# Tag Info

Accepted

### Are diffeomorphic smooth manifolds truly equivalent?

The counterexample just shows that two diffeomorphic smooth structures on the same set $X$ do not need to share a common atlas. However, in any case, two diffeomorphic structures cannot be ...
• 4,237
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### Can every manifold be turned into a Lie group?

There is an easy counterexample: $S^2$ cannot be given a Lie group structure (this is a consequence of the hairy ball theorem). The problem with your construction is that it doesn't offer how to ...
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### Are diffeomorphic smooth manifolds truly equivalent?

This is nothing specific to differentiable manifolds. If you have some set $X$ and you define some structure on $X$ (e.g. group, vector space, topological space, differentiable manifold, ...) there ...
• 29.8k
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### The integral of a function on manifold and differential form

Differential forms are not introduced to answer the question "How do I integrate functions on manifolds?" Instead, they are introduced to answer a different question, namely "What are ...
• 119k
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• 12.3k
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### What is the structure group of the tangent bundle?

Structure group: Given a fiber bundle with total space $E$, base space $M$, model fiber $F$ and projection $\pi$, the local trivialization condition says that given any $u\in E$ such that $u$ ...
• 7,269

### Can every manifold be turned into a Lie group?

To add to the previous answers, topological groups have abelian fundamental groups. $G$ is Topological $\implies$ $\pi_1(G,e)$ is Abelian Orientable surfaces of genus at least two are not ...
• 1,467
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### Manifold has uncountable many smooth stuctures if it has one

It took me a while to understand the great idea proposed by Anthony Carapetis since I think that other people may have the same doubt that I had, I decided to write a more detailed answer using his ...
• 3,179
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### Origins of Differential Geometry and the Notion of Manifold

[2016-07-25]: Section Differential Geometry added. Although OP narrowed down the post, there are still many more important historical facts which should be addressed to adequately answer the question,...
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### Showing that $\bar{\mathbb{B}}^n$ is a manifold with boundary (Lee ITM Probelm 3-4)

I've been worked on this problem for some time, and i think i probably solved it based on the hint given on the book. Maybe this seems a little long, but it is really not. I tried my best to make this ...
• 7,000
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### Understanding Takens' Embedding theorem

Practical meaning of Takens’ Theorem using your example The butterlfly-like structure traced out by the trajectories of the Lorenz system is the attractor of this dynamics. Its properties contain ...
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### The Pullback Bundle is an Embedded Submanifold of its Parent Space

$\newcommand{\M}{M}$ $\newcommand{\N}{N}$ $\newcommand{\brk}[1]{\left(#1\right)}$ $\newcommand{\be}{\beta}$ $\newcommand{\al}{\alpha}$ $\newcommand{\til}{\tilde}$ The pullback bundle is indeed an ...
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I believe this is proven in Chapter 7 of Nestruev's Smooth Manifolds and Observables, but I haven't checked carefully. More precisely, the functor $M \to C^{\infty}(M)$ from smooth manifolds to the ...