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Sobre la fórmula de Gauss-Bonnet para poliedros en espacios de curvatura constante

1962; Unión Matemática Argentina; Volume: 20; Linguagem: Inglês

1669-9637

Luis Antonio Santaló Sors,

Mathematics and Applications

For spaces Sn of constant curvature K, the generalized formula of GaussBonnet for a closed, orieiltable surface of class !1:; 3 which is the boundary of a body Q, takes theforms (1.4) and (1.5) according to the parity of n. In this paper we consider the case in which Q is a polyhedron. Then the formula takes the form (3.4), (3.5), where the terms IXn_h_,LJ, are the sums (3.2) between the h-dimensional measures Lh' of the h-dimensional edges of Q and the (n-h-l )-dimensional polyhedral angles polar of those formed by the faces of Q which are incident in La'. If Q is a simplex of S these formulae must contain the Poincare's relations between the angies of spherical simplexes [11], [8], [10], [15]. This is computed for n=2, 3,4 in N.o 4 and 5. However the computation in the general case it seems to be far from obvious.