2 - Polar spaces

Summary

The second chapter contains some theory of polar spaces. The latter provide the main examples of rank 3 graphs, and, more generally, many examples of strongly regular graphs. Polar spaces are introduced in general as embedded in a projective space, and a full proof of the classification of finite embedded polar spaces is provided.In this chapter, the emphasis is on the graphs whose vertices are objects of a polar space such as the point graph and graphs on (one type of) (maximal) singular subspaces. We review the three types of polar spaces, orthogonal, unitary and symplectic using reflexive forms and discuss Witt’s theorem. For the collinearity graphs of (finite) polar spaces, we discuss, sometimes in detail, sometimes merely surveying, maximal cliques, maximal co-cliques (ovoids), intriguing sets (tight sets and h-ovoids), spreads, partial spreads, hemisystems,…