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)

A classical theoremof Kervaire states that products of spheres are parallelizable if and only if at least one of the factors has odd dimension. Two explicit parallelizations on S^m × S^{2h−1} seem to be quite natural, and have been previously studied by the first named author. The present paper is devoted to the three choices G = G_2, Spin(7), Spin(9) of G-structures on S^m × S^{2h−1}, respectively with m + 2h − 1 = 7, 8, 16 and related with octonionic geometry.