Octonions and affine connections on spheres

The interplay of the theory of octonions with affine connections on the spheres S ⁶, S ⁷ and S ¹⁵ will be presented. Specifically, when these spheres are realized as the reductive homogeneous spaces S ⁶ = G ₂/SU(3), S ⁷ = Spin(7)/G ₂ and S ¹⁵ = Spin(9)/Spin(7), the structure of octonions can be effectively used to determine all left invariant affine connections on them. These connections on S ⁶ and S ⁷ are respectively obtained from a two parameters family of the compact vector color algebra and a one parameter family of the compact simple non-Lie Malcev algebra. If O = ℝ ⊕ O 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187674/f8337d7b-6b74-4da5-ae84-c225ed353b0a/content/pg43_1.tif"/> is the algebra of octonions (or Cayley algebra), then all Spin(9)-invariant affine connections on S ¹⁵ are given by a three parameters family of ℤ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187674/f8337d7b-6b74-4da5-ae84-c225ed353b0a/content/pg43_2.tif"/> -graded products on O 0 × O https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187674/f8337d7b-6b74-4da5-ae84-c225ed353b0a/content/pg43_3.tif"/> , with even part O 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187674/f8337d7b-6b74-4da5-ae84-c225ed353b0a/content/pg43_4.tif"/> and odd part O https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187674/f8337d7b-6b74-4da5-ae84-c225ed353b0a/content/pg43_5.tif"/> .