In this project, we propose to investigate various structures and invariants of manifolds, especially in low dimension. Our objectives cover several topics of current interest in geometry and topology. The proposal is structured around the following lines: Line A: Comparison theorems in Riemannian geometry Line B: Integral geometry and valuations Line C: Projective structures on low dimensional manifolds and orbifolds Line D: Holomorphic foliations and webs on complex surfaces Line E: Invariants of integral homology spheres in dimension