Page - 205 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 then there exists auniqueγUU∗ ∈Γ such that [γUU∗ ·Φ](e)=Φ∗(e) ∀e∈EU∩U∗. Comments.Requirements (3) and (4)meanthat [ΦU(e),φU(π(e)]= [[θU(e),ξU(e)],θU(e)] Bothrequirements (4) and (5)yield the followingremarks: the followingaction isdifferentiable Γ×M (γ,x)→γ ·x∈M, the followingaction is anaffineaction Γ×Rm (γ,θ)→ γ˜ ·θ, both the left sidememberand the right sidememberof (5)have the followingmeaning. γUU∗ · [θU(e),ξU(e)]= [θU∗(e),ξU∗(e)]. Consequently (5) implies that for all x∈U∩U∗ onehas γ˜UU∗ ·φ(x)=φ∗(x). Thereforeweget γUU∗=φ ∗◦φ−1. Suppose thatU,U∗ andU∗∗ aredomainsof local chartwith U∩U∗∩U∗∗ =∅ then γU∗U∗∗ ◦γUU∗=γUU∗∗. Therequirement (3)means that thefibrationπ isΓequivariant. TheFigure2expresses therequirementproperty (3). E E MM γ γ π π Figure2.Fibration. Werecall that thegroupΓacts inbothE andM. Figure2expresses that theprojectionπofE on M isΓ-equivariant. 205