Geometrie du plan projectif des octaves de Cayley

Abstract

In this article we study certain geometric aspects of the projective plane P2(O) over the octaves (Cayley numbers) over the reals. First, we use the explicit representation of points of P2(O) by Hermitian 3×3 matrices over the octaves to determine homogeneous coordinates on the projective line with the help of the fibration of S15 with basis S8 and fiber S7. Next we give a table of Lie products in the Lie algebra F4, which enables us to explicitly compute the curvature tensor of P2(O) as a symmetric space. Finally we exhibit a non-zero skew symmetric 8-form which is invariant under the holonomy group Spin(9). The expression we obtain is the analog of the Kahler form and the fundamental 4-form on the complex and quaternion projective plane, respectively.