Isometric full embeddings of [formula omitted] into [formula omitted

Abstract

We show that there is up to isomorphism a unique isometric full embedding of the dual polar space DW ( 2 n − 1 , q ) into the dual polar space DH ( 2 n − 1 , q 2 ) . We use the theory of valuations of near polygons to study the structure of this isometric embedding. We show that for every point x of DH ( 2 n − 1 , q 2 ) at distance δ from DW ( 2 n − 1 , q ) the set of points of DW ( 2 n − 1 , q ) at distance δ from x is a so-called SDPS-set which carries the structure of a dual polar space DW ( 2 δ − 1 , q 2 ) . We show that if n is even, then the set of points at distance at most n 2 − 1 from DW ( 2 n − 1 , q ) is a geometric hyperplane of DH ( 2 n − 1 , q 2 ) and we study some properties of these new hyperplanes.