Joachim Hilgert Geordnete symmetrische R\"aume Journal of Lie Theory, vol. 1 (2) We study the geometry of the one sheeted hyperboloid viewed as an ordered symmetric space. The geometric results are indispensible prerequesites for the harmonic analysis of causal integral equations on such spaces. The present note may be used as an introduction to the geometric parts of the two forthcoming papers [\FHO, \HN].