Higher Schläfli Formulas and Applications Ii Vector-valued Differential Relations

The classical Schläfli 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. 1. Results 1.1. Scalar formulas. The celebrated Schläfli formula (see e.g. [Mil94]) relates in a simple way the variations of the volume and of the dihedral angles, at codimension 2 faces, of a polyhedron under a first-order deformation. In a n+ 1-dimensional space of constant curvature K, the formula reads as: KdV = 1 n ∑ e Wedθe , where the sum is over the codimension 2 faces of the polyhedron, We is the volume of the face, and θe is the interior dihedral angle at that face. In a previous paper [SS03] we gave “higher” analogs of the Schläfli formula: for each 1 ≤ p ≤ n− 1, K(n− p) ∑