Invariant connections on symmetric spaces

Resumo

A Classical problem in Differential Geometry, the determination of the invariant affine connections in the simply connected irreducible symmetric spaces, is equivalent to the algebraic problem of computing the set Hom(S)(T circle times (R) T, T) for any Zz-graded simple Lie algebra L = S circle plus T. The algebraic problem is solved using known information about the Lie triple system structure on T, because the simple Z(2)-graded Lie algebra L = S circle plus T is just the embedding for the simple Lie triple system T. It turns out that the set of homomorphisms contains non trivial elements if and only if T is related to a simple Jordan algebra. Now it is possible to come back to the geometric context to describe the affine connections and express the holonomy and torsion and curvature tensors in algebraic terms.