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In this article is given a simple expression for the \textit{ center of mass}for a system of material points in a two-dimensional surface of constantnegative Gaussian curvature. Using basic techniques of Geometry, an expressionin intrinsic coordinates is obtained, and it is showed how it extends thedefinition for the Euclidean case. The argument is constructive and also servesfor defining center of mass of a system of particles on the one-dimensionalhyperbolic space $\mathbb{L}^1_R$. Finally, is showed some applications to thecurved $2$- and $3$-body problems.

Hyperbolic Center of Mass for a System of Particles on a two-dimensional  Space with Constant Negative Curvature: An Application to the Curved $2$- and  $3$-Body Problems