Global theory of geodesically equivalent metrics

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Two Riemannian metrics g and g on one manifold Mn are called geodesically equivalent, if every geodesic of g, considered as an unparameterized curve, is a geodesic of g. An autodiffeomorphism of a Riemannian manifold is called a projective transformation, if it takes (unparameterized) geodesics to geodesics. My aim is to - solve the Beltrami Problem for closed 3-manifolds ... - prove the Projective Lichnerowicz-Obata-Solodovnikov Conjecture ... - and to prove the Geodesic Rigidity Problem ...