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The Finsler Type of Space-time Realization of Deformed Very Special Relativity

We investigate here all the possible invariant metric functions under the action of various kinds of semi-direct product Poincaré subgroups and their deformed partners. The investigation exhausts the possible theoretical frameworks for the spacetime realization of Cohen-Glashow's very special relativity and the deformation very special relativity approach by Gibbons-Gomis-Pope. Within Finsler-Minkowski type of spacetime, we find that the spacetime emerge a Finsler type of geometry in most cases both for undermed Poincaré subgroup and for deformed one. We give an explanation that the rotation operation should be kept even in a Lorentz violating theory from geometrical view of point. We also find that the admissible geometry for $DTE3b$, TE(2), ISO(3) and ISO(2,1) actually consists of a family in which the metric function vary with a freedom of arbitrary function of the specified combination of variables. The only principle for choosing the correct geometry from the family can only be the dynamical behavior of physics in the spacetime.