A comparison theorem on the Ricci curvature in projective geometry (Q1811006)

This paper starts with the following theorem in the spray manifold: Let \((M,G)\) be a complete spray manifold and \(\widetilde G\) another spray which is pointwise projectively equivalent to \(G.\) If the Ricci curvatures of \(G\) and \(\widetilde G\) satisfy \(\widetilde{Ric} \leq Ric\) then the projective equivalence is trivial. In the proof of the above theorem in the expression of \(P_0(t)\), \(y\) is \(\dot c(0).\) After this, they assert that this theorem actually holds for more general geometric structures, such as Riemannian metrics and Finsler metrics. A manifold is called a spray manifold if the spray vector field \(S\) is given in the tangent bundle and satisfies the following: the coefficients of the spray \(S\) are positively homogeneous with respect to \(y\) and differentiable on \(\pi^{-1}(U)\setminus \{ 0\}.\) Geodesics of \(G\) in \(M\) are defined as the projections of the integral curves of \(G.\) The assumption of completeness cannot be dropped. The term ``projective'' of the Beltrami theorem should be read ``projectively flat''. The projective equivalence is trivial if and only if the parameter transformation is always affine.