Nontrivial copies of SO(r) in SO(n) - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T10:44:05Z http://mathoverflow.net/feeds/question/119876 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119876/nontrivial-copies-of-sor-in-son Nontrivial copies of SO(r) in SO(n) Marcos Cossarini 2013-01-25T18:54:30Z 2013-01-25T18:54:30Z <p>If $G=SO(n)=SO(\mathbb R^n)$ and $r\leq n$, it is easy to find a closed subgroup $H\leq G$ that is isomorphic to $SO(r)$, just let $S\subseteq\mathbb R^n$ be an $r$-dimensional subspace and let <code>$H=\{g\in G:(\forall x\in S^\bot)g(x)=x\}$</code>. These are the trivial examples.</p> <p>If $r=2$ and $m\geq 4$ we can find nontrivial embeddings $SO(r)\to SO(n)$, for example:</p> <p><code>$\begin{pmatrix} \cos t&amp;-\sin t\\ \sin t&amp;\cos t\end{pmatrix} \mapsto \begin{pmatrix} \cos(2t)&amp;-\sin (2t)&amp;0&amp;0\\ \sin(2t)&amp;\cos(2t)&amp;0&amp;0\\ 0&amp;0&amp;\cos(3t)&amp;-\sin(3t)\\ 0&amp;0&amp;\sin(3t)&amp;\cos(3t)\end{pmatrix}$</code></p> <p>Are there examples for $r=3$ (or more)?</p>