Generalized euclidean and elliptic geometries, their connections and automorphism groups

The notion of an elliptic plane given 1975 by K. Sörensen [S1] will be extended to the notion of a "generalized elliptic space". Each such elliptic space is derivable from a generalized euclidean space in the sense of H.-J. Kroll and K. Sörensen [KS]. For the case that the euclidean resp. elliptic space has the dimension 3 resp. 2 there is a one to one correspondence between these structures and quaternion fields. Each quaternion field of characteristic ≠ 2 defines in a natural way a 4-dimensional euclidean and a 3-dimensional elliptic space. But, in general, we do not obtain in this way all 4- resp. 3-dimensional geometries. The geometries derivable from quaternion fields will be characterized. Both of these two classes of geometries are provided with different structures, so that there are different automorphism groups, which will be studied. © 1993 Birkhäuser Verlag