Projectively Invariant Objects and the Index of the Group of Affine Transformations in the Group of Projective Transformations

The paper is based on the lecture course "Metric projective geometry" which I conducted at the summer school "Finsler geometry with applications" at Karlovassi, Samos, in 2014, and at the workshop before the 8th seminar on Geometry and Topology of the Iranian Mathematical society at the Amirkabir University of Technology in 2015. The goal of this lecture course was to show how effective projectively invariant objects can be used to solve natural and named problems in differential geometry, and this paper also does it: I give easy new proofs to many known statements and also prove the following new statement: on a complete Riemannian manifold on nonconstant curvature, the index of the group of aline transformations in the group of projective transformations is at most two.