Partial linear spaces are incidence structures that are generalizations of the generalized quadrangles. These latter structures are well-studied objects in finite incidence geometry. A partial linear space of order (s,t) is a point-line geometry such that each line contains exactly s+1 points, where s > 0; each point is incident with exactly 1+t lines, where t > 0; and two distinct points are incident with at most one line. A partial linear space is an (&agr;, &bgr;)-geometry if, for each point p and each line L, p not incident with L, there exist exactly &agr; or &bgr; lines, each of them incident with p and a point of L.

The point-line geometry NQ⁺(3,q) is defined as the geometry consisting of the points, respectively the lines, not contained in, respectively not intersecting, a hyperbolic quadric Q⁺(3,q), together with the natural incidence. This geometry is a (0, q/2)-geometry when q is even. An (&agr;,&bgr;)-geometry is called fully embedded in PG(n,q) if the pointset, respectively lineset, is a subset of the pointset, respectively lineset, of PG(n,q), if the incidence is the inherited incidence from PG(n,q) and if s=q.

The main theorem states: let S be a ((q-1)/2,(q+1)/2)-geometry fully embedded in PG(n,q), for q odd and q > 3. Then, S=NQ⁺(3,q). By this interesting result, the paper contributes not only to the study of partial linear spaces, but also to the study of classical Galois geometry, by providing a characterization of a special pointset of PG(3,q) related to the hyperbolic quadric.