# Tag Info

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### Paraboloid and parametrizations

As mentioned in the comments, the projection on the xy plane gives you a global parametrization for the paraboloid, which is also a local parametrization around every point. The open $U$ is the whole ...

### How to show there exists a smooth local frame $(\tilde{X_1},\cdots,\tilde{X_n})$ on some neighborhood of $A$ such that $\tilde{X_i}|_A=X_i$?

From J. M. Lee's book "Introduction to smooth manifolds" we have the next proposition: 10.12 Let $\pi:E\to M$ be a (smooth) vector bundle. Let $C$ be a closed subset of $M$ and $s:C\to E$ a (...
1 vote

### Is the subset of symmetric $3\times 3$ matrices with given eigenvalues a manifold?

This second answer gives a shorter proof, more algebraic, which does not expand on your ideas. $\newcommand{\O}{\mathcal{O}}$ We will use the following fact twice, whose proof is given by elementary ...
1 vote
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### Manifold of lines in $\mathbb{R}^3$

That is $\mathbb {RP}^2 \times \mathbb R^2$. $\mathbb {RP}^2$ is for the line orientation noted $m$ (unit vector) in the OP. I first wrongly commented it was $S^2$, but we take the quotient by the ...
1 vote
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### A question on the manifold $\{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\}$.

This is not true. There's an action of $SO(3)$ on such pairs $(n, m)$ which for $O\in SO(3)$ sends the pair $(n, m)$ to $(On, Om)$. The action is transitive. You've correctly identified that the ...
1 vote
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The inclusion of the embedded submanifold $\iota : S \to M$ is a smooth map. The composition $\Psi = \Phi \circ \iota$ is defined on set $\iota^{-1}(U) = S \cap U$. Let us be more precise concerning $... 1 vote Accepted ### On the proof of the uniqueness of smooth structures on submanifolds by John Lee I think you are getting confused by a common abuse of notation. Call$\tilde i:\tilde S\to M$and$i:S\to M$to the inclusions. And write$\tilde i':\tilde S\to S$for the co-restriction, so$i\circ\...

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