Groups of diffeomorphisms and their subgroups

This paper has two purposes. The first is to prove the existence of a normal coordinate with respect to a connection defined on the group of diffeomorphisms of a closed manifold, relating to an elliptic complex. The second is to prove a Frobenius theorem with respect to a right invariant distribution defined on the group of diffeomorphisms of a closed manifold, relating to an elliptic complex. Consequently, the group of all volume preserving diffeomorphisms and the group of all symplectic diffeomorphisms are Fréchet Lie groups.

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