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Tagged with orthogonal-matrices riemannian-geometry
6 questions
8
votes
2
answers
440
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Show $\langle \log(R), \log(R^{-1}S) \rangle \geq \langle \log(R), \log(S) - \log(R) \rangle $ for all $R,S \in \mathrm{SO}(3)$
$\DeclareMathOperator\SO{SO}$I have a similar question to one I asked a few days ago. Lately, I've been researching Lie groups equipped with bi-invariant Riemannian metrics. One common object is $\SO(...
- 108k
8
votes
2
answers
645
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Is it true $\left\|\log(RS)\right\|≤\left\|\log(R)+\log(S)\right\|$ for all $R,S \in \mathrm{SO}(3)$, where $\|\cdot\|$ is the Frobenius norm?
$\DeclareMathOperator\SO{SO}$I asked this initially in math stack exchange, but thought to ask it here since it is more advanced and related to my research topic. I study optimization on Lie groups ...
- 108k
3
votes
1
answer
275
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Is the Cayley distance on permutation (matrices) equivalent to the Riemannian metric on $O(n)$?
Denote by $d_C(\sigma,\mu)$ the minimal number of transpositions needed to go from a permutation $\sigma$ to a permutation $\mu$. E.g. if $d_C(\sigma,\mu)=0$, then $\sigma=\mu$, if $d_C(\sigma,\mu)=1$,...
CommunityBot
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2
votes
1
answer
101
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Characterization of extrinsic distance prevserving embedding (see the definition given!) from low dimensional Euclidean spaces to high dimensions
P.S. I asked the question on MSE more than a week ago, but didn't get any desired answer, so asking here.
Let $m < n \in \mathbb{N}$. Let us equip $\mathbb{R}^m, \mathbb{R}^n $ with their ...
- 2,479
3
votes
1
answer
687
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Upper bound on the sectional curvature of the orthogonal group
Consider the orthogonal group $O(n)$ as a Riemannian manifold endowed with the usual (bi-invariant) metric $\langle P, Q \rangle_A = \textrm{Tr}\ P^\top Q$ for tangent vectors $P, Q$, with $$T_A O(n) =...
- 2,139
6
votes
1
answer
509
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Principal curvatures of $\mathbb{R}^{n^2}$-embedded SO(n)
It's well known that the sectional curvatures of a Lie group, endowed with a left-invariant metric have a nice closed-form formula $k(X,Y) = \frac{1}{4} \|[X Y]\|^2$.
I'm wondering if the following (...
- 108k