Ricerc@Sapienza

Starting from the 2001 Thomas Friedrich’s work on Spin(9), we review some interactions
between Spin(9) and geometries related to octonions. Several topics are discussed in this respect:
explicit descriptions of the Spin(9) canonical 8-form and its analogies with quaternionic geometry
as well as the role of Spin(9) both in the classical problems of vector fields on spheres and in the
geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spin(9)
manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems
and Clifford structures to Cayley–Rosenfeld planes and to three series of Grassmannians.