Minimizing Curvature in Euclidean and Lorentz Geometry

In this paper, an interesting symmetry in Euclidean geometry, which is broken Lorentz studied. As it turns out, attempting to minimize the integral of square scalar curvature leads completely different results these two cases. The main concern paper about metrics R3, are close being invariant under rotation. If we add a time-axis and let metric start rotate with time, out that, case (locally) (four-dimensional) will increase speed rotation as expected. However, instead initially decrease. other words, rotating can, case, be said less curved than non-rotating ones. This phenomenon seems very general, but because enormous amount computations required, only proved for class flat one, (symbolic) have been carried on computer. Although here purely mathematical, there also connection physics. deeper understanding geometry fundamental importance many applied problems.