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It is an important question whether it is possible to put a geometry on a given manifold or not. It is well known that any simply connected closed manifold admitting a real projective structure must be a sphere. Therefore, any simply connected manifold M which is not a sphere (dim M ≥ 4) does not admit a real projective structure. Cooper and Goldman gave an example of a 3-dimensional manifold not admitting a real projective structure and this is the first known example. In this article, by generalizing their work, we construct a manifold M n with the infinite fundamental group Z 2 ∗ Z 2 , for any n ≥ 4 , admitting no real projective structure. [ABSTRACT FROM AUTHOR]

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 affine transformations in the group of projective transformations is at most two. [ABSTRACT FROM AUTHOR]