(M)\\ \omega &\longmapsto f^*\omega \end{align*} where $$(f^*\omega )_p(X_1,...,X_k)=\omega _{f(p)}(\mathrm d …
user301068
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Show that $\mathcal D_p(M)=\text{span}\left(\left.\frac{\partial }{\partial x^1}\right|_p,...,\left.\frac{\partial }{\partial x^n}\right|_p\right)$

Let $M$ a manifold of dimension $n$. I have to show that $$\mathcal D_p(M)=\text{span}\left(\left.\frac{\partial }{\partial x^1}\right|_p,...,\left.\frac{\partial }{\partial x^n}\right|_p\right)$$ where $\mathcal D_p(M)$ is the set of the derivation…
user301068
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Prove that $Q=\{(x_1,...,x_n)\in \mathbb R^n\mid \forall i, x_i\geq 0\}$ is a topological manifold with boundary.

I have to prove that $Q=\{(x_1,...,x_n)\in \mathbb R^n\mid \forall i=1,...,n,\ x_i\geq 0\}$ is a topological manifold with boundary. The fact that the topology is second countable and hausdorff is a consequence of the fact that it's a subset of…
user301068
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exactness of product of forms

Given two forms α and β, let α be closed and β be exact, how do you prove that αβ is exact? I can see that αβ is closed, but that is only sufficient fact from the product being exact. Any suggestions?
jfcjohn
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