Questions tagged [smooth-manifolds]

For questions about smooth manifolds, a topological manifold with a maximal smooth atlas.

A topological manifold of dimension $n$ is a Hausdorff, second countable topological space $X$ such that each $x \in X$ has a neighbourhood homeomorphic to an open subset of $\mathbb{R}^n$.

A smooth atlas on a topological manifold $X$ is a collection of pairs $\{(U_{\alpha}, \varphi_{\alpha})\}_{\alpha\in A}$ where $U_{\alpha}$ is an open subset of $X$ and $\varphi_{\alpha} : U_{\alpha} \to \varphi_{\alpha}(U_{\alpha}) \subseteq \mathbb{R}^n$ is a homeomorphism such that $\bigcup\limits_{\alpha\in A}U_{\alpha} = X$.

A topological manifold with a maximal smooth atlas is called a smooth manifold.

5258 questions
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Properly discontinuous action: equivalent definitions

Let us define a properly discontinuous action of a group $G$ on a topological space $X$ as an action such that every $x \in X$ has a neighborhood $U$ such that $gU \cap U \neq \emptyset$ implies $g = e$. I would like to prove that this property is…
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Origins of Differential Geometry and the Notion of Manifold

The title can potentially lend itself to a very broad discussion, so I'll try to narrow this post down to a few specific questions. I've been studying differential geometry and manifold theory a couple of years now. Over this time the notion of a…
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Are diffeomorphic smooth manifolds truly equivalent?

It seems to be an often repeated, "folklore-ish" statement, that diffeomorphism is an equivalence relation on smooth manifolds, and two smooth manifolds that are diffeomorphic are indistinguishable in terms of their smooth atlases. There is a…
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Meaning of derivatives of vector fields

I have a doubt about the real meaning of the derivative of a vector field. This question seems silly at first but the doubt came when I was studying the definition of tangent space. If I understood well a vector is a directional derivative operator,…
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Number of Differentiable Structures on a Smooth Manifold

On John Lee's book, Introduction to Smooth Manifolds, I stumbled upon the next problem (problem 1.6): Let $M$ be a nonempty topological manifold of dimension $n \geq 1$. If $M$ has a smooth structure, show that it has uncountably many distinct…
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Elementary proof of the fact that any orientable 3-manifold is parallelizable

A parallelizable manifold $M$ is a smooth manifold such that there exist smooth vector fields $V_1,...,V_n$ where $n$ is the dimension of $M$, such that at any point $p\in M$, the tangent vectors $V_1(p),...,V_n(p)$ provide a basis for the tangent…
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how to compute the de Rham cohomology of the punctured plane just by Calculus?

I have a classmate learning algebra.He ask me how to compute the de Rham cohomology of the punctured plane $\mathbb{R}^2\setminus\{0\}$ by an elementary way,without homotopy type,without Mayer-Vietoris,just by Calculus. I have tried and failed.Is…
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Orientability of $m\times n$ matrices with rank $r$

I know that $$M_{m,n,r} = \{ A \in {\rm Mat}(m \times n,\Bbb R) \mid {\rm rank}(A)= r\}$$is a submanifold of $\Bbb R^{mn}$ of codimension $(m-r)(n-r)$. For example, we have that $M_{2, 3, 1}$ is non-orientable, while some others are, such as…
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Proving that the pullback map commutes with the exterior derivative

I'm trying to prove that the pullback map $\phi^{\ast}$ induced by a map $\phi:M\rightarrow N$ commutes with the exterior derivative. Here is my attempt so far: Let $\omega\;\in\Omega^{r}(N)$ and let $\phi :M\rightarrow N$. Also, let…
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Can every manifold be turned into a Lie group?

I am studying Lie theory and just thought of this random question out of curiosity. Can any manifold be turned into a Lie group? More precisely, given a manifold $G$, can we always construct (or prove the existence of) some smooth map $m:G\times…
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Why is the tangent bundle orientable?

Let $M$ be a smooth manifold. How do I show that the tangent bundle $TM$ of $M$ is orientable?
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Explain densities to me please!

When it comes to integration on manifolds, I speak two languages. The first is of course the language of differential forms, which is something I am relatively well acquinted with. The second language is what is often used in general relativity…
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Find $f$ such that $f^{-1}(\lbrace0\rbrace)$ is this knotted curve (M.W.Hirsh)

I would like to solve the following problem (it comes from Morris W. Hirsh, Differential Topology, it's exercise 6 section 4 chapter 1): Show that there is a $C^\infty$ map $f:D^3\to D^2$ with $0\in D^2$ as a regular value such that…
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What hyperbolic space *really* looks like

There are several models of hyperbolic space that are embedded in Euclidean space. For example, the following image depicts the Beltrami-Klein model of a hyperbolic plane: where geodesics are represented by straight lines. The following image, on…
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Can we recover a compact smooth manifold from its ring of smooth functions?

It is well-known that if $X$ is a reasonably nice topological space (compact Hausdorff, say) then we can recover $X$ from the ring $C(X)$ of continuous functions $X\to\mathbb R$; see this MSE question for a discussion and problem 26 in the first…
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