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N, where, as usual, {x k } are local coordinates on M, and let us examine the linear action of dx k on the basis vectors ∂h dx k : ∂h → (dx k )(∂h ) = ∂x k = δkh , ∂x h which may be written as dx k , ∂h = δkh . Thus, {dx k } is the dual basis of {∂h }. An element ω ∈ Tp∗ M is called a covector and is written in coordinates ω = ωk dx k . The ωk are called covariant components, because they change like the natural basis. Push-forward and Pull-back Given two vector spaces V and W , not necessarily of the same dimension, and a linear map T : V → W , one defines the dual map T∗ : W ∗ → V ∗ as that satisfying T∗ ω, v = ω, Tv , ∀v ∈ V and ∀ω ∈ W ∗ .

First, the group SO(4), which acts isometrically on the 3-dimansional sphere, is the symmetry group of the Kepler problem; then, the SO(2) group generates the motion on the geodesic circle; lastly, the dynamical evolution of position and velocity can be parametrized with two orthogonal vectors spanning the circle itself. Roughly speaking, the two groups and the couple of vectors fit together to form the dynamical group SO(2,4). Taking the two orthogonal vectors as dynamical variables also turns out to be suited for studying the perturbed case, for example the hydrogen atom in electric and magnetic fields.

0. n times Notice the differences and similarities with the Riemannian case: skew symmetry of the tensor Ωμν versus symmetry of the tensor ghk , but nondegeneracy in both cases. This last property ensures that a bivector Ωαβ , called a β Poisson bivector , exists such that Ωμν Ωνβ = δμ . As in the Riemannian case, the bilinear nondegenerate form Ω defines an inner product in Tx P , and thus a canonical isomorphism between Tx P and its dual Tx∗ P . This isomorphism is sometimes denoted with the symbols4 and : Tx∗ P → Tx P by ω → ω = v or v μ = Ωμν ων , : Tx P → Tx∗ P by v → v = ω or ωμ = Ωμν v ν .