Dual connections and affine geometry (Q910722)

Two torsion free affine connections \(\nabla\) and \({\bar \nabla}\) of a pseudo-Riemannian manifold \((M,g)\), \(\dim M=n\) are called dual if \(Xg(Y,Z)=g(\nabla_ XY,Z)+g(Y,{\bar \nabla}_ XZ)\) for any vector fields X, Y, Z. Such an \((M,g,\nabla,{\bar \nabla})\) is called a statistical manifold. It is proved that if an \((M,g,\nabla,{\bar \nabla})\) has constant curvature (defined in the paper), then there exist affine immersions \((x,\xi)\) of \((M,\nabla)\) and \((\bar x,{\bar \xi})\) of \((M,{\bar \nabla})\) into an affine \({\mathbb{R}}^{n+1}\) such that their second fundamental forms are equal to g; and conversely. Corollary 1 gives a local description for explicit calculation. In Corollary 2 the case of positive definite g is discussed and a theorem of S.-T. Yau concerning complete affine hyperspheres and affine mean curvature is involved.