Special affine connections on symmetric spaces

Let (G, H, σ) be a symmetric pair and g = m ⊕ h the canonical decomposition of the Lie algebra g of G.We denote by ∇ 0 the canonical affine connection on the symmetric space G/H.A torsion-free G-invariant affine connection on G/H is called special if it has the same curvature as ∇ 0 .A special product on m is a commutative, associative, and Ad(H)-invariant product.We show that there is a one-to-one correspondence between the set of special affine connections on G/H and the set of special products on m.We introduce a subclass of symmetric pairs called strongly semi-simple for which we prove that the canonical affine connection on G/H is the only special affine connection, and we give many examples.We study a subclass of commutative, associative algebra which allows us to give examples of symmetric spaces with special affine connections.Finally we compute the holonomy Lie algebra of special affine connections.