Characterization of symmetric planes in dimension at most 4

Abstract

A stable plane is a topological geometry with the properties that (i) any two points are joined by a unique line, and (ii) the operations of join and intersection are continuous and have open domains of definition. A stable plane is called symmetric if its point space is a (differentiable) symmetric space whose symmetries are automorphisms of the plane. Among locally compact stable planes of positive (topological) dimension ≦ 4, we determine those which admit a reflection at each point (i.e., an involutory automorphism fixing this point line-wise), and we list the possible groups containing reflections at all points. Together with an additional, purely geometric condition, this yields a characterization of symmetric planes and, indirectly, of the plane geometries defined by real and complex hermitian forms. No differentiability hypotheses and no algebraic axioms ruling the reflections are needed.