# Questions tagged [smooth-manifolds]

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

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### Question about Proposition 12.9 of Tu's "Introduction to Manifolds" [closed]

I'm reading through Tu's book and having trouble understanding Proposition 12.9, where they prove the sum of two $C_{\infty}$ sections $s(p), t(p)$ is $C_{\infty}$. Specifically, how they use ...
1 vote
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### Details in the proof of quotient manifold theorem

The theorem is stated as: Let $G$ be a Lie group acting smoothly, freely and properly on a smooth manifold $M$. Then $M/G$ is a topological manifold of dimension dim$M$ - dim$G$, and has a unique ...
1 vote
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### Change of coordinates in a regular surface $S$.

I'm quite confused in the following situation: Suppose that $S \subset \mathbb{R}^3$ is a regular surface. Let $U_1$ and $U_2$ be open sets in $\mathbb{R}^2$, and consider the pair of ...
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### Non-parametric and parametric statistical manifolds: Equivalence of score functions in tangent spaces [closed]

Below is the framework to give the manifold structure to the space $M_{\mu}=\{f \in L^{1}(\mu): f>0 , \mu a.e , \int f d\mu=1\}$ Statistical Model and its Topology: The Statistical Model and ...
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### Is this a valid definition of tangent space?

Given a smooth manifold $M$ and a point $p\in M$, I wonder if the tangent space of $M$ at $p$, denoted $T_pM$, can be defined in the following way. First define tangent space for $\mathbb{R}^n$ in any ...
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### Suppose $M/G$ is a smooth quotient manifold and $N$ is a $G$-invariant submanifold of $M$. Then is $N/G$ a submanifold of $M/G$?

Let $M$ be a smooth manifold equipped with a, not necessarily proper, smooth Lie group action $G$. Suppose $M/G$ is a smooth quotient manifold. That is, there exists a smooth structure on the quotient ...
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### Curves on a trousers space. [closed]

How does one go about defining curves on a trousers space? I want to define two curves evolving cyclically around a cylinder and then at some time let one of the curves evolve on the other cylinder. ...
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### Any smooth mapping from $\mathbb{R}^n$ into $S^1$ is of the form $e^{if(x)}$?

Let $F\colon \mathbb{R}^n \to S^1$ be a smooth mapping. Then, I strongly suspect that there must be a smooth function $f\colon \mathbb{R}^n \to \mathbb{R}$ such that F(x) = \exp \big( ...
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### Non-proper smooth embedding

Does all smooth embedding $\iota: S\to M$ is proper if $\dim S < \dim M$? I thought then the slice criterion implies $S$ is locally closed, so closed in $M$. But this is equivalent to saying that ...
1 vote
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### Computations in coordinates of Hamiltonian vector fields

I just need someone to snap me out of a (hopefully small) misunderstanding: in p.574 of (the 2nd edition of) Lee's Introduction to Smooth Manifolds it is written that a Hamiltonian vector field $X_f$ ...
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### Images of structure preserving maps are structures?

Context: I have been working with (mostly concrete) categories like Sets, Groups, Rings, Vect$_k$, Ab, $R$-Mod, Top, Diff and so on. As I understand it, images of group maps are subgroups of their co-...
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### Chain rule to differentiate vector fields with a connection or Lie derivative

Let $X : M \rightarrow TM$ be a smooth vector field, $f : M \rightarrow \mathbb{R}$ a smooth real valued function and $\gamma : [0,T] \rightarrow M$ a smooth curve in $M$. I am interested in ...
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### A curve in a single orbit in the space of Riemannian metrics

Let $M$ be a smooth manifold. Let $\mathcal M$ denote the collection of all Riemannian metrics on $M$. There is a right action of the product group $\mathbb R^+ \times \text{Diff}(M)$ on $\mathcal M$ ...
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1 vote
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### Is $\int_{M_1\sqcup M_2}\omega= \int_{M_1}i_1^*\omega + \int_{M_2}i_2^*\omega$?

If I want to compute the integral of a smooth compactly supported differential form over a disconnected oriented smooth manifold $M=M_1\sqcup M_2$ with two connected components $M_1$ and $M_2$, is it ...
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### What's the continuous $\mathbb C^0$ version for immersion? I'm trying to see if topological embeddings are 'topological immersions.'
A 'smooth embedding' $f: M \to N$ between smooth ($m$,$n$)-manifolds ($M$,$N$) is a smooth map that is both a smooth immersion and a topological embedding, which is simply defined as that the range-...