By O I Mokhov
This evaluation offers the differential-geometric idea of homogeneous constructions (mainly Poisson and symplectic structures)on loop areas of tender manifolds, their typical generalizations and functions in mathematical physics and box conception.
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Extra resources for Symplectic and Poisson geometry on loop spaces of smooth manifolds, and integrable equations
146). 152) where Σ(i, j) means summation over all permutations of the corresponding elements (i, j) in parentheses under the summation sign. 150) corresponds exactly to the condition that the coefficients define a symplectic connection on (M, gij). 146). 145), m=2, on the correspondence between symplectic connections loop space ΩM. 153) where Σ(η, ξ, ν) means summation over all cyclic permutations of the elements (η, ξ, ν), ξ(u(x)), η(u(x)) are smooth vector fields along the loop and the vector field v is the velocity vector field of the loop γ(x), that is, is the covariant differentiation along the loop γ; is the torsion tensor of the symplectic connection; is the curvature tensor of the symplectic connection.
This reduction preserves skewsymmetry and the Jacobi identity. 4) is transformed to an arbitrary linear combination of the corresponding onedimensional Poisson brackets. 4) is a multidimensional Poisson bracket is a considerably stronger condition for the coefficients than the necessary requirement that all the corresponding one-dimensional Poisson brackets are compatible. The problem of describing or classifying multidimensional homogeneous Poisson brackets of hydrodynamic type corresponds to the problem of describing or classifying an important subclass of compatible one-dimensional homogeneous Poisson brackets of hydrodynamic type.
In Section 4 we shall concentrate on the Dubrovin-Novikov Poisson structures, their generalizations and applications. 145) were introduced and completely studied for small m (m=1, 2) by the present author in , , . 2). SYMPLECTIC AND POISSON GEOMETRY 33 It follows from the skew-symmetry of the operator that the higher coefficient which is always transformed as a metric on the manifold under local changes of coordinates, must be a symmetric matrix whenever m is odd, m=2k+1, and a skewsymmetric matrix whenever m is even, m=2k.
Symplectic and Poisson geometry on loop spaces of smooth manifolds, and integrable equations by O I Mokhov