Nontrivial copies of SO(r) in SO(n) - MathOverflow most recent 30 from http://mathoverflow.net2013-06-19T10:44:05Zhttp://mathoverflow.net/feeds/question/119876http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/119876/nontrivial-copies-of-sor-in-sonNontrivial copies of SO(r) in SO(n)Marcos Cossarini2013-01-25T18:54:30Z2013-01-25T18:54:30Z
<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>$\begin{pmatrix}
\cos t&-\sin t\\
\sin t&\cos t\end{pmatrix} \mapsto \begin{pmatrix}
\cos(2t)&-\sin (2t)&0&0\\
\sin(2t)&\cos(2t)&0&0\\
0&0&\cos(3t)&-\sin(3t)\\
0&0&\sin(3t)&\cos(3t)\end{pmatrix}$</code></p>
<p>Are there examples for $r=3$ (or more)?</p>