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Is the subset of symmetric $ 3\times 3 $ matrices with given eigenvalues a manifold?
This answer expands on your idea, although is not the simplest way to prove the statement.
We consider the general dimension $n$.
Let us first show the following general statement:
Let $M \in M_n(\...
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Manifold with the same dimension as $\mathbb{R}^n$
It is true that if a subset of $\mathbb R^n$ is an $n$-dimensional manifold then that subset must be open. This is a deep theorem of algebraic topology, known as the Invariance of Domain Theorem.
- 117k
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Question on equation of dimensions of tangent spaces in Proposition 5.38 of Lee's Smooth Manifolds
The intersection $S\cap U$ is a regular level set of $\Phi$, so the codimension of $S\cap U$ equals the dimension of $N$, by Corollary 5.13 (Submersion Level Set Theorem). In other words, $\dim M - \...
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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 ...
2
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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 (...
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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 ...
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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 ...
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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 ...
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Characterization of the tangent space of an embedded submanifold via defining map
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 $...
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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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