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By Miller G. A.

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The rest of the proof of the theorem is to check the gluing of the so defined metric and almost complex structure on the common definition set of any two k-orthogonal basis {X1 , . . , X2n } and {Y1 , . . , Y2n } which is so because of the construction made above and because of the uniqueness of the polar decomposition. We remark also that if (J, h) is a starting structure for the polarization of the fundamental form ω(X, Y ) = h(X, JY ), then the result of polarization is the same metric h and almost complex structure J.

Ti+1 . 2. Let M be real analytic manifold, ε > 0 and H(p, t) = ωt (p) homotopy operator for ω0 and ω1 consisting of closed differential 2-forms on M . Let B ⊆ E 1 (M ) be an arbitrary small convex Whitney neighbourhood ˜ t) = of the zero one-form on M . Then there exist a homotopy operator H(p, ˜ t)) < ε and a family of ω ˜ t (p) of the two-forms ω0 and ω1 , D(H(p, t) − H(p, ˜ one-forms βt = B(p, t) ∈ B, for each t ∈ I, continuous on the parameter t, ˜ t + dβt is real analytic for each t ∈ I.

Camassa-Holm equation – right invariant metric on the diffeomorphism group The construction described briefly in the previous section can be easily generalized in cases where the Hamiltonian is a left- or right-invariant bilinear form. Such an interesting example is the Camassa-Holm (CH) equation [2,12,17]. This geometric interpretation of CH was noticed firstly by Misiolek [23] and developed further by several other authors [7–9,13,14,18]. Let us introduce the notation u(g(x)) ≡ u ◦ g and let us consider the H 1 Sobolev inner product H(u, v) ≡ 1 2 M (uv + ux vx )dµ(x), with µ(x) = x (5) The manifold M is S1 or in the case when the class of smooth functions vanishing rapidly at ±∞ is considered, we will allow M ≡ R.

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