Resumen de Special affine connections on symmetric spaces

Othmane Dani, Abdelhak Abouqateb

• 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.