The Minkowski metric restricts to a Riemannian metric on M, giving M the metric structure of a hyperbolic space, in particular it is the hyperboloid model of hyperbolic space, see geometry of Minkowski space for proof. The Poincare group P acts on M because (forgetting the action of the translation subgroup ℝ4 with addition inside P) it preserves the Minkowski inner product, and an element x of the translation subgroup ℝ4 of the Poincare group acts on L 2 ( M ) {\displaystyle ~L^{2}(M)~} by multiplication by suitable phase multipliers exp ⁡ ( − i p → ⋅ x → ) , {\displaystyle ~\exp \left(-i{\vec {p}}\cdot {\vec {x}}\right)~,} where p ∈ M . {\displaystyle ~p\in M~.} These two actions can be combined in a clever way using induced representations to obtain an action of P acting on L 2 ( M ) , {\displaystyle ~L^{2}(M)~,} that combines motions of M and phase multiplication.