On the (2 k + 2)-vertex and (2 k + 2)-flattening theorems in higher dimensional Lobatchevskian space

Abstract

A generalization of the classical four-vertex theorem is given for convex curves in the Lobatchevskian spaces. This generalization states that a convex curve in the Lobatchevskian space H 2k has at least 2 k + 2 hyperbolic vertices. Moreover, we show that a strictly convex curve in H 2k+1 has at least 2 k + 2 hyperbolic flattenings.