Regular level set
= Regular level set
{title2=$M=f^{-1}(c),\quad Df_p\text{ surjective for every }p\in M$}
A <level set> at a <regular value> is a smooth <submanifold>. By the <regular level set theorem>, its codimension is the dimension of the target. For determinant-one real <symmetric matrices>, $D(\det)_B(C)=\operatorname{tr}(B^{-1}C)$ is nonzero because $C=B$ gives $n$, so the constraint has codimension one.