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Vector field being weakly tangent to diffeomorphism subgroup

Source: Schweitzer 2009

November 13, 2009
vectorgroup theoryabstract algebratopologyfunctionadvanced fieldsadvanced fields unsolved

Problem Statement

Let H H be an arbitrary subgroup of the diffeomorphism group Diff∞(M) \mathsf{Diff}^\infty(M) of a differentiable manifold M M. We say that an C∞ \mathcal C^\infty-vector field X X is weakly tangent to the group H H, if there exists a positive integer k k and a C∞ \mathcal C^\infty-differentiable map \varphi \mathrel{: } \mathord{]} \minus{} \varepsilon,\varepsilon\mathord{[}^k\times M\to M such that (i) for fixed t1,…,tk t_1,\dots,t_k the map φt1,…,tk:x∈M↦φ(t1,…,tk,x) \varphi_{t_1,\dots,t_k} : x\in M\mapsto \varphi(t_1,\dots,t_k,x) is a diffeomorphism of M M, and φt1,…,tk∈H \varphi_{t_1,\dots,t_k}\in H; (ii) \varphi_{t_1,\dots,t_k}\in H \equal{} \mathsf{Id} whenever t_j \equal{} 0 for some 1≤j≤k 1\leq j\leq k; (iii) for any C∞ \mathcal C^\infty-function f:M→R f: M\to \mathbb R X f \equal{} \left.\frac {\partial^k(f\circ\varphi_{t_1,\dots,t_k})}{\partial t_1\dots\partial t_k}\right|_{(t_1,\dots,t_k) \equal{} (0,\dots,0)}. Prove, that the commutators of C∞ \mathcal C^\infty-vector fields that are weakly tangent to H⊂Diff∞(M) H\subset \textsf{Diff}^\infty(M) are also weakly tangent to H H.
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