We are working on concrete applications of the relativistic Schwarzschild metric to the cosmos. In this report we calculate the Gaussian curvature of space-time. The relativistic Schwarzschild metric solves Einstein's equations exactly assuming a point gravitational mass and empty space in its vicinity. This metric leads to a static and symmetric solution 2D of the mathematical equation of space-time that allows to calculate the Gaussian curvature at each point. We have calculated some curvature values and found a simple equation to calculate them which allows us to extend the results to a wider range of distances. Finally using this equation and the Birkhoff—Jebsen theorem we have studied the curvature in a homogeneous and isotropic universe with a constant energy density and obtain a value very close to zero for the curvature at any inner point of that universe.