Symmetry and special relativity in Finsler spacetime with constant curvature

Within the framework of projective geometry, we investigate kinematics and symmetry in $(α,β)$ spacetime-one special types of Finsler spacetime. The projectively flat $(α,β)$ spacetime with constant flag curvature is divided into four types. The symmetry in type A-Riemann spacetime with constant sectional curvature is just the one in de Sitter special relativity. The symmetry in type B-locally Minkowski spacetime is just the one in very special relativity. It is found that type C-Funk spacetime and type D-scaled Berwald's metric spacetime both possess the Lorentz group as its isometric group. The geodesic equation, algebra and dispersion relation in the $(α,β)$ spacetime are given. The corresponding invariant special relativity in the four types of $(α,β)$ spacetime contain two parameters-the speed of light and a geometrical parameter which may relate to the new physical scale. They all reduce to Einstein's special relativity while the geometrical parameter vanishes.