The paper is grown from the lecture course "Metric projective geometry" whichI hold at the summer school "Finsler geometry with applications" at Karlovassi,Samos, in 2014, and at the workshop before the 8th seminar on Geometry andTopology of the Iranian Mathematical society at the Amirkabir University ofTechnology in 2015. The goal of this lecture course was to show how effectiveprojectively invariant objects can be used to solve natural and named problemsin differential geometry, and this paper also does it: I give easy new proofsto many known statements, and also prove the following new statement: on acomplete Riemannian manifold of nonconstant curvature the index of the group ofaffine transformations in the group of projective transformations is at mosttwo.
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