Numerical properties of Koszul connections

We use the notation EX(S>M), EXF(S>M) and DL(S>M), where M is a smooth manifold and S is a geometric structure. EX(S>M) is the question whether S exists in M. EXF(S>M) is the question whether M admits S-foliations. DL(S>M) is the search of an invariant measuring how M is far from admitting S. For many major geometric structures, those questions are widly open. In this paper, we address EX(S>M), EXF(S>M) and DL(S>M) for affine structure and symplectic structure, left invariant affine structure, left invariant symplectic structure and bi-invariant riemanniann structure in Lie groups