On affine connections admitting tangent Levi-Civita connections (Q2366393)

Given a torsion free affine connection \(\nabla\), first the conditions of the existence of a germ of a Riemannian metric \(g_{x_ 0}\) (i.e. of a second order jet) are determined, such that \(\nabla\) and the Levi-Civita connection of \(g_{x_ 0}\) have first order tangency at \(x_ 0\); then the enveloping of \(\nabla\) on a region \(U \subset M^ n\) by a family of such Levi-Civita connections is investigated.