Affine hypersurfaces with the Ricci semi-symmetry (Q1781911)

A differentiable manifold \(M\) with the curvature tensor field \(R\), defined by a torsion-free affine connection \(\nabla\) on \(M\), is called locally symmetric resp. semi-symmetric if \(\nabla R= 0\) resp. \(RR:= R(X, Y)R= \nabla_X\nabla_Y R-\nabla_Y\nabla_X R-\nabla_{[X,Y]}R= 0\) for all smooth tangent vector fields \(X\), \(Y\) on \(M\). In the same way one speaks of local Ricci symmetry, resp., Ricci semi-symmetry of \(M\) if \(\nabla\,\text{Ric}= 0\), resp., \(R\text{\,Ric}= 0\) for the Ricci-tensor field Ric of \(M\). The author applies these definitions to the induced connnection \(\nabla\) of an affine hypersurface \(f: M^n\to \mathbb R^{n+1}\) which is induced by an affine normal vector field of \(f\). His main result is: If \(f\) with \(n\geq 3\) is a (nondegenerate) Blaschke immersion, then \(M^n\) is Ricci semi-symmetric with respect to the induced connection \(\nabla\) of \(f\) iff \(f\) is either a proper affine hypersphere or an affine cylinder where the rank of the shape operator \(S\) of \(f\) is \(\leq 1\). In the first case semi-symmetry is equivalent to Ricci semi-symmetry.