Clifford systems, Clifford structures, and their canonical differential forms

A comparison among different constructions in $$\mathbb {H}^2 \cong {\mathbb {R}}^8$$ H 2 ≅ R 8 of the quaternionic 4-form $$\Phi _{\text {Sp}(2)\text {Sp}(1)}$$ Φ Sp ( 2 ) Sp ( 1 ) and of the Cayley calibration $$\Phi _{\text {Spin}(7)}$$ Φ Spin ( 7 ) shows that one can start for them from the same collections of “Kähler 2-forms”, entering both in quaternion Kähler and in $$\text {Spin}(7)$$ Spin ( 7 ) geometry. This comparison relates with the notions of even Clifford structure and of Clifford system. Going to dimension 16, similar constructions allow to write explicit formulas in $$\mathbb {R}^{16}$$ R 16 for the canonical 4-forms $$\Phi _{\text {Spin}(8)}$$ Φ Spin ( 8 ) and $$\Phi _{\text {Spin}(7)\text {U}(1)}$$ Φ Spin ( 7 ) U ( 1 ) , associated with Clifford systems related with the subgroups $$\text {Spin}(8)$$ Spin ( 8 ) and $$\text {Spin}(7)\text {U}(1)$$ Spin ( 7 ) U ( 1 ) of $$\text {SO}(16)$$ SO ( 16 ) . We characterize the calibrated 4-planes of the 4-forms $$\Phi _{\text {Spin}(8)}$$ Φ Spin ( 8 ) and $$\Phi _{\text {Spin}(7)\text {U}(1)}$$ Φ Spin ( 7 ) U ( 1 ) , extending in two different ways the notion of Cayley 4-plane to dimension 16.

The Role of Spin(9) in Octonionic Geometry

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.

The Role of Spin(9) in Octonionic Geometry

Starting from Thomas Friedrich’s work “Weak Spin(9) structures on 16-dimensional Riemannian manifolds”, we review several 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, 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.