/ 03 06 01 0 v 1 2 J un 2 00 3 String theory in Lorentz - invariant light cone gauge - III

Definition 1: let the phase space M be a smooth orientable manifold. Let the orbit of gauge group G and the surface of gauge fixing condition F be its smooth orientable submanifolds with dim(G) = codim(F ). Let P be the point of their transversal intersection. This means that in point P the tangent spaces to G and F span the tangent space to M . Let ~τ(F, P ), ~τ (G,P ) and ~τ(M,P ) be the bases in the tangent spaces, defining the orientation of F , G and M , evaluated in point P . The index of intersection of F and G in point P is defined as a number ν, equal to (+1) if the basis (~τ(F, P ), ~τ (G,P )) has the same orientation as the basis ~τ(M,P ), and equal to (−1) if the orientations are opposite.