 Orthogonal group as a regular level set (source code)

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= Orthogonal group as a regular level set
{title2=$\dim_{\mathbb R}O(n)=n(n-1)/2$}

The smooth Gram map $F(A)=A^TA$ takes real matrices to symmetric matrices. At an orthogonal $A$, its derivative sends $AH/2$ to any prescribed symmetric matrix $H$. The regular level set $F^{-1}(I)$ is therefore a <smooth manifold> of the displayed dimension.

// Target: geometry-and-topology.bigb

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