By Pieter Thijs Eendebak
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Additional info for Contact Structures of Partial Differential Equations
The standard hyperbolic structure on V is given by the diagonal matrix K with entries K ii = (−1)i+1 . The standard hyperbolic structure on R2 is given by K0 = 1 0 0 . 7. A hyperbolic structure on an even dimensional vector space does not determine an orientation. This is in contrast with a complex structure that does determine an orientation. Consider for example the standard hyperbolic structure on R4 given by 1 0 0 0 0 −1 0 0 K0 = 0 0 1 0 . , AK = K A, but is orientation reversing.
The form α˜ defines a map from the vector fields on P to sections of T P/C by X → α(X ˜ ). We claim that this map is an isomorphism from the infinitesimal automorphisms of the contact structure to the sections of the line bundle T P/C. 3 Contact transformations 31 Proof. Let X be an infinitesimal symmetry for which α(X ˜ ) = 0. Then X must be contained in C. Since X is a symmetry we must have [X, Y ] ⊂ C for all Y ⊂ C. But the Lie brackets modulo C are non-degenerate, hence X = 0. This proves the map is injective.
The converse is also true, with some regularity assumptions on the coframe: if all the structure invariants are the same, then the corresponding coframes are equivalent. There are many relations between the structure functions. The knowledge of all structure equations Tσ for which s ≤ S and j < k, 1 ≤ l1 ≤ . . 18) is sufficient to determine all structure functions of order at most S. This follows from the fact i are anti-symmetric in j, k and the commutation relations [∂ , ∂ ] = −2T i ∂ . 18) is denoted by I S .