Second order local affine invariants (Q1909831)

Let \(g\) be a Riemannian metric and \(\nabla\), \(\nabla^*\) two torsion free Ricci symmetric connections. A triple \((\nabla, g, \nabla^*)\) is said to be conjugate if for all vector fields \(U, V, W\) the equality \[ U_g(V,W) = g(\nabla_U V,W) + g(V,\nabla^*_UW) \] holds. For a conjugate triple, the tensors \(\nabla g\) and \(\nabla^*g\) satisfy the classical Codazzi equations. Also, in this case, the Levi Civita connection of the metric \(g\) is given by \(\nabla^g ={1\over 2}(\nabla + \nabla^*)\). Next, the authors define the tensor \(C = \nabla - \nabla^g\) and a special transformation of \(C\) associated with a conformal transformation of \(g\). This transformation has a number of properties permitting construction of local invariants. The authors describe and study such invariants. They also study various special cases in which the assumption that these invariants have specific values forces the underlying manifold to be of special type, such as a sphere, a hyperovaloid, etc.