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We aim to investigate the theory of Lorentz violation with an invariantminimum speed so-called Symmetrical Special Relativity (SSR) from the viewpointof its metric. Thus we should explore the nature of SSR-metric in order tounderstand the origin of the conformal factor that appears in the metric bydeforming Minkowski metric by means of an invariant minimum speed that breaksdown Lorentz symmetry. So we are able to realize that there is a similaritybetween SSR and a new space with variable negative curvature ($-\infty<\mathcalR<0$) connected to a set of infinite cosmological constants($0<\Lambda<\infty$), working like an extended de Sitter (dS) relativity, sothat such extended dS-relativity has curvature and cosmological "constant"varying in the time. We obtain a scenario that is more similar to dS-relativitygiven in the approximation of a slightly negative curvature for representingthe current universe having a tiny cosmological constant. Finally we show thatthe invariant minimum speed provides the foundation for understanding thekinematics origin of the extra dimension considered in dS-relativity in orderto represent the dS-length.

Lorentz violation with a universal minimum speed as foundation of de Sitter relativity