Abstract
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.
Highlights
One of the oldest evidences of interest for the Spin(9) group in geometry goes back to the1943 Annals of Mathematics paper by D
An application of Spin(9) structures is the possibility of writing a maximal orthonormal system of tangent vector fields on spheres of any dimension
The invariance of the octonionic Hopf fibration under Spin(9) shows that all its fibers are characterized as being orthogonal to the vector fields I1 N, . . . , I9 N in R16
Summary
One of the oldest evidences of interest for the Spin(9) group in geometry goes back to the. The condition Γ = 0 is equivalent to the inclusion Hol ⊂ G for the Riemannian holonomy, and G-structures with Γ 6= 0 are called non-integrable This scheme can be used, in particular, when G is the stabilizer of some tensor η in Rn , so that the G-structure on M defines a global tensor η, and here, Γ = ∇η can be conveniently thought of as a section of the vector bundle:. The proportionality factor, computed by looking at any of the terms of ΦSpin(9) and τ4 (ψ) turns out to be 360 This can be rephrased in the context of Spin(9) structures on Riemannian manifolds M16 and gives the following two (essentially equivalent) algebraic expressions for the the 8-form, ΦSpin(9) : Theorem 1. When Spin(9) is the holonomy group of the Riemannian manifold (M16 ), the Levi–Civita connection (∇) preserves the E9 vector bundle, and the local sections I1 , .
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