In this paper, by using the dual problem which was solved by Feng Luo (Geom. Dedicata 64 (1997), 277-282) and a new method, we give necessary and sufficient conditions for given ðnðn þ 1ÞÞ=2 positive real numbers to be the edge lengths of a hyperbolic n-simplex. By using determinants, we also give necessary and sufficient conditions for given ðnðn þ 1ÞÞ=2 positive real numbers to be the edge lengths of a spherical n-simplex. (2000). 51M04, 51M05, 51M20, 51M25, 52A38, 52A37, 52B10.
Mathematics Subject Classifications
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.
In this paper; using the angle between unit normal vector field of surfaces and a fixed spacelike axis in R₁⁴, we develop two class of spacelike surface which are called constant timelike angle surfaces with timelike and spacelike axis in de Sitter space S₁³.
We show that geodesics with their causal characters are invariant under stereographic projection of n-dimensional pseudosphere and pseudohyperbolic space with v-index.
In this paper, we study a special class of timelike surface which is called constant timelike angle surfaces in de Sitter space 3 1 S. In 3 1 S , conditions being a constant angle timelike surface have been determined and invariants of these surface have been investigated. In here, we use the angle between unit normal vector field of surfaces and a fixed spacelike axis in ambient space.
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