An Operator Approach to Tangent Vector Fields Processing Supplemental Material

Lemma 2.1 Let V a vector field on M and let T t F , t ∈ R be the functional representations of the diffeomorphisms Φt V : M→M of the one parameter group associated to the flow of V . If D is a linear partial differential operator then DV ◦D = D◦DV if and only if for any t ∈ R, T t F ◦D = D◦T t F . Proof Let p ∈ M and f ∈ C∞(M) be a smooth function. If V (p) = 0, then Φt V (p) = p and DV ( f )(p) = 0. It immediately follows that DV ◦D( f )(p) = D◦DV ( f )(p) if and only if T t F ◦D( f )(p) = D◦T t F ( f )(p) because the right hand side of both equation is equal to 0.