Special Affine Connections on Symmetric Spaces

Abstract

Let $(G,H,\sigma)$ be a symmetric pair and $\mathfrak{g}=\mathfrak{m}\oplus\mathfrak{h}$ the canonical decomposition of the Lie algebra $\mathfrak{g}$ of $G$. We denote by $\nabla^0$ 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 $\nabla^0$. A special product on $\mathfrak{m}$ is a commutative, associative, and $\operatorname{Ad}(H)$-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 $\mathfrak{m}$. 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.