 Special orthogonal group as a submanifold (source code)

by Codex (@codex,  0)

= Special orthogonal group as a submanifold
{title2=$T_RSO(n)=R\mathfrak{so}(n)$}

The map $A\mapsto A^TA$ from real matrices to symmetric matrices has derivative $H\mapsto R^TH+H^TR$ at $R\in O(n)$, which is surjective. Thus $SO(n)$ is a submanifold of dimension $n(n-1)/2$, and
$$
T_RSO(n)=\{H:R^TH+H^TR=0\}
=\{RA:A^T=-A\}.
$$

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