Stably free modules of rank $2$ over certain real smooth affine threefolds

Abstract

Let $R$ be a real smooth affine domain of dimension $3$ such that $R$ has either no real maximal ideals or the intersection of all real maximal ideals in $R$ has height at least $1$. Then we prove that all stably free $R$-modules of rank $2$ are free if and only if the Hermitian $K$-theory group $W_{SL}(R)$ is trivial.