Invariant connections for conformal and projective changes (Q2367472)

Let \(M(a)\) be a fixed Riemannian manifold and consider another metric tensor \(g\) on \(M\). Then, using the Riemannian connection of \(g\) and the scalar function \(C= {1\over n}\log {\sqrt{G}\over \sqrt{A}}\), where \(G = \text{det}(g)\) and \(A = \text{det}(a)\), the author defines a linear connection \(C_ \Gamma\) on \(M\) which is symmetric but not metric. This connection is invariant under a conformal change of \(g\) and its curvature tensor is conformally invariant but it is different from the Weyl conformal curvature tensor. Therefore, the notion of relative conformal flatness is studied. An analogous construction is given in a Finsler space, obtaining a projective invariant nonlinear connection. As an application, the author reobtains known results on Randers spaces. Furthermore, as a special case, he obtains a Riemannian projective theory, studying projective changes of Riemannian metrics and projective parameters of extremals.