Calibrated Submanifolds

Clifford systems, Clifford structures, and their canonical differential forms

“…It is now worth to remind the following parallel situations in complex, quaternionic and octonionic geometry. The groups U(2) ⊂ SO(4) , Sp(2) ⋅ Sp(1) ⊂ SO(8) , Spin(9) ⊂ SO (16) are the stabilizers of the vector subspaces spanned respectively by the Pauli, quaternionic Pauli, octonionic Pauli matrices.…”

“…7 Rank 8, 7 and 6 Clifford systems on ℝ 16 Look now closer at the nine octonionic Pauli matrices, that define a rank 9 Clifford system in ℝ 16 , and at the choices among them that give rise to ranks r = 8, 7, 6 (cf. previous Section).…”

“…where boldface notations have the same meaning as before. The presence of the factor U(1) in the group Spin(7)U(1) is here due to a computation showing that matrices in SO (16) commuting with the seven involutions I 2 , … I 8 are a U(1) subgroup, well identified in [3, Chapter 6, p. 44] (cf. also proof of Theorem 1.4 below).…”

“…[10, p. 278-279]). It follows that Spin(8) contains the diagonal Spin(7) Δ (characterized by choices a + = a − ) and acts transitively on transversal 4-planes of ℝ 16 . On the other hand the 4-form Φ Spin(8) is invariant under the action of Spin(8) .…”

“…On the other hand the 4-form Φ Spin(8) is invariant under the action of Spin(8) . Thus, since Φ Spin(8) takes value 1 on the 4-plane spanned by the coordinates 121 ′ 2 ′ , Φ Spin(8) takes value 1 on any tranversal Cayley 4-plane in ℝ 16 . ◻…”