On the geometry on the nondegenerate subspaces of orthogonal space (Q2455392)

Based on the first author's Diplomarbeit of 2003, this is a contribution to the program of Phan-theoretic study of geometries described in \textit{C. D. Bennett}, \textit{R. Gramlich}, \textit{C. Hoffman} and \textit{S. Shpectorov} [A. A. Ivanov et al. (eds), Groups, combinatorics and geometry. Proceedings of the L.M.S. Durham symposium, Durham, UK, July 16--26, 2001. River Edge, NJ: World Scientific. 13--29 (2003; Zbl 1063.20012)]. The paper contains six main very technical theorems (ranging from the simple connectedness of an incidence geometry to amalgam results in the spirit of Phan's 1977 theorems) on the incidence geometry of vector spaces over fields of characteristic \(\neq 2\) endowed with a non-degenerate bilinear form (with symmetrized containment as incidence), expressed (with one exception) in terms of the associated special orthogonal group. The action of this group turns out to be flag-transitive \textit{if} the coordinate field does not admit a quadratic extension (\textit{if and only if} in case the bilinear form has Witt index \(\geq 1\)). This fact prompts the authors to study these geometries over real closed fields as well.