Higher Schl{\"a}fli Formulas and Applications II. Vector-valued   differential relations
  The classical Schl\"afli formula, and its ``higher'' analogs given in [SS03], are relations between the variations of the volumes and ``curvatures'' of faces of different dimensions of a polyhedra (which can be Euclidean, spherical or hyperbolic) under a first-order deformation. We describe here analogs of those formulas which are vector-valued rather than scalar. Some consequences follow, for instance constraints on where cone singularities can appear when a constant curvature manifold is deformed among cone-manifolds. 
Jean-Marc Schlenker and Rabah Souam