Degenerate torsion-free G3-connections revisited

between any two coordinates ( i) and ( i). Hence any choice of Γ k ij in one coordinate chart suffices to define the connection in another. Let T k ij := Γkij − Γkji be the torsion tensor of the connection. The case when torsion is zero (torsion-free) is of particular interest in geometry. In Hermann Weyl’s terms [11], the very existence of inertia systems in the Universe warrants that its linear connection is torsion-free. Cartan [3] first introduced the holonomy group of a linear connection to study symmetric spaces. By definition, the holonomy group of a manifold M at a point p is the Lie group generated by the parallel translations along all loops starting and ending at p. It is thus a subgroup of the general linear group of the tangent space at p. This group is in general not connected; however, its connected component containing the identity is generated by small loops at p. Therefore, a simply connected manifold always has a connected holonomy group, which we will assume henceforth.