On a differential structure for the group of diffeomorphisms

Groups of Diffeomorphisms and their Subgroups

“…Moreoever, there exists an open neighborhood V of 0 in E° such that (a) Exp(F n Ek) = (Exp F) n Ek, (b) putting Vk = (Exp V) n Ek, the exponential mapping is a smooth diffeomorphism of V n Ek onto Vk. Therefore, to prove Theorem B(ii), it is enough to prove the inequalities (I) and (II) in a coordinate expression of the connection $ defined by (4). This will be proved in §6.…”

“…(Cf. Lemma 4 and (G,[1][2][3][4][5][6][7] (u,v) and diu is the derivative of £ at u. Of course, these are only definitions and have nothing to do with the inequalities (I), (II) or (I"), (II") directly.…”

“…Although it is well known that the group ià of the smooth diffeomorphisms of a compact manifold M is a so-called Fréchet Lie group [2], [4], [7], the category of Fréchet Lie groups seems to be still too huge to treat and get some useful results. Compared with Hubert Lie groups, Fréchet Lie groups have not so nice a property.…”

On the group of real analytic diffeomorphisms of a compact real analytic manifold

“…In [12] we defined a differential structure for the set of Coo mappings of a compact manifold M into a compact manifold N that we shall designate by Coo(M, N). That manifold was defined analogously to the one we are discussing here, but was modelled on the Coo vector fields with the Coo topology.…”

“…In this section we shall exhibit a differential structure on the group of analytic diffeomorphisms (see Theorem 4.1). The methods we use are essentially the same as the ones we used in [12]. is the composite…”