Vertex Angles of a Simplex in Hyperbolic Space H n

Abstract

For an n-simplex in H n or S n , we give sine theorems and their results for kth−order vertex angles which are defined in this paper. The Gram and edge matrices, and polar simplex of n-simplex in H n or S n are used to prove the theorems and their results.

On the Schlafli differential formula based on edge lengths of tetrahedron in H 3 and S 3

“…The proofs of (i) and (ii) can be obtained by the same arguments in [12,24]. To prove (iii), Jacobi identity can be used (see [25]).…”

“…As an immediate fundamental consequence of this Theorem which will be used in our future calculations, we have the following result. Proof The theorem can be proved by following similar steps to those in [11,12], but based on edge lengths.…”

“…The proofs of (i) and (ii) can be obtained using the same arguments in [11,12,24]. To prove (iii) the Jacobi identity can be used (see [25] Proof The proof may be obtained using a similar argument to that in [11,12], but using edge lengths.…”

“…To prove (iii) the Jacobi identity can be used (see [25] Proof The proof may be obtained using a similar argument to that in [11,12], but using edge lengths. …”

“…The sine theorem and volume formula based on dihedral angles of hyperbolic and spherical tetrahedra are given by Derevnin et al [11]. On the other hand, the sine theorems for vertex angles of hyperbolic and spherical tetrahedra have been studied by Karlıga and Yakut [12]. The volume of an ideal hyperbolic polyhedron was found by Vinberg [13].…”