Codazzi-equivalent affine connections (Q2655723)

On a differentiable manifold \(M\) of \(\dim \;M \geq2\) two Riemannian metrics \(g, g^*\) are called \textit{Codazzi-equivalent}, if \(g^*(u,v)=g(Lu,Lv)\) for all tangent fields \(u,v\), where \(L\) is an operator of maximal rank on \(M\) that satisfies Codazzi equations with respect to the Levi-Civita connection \(\nabla(g)\) of \(g\). This notion was introduced by the authors and L. Vrancken in the paper \textit{Codazzi-equivalent Riemannian metrics} (to appear). The above concept implies that the Levi-Civita connections satisfy \(\nabla(g^*)_v w=L^{-1}\nabla(g)_v L(w)\). In the present paper the above considerations are extended to affine connections. In case of relative normalized hypersurfaces with parallel normals this simplifies the investigations and leads to a better understanding of the affine Gauß\ maps. At last a new proof of E. Calabi's global \textit{affine Minkowski problem} is given.