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