2 00 4 Continuity of the Volume of Simplices in Classical Geometry

It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices. §1. Introduction 1.1. It is well known that the area of a spherical or a hyperbolic triangle can be expressed as an affine function of the inner angles by the Gauss-Bonnet formula. In particular, the area considered as a function of the inner angles can be extended continuously to degenerated spherical or hyperbolic triangles. The purpose of the paper is to show that the continuous extension property holds in any dimension. Namely, if a sequence of spherical (or hyperbolic) n-simplices has the property that their corresponding dihedral angles at codimension-2 faces converge, then the volumes of the simplices converge. Note that if we consider the area as a function of the three edge lengths of a triangle, then there does not exist any continuous extension of the area to all degenerated triangles. For instance, a degenerated spherical triangle of edge lengths 0, π, π is represented geometrically as the intersection of two great circles at the north and the south poles. However, its area depends on the intersection angle of these two geodesics and cannot be defined in terms of the lengths. This 2-dimensional simple phenomenon still holds in high dimension for both spherical and hyperbolic simplices. To state our result, let us introduce some notations. Given an n-simplex with vertices v n+1. The dihedral angle between the i-th and j-th codimension-1 faces is denoted by a ij. As a convention, we define a ii = π and call the symmetric matrix [a ij ] (n+1)×(n+1) the angle matrix of the simplex. It is well known that the angle matrix [a ij ] (n+1)×(n+1) determines the simplex up to isometry in spherical and hyperbolic geometry. Let R m×m be the space of all real m × m matrics. Our main result is the following. (n+1)×(n+1) be the spaces of angle matrices of all n-dimensional spherical and hyperoblic simplices respectively. The volume function V : X n (k) → R can be extended continuously to the closure of X n (k) in R (n+1)×(n+1) for k = 1, −1. Note that both spaces X n (1) and X n (−1) are fairly explicitly known. Topologically, both of them are homeomorphic to the Euclidean space of dimension n(n + 1)/2. We do not know if …