Symple&poiss Geom Loop Smooth by Mokhov
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Proof. 4) Let be the linear space of all closed homogeneous k-forms ω of order m on ΩM (dω=0), and let be the linear subspace of all exact homogeneous k-forms ω of the order m on ΩM (ω=dα). 2 (, ) We shall call the groups the homogeneous cohomology groups of order m of the loop space ΩM of the smooth manifold M. 3) is a very natural generalization of the classical de Rham complex on a smooth manifold to the case of loop spaces of smooth manifolds. 3) of homogeneous forms on the loop space of a smooth manifold can be expressed explicitly via the de Rham cohomology groups of the manifold M.
4) can be reduced to a constant form by a local change of coordinates, and consequently defines a Poisson bracket. 2 shows, even in the twodimensional case not all necessary relations can be obtained on functionals of hydrodynamic type. 11) is essential for the case of degenerate metrics). For N=1 and any n all the obstacle tensors are equal to 0. In fact, in the one-component case relations (4. 8). Thus, in the onecomponent case for any n and for any a we have gα=cαg(u), where gα(u) is an arbitrary function, and cα is an arbitrary constant.
44) where fij(u) is an arbitrary smooth tensor field on the manifold M. Let us introduce a new skew-symmetric tensor field βij=fji− fij. 45) where βij(u) is an arbitrary smooth skew-symmetric tensor field on the manifold M. 1 are thereby proved. MOKHOV We note that homogeneous k-forms of order 1 are invariant with respect to the action of the group Diff+ (S1) of diffeomorphisms of the circle S1 preserving the orientation. This is important, in particular, from the viewpoint of applications in the theory of closed boson strings in curved N-dimensional space-time M with the metric (gravitational field) gij.