By Pieter Thijs Eendebak
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Extra info for Contact Structures of Partial Differential Equations
C In the two examples above we could describe a geometric structure by specifying a particular coframe θ on a manifold M and a structure group G. The structure group is assumed to be a Lie subgroup of GL(n, R). The coframe and the structure group G together determine a G-structure on M. The set of all coframes obtained from θ by multiplication with an element in G forms a principal G-bundle in the frame bundle FM. An equivalence of two geometric structures defined by a coframe θ on M and a coframe θ˜ ˜ This is a diffeomorphism on M˜ is an equivalence of the corresponding G-structures B and B.
From the form in local coordinates we see that the conformal structure on the Grassmannian is equivalent to the flat conformal structure on R4 . 5. The conformal isometry group of Gr2 (V ) is equal to P GL(4, R). The group P GL+ (4, R) of orientation preserving projective linear transformations is the conformal isometry group of Gr2 (V ). Proof. 2). These transformations are conformal and the action of A ∈ GL(4, R) is non-trivial on the Grassmannian if and only if A ∈ RI . 4 we proof that every conformal transformation can be obtained from P GL(4, R).
55. A coframe θ on M is regular of rank r if for some s ≥ 0 the ranks of T (s) and T (s+1) are equal to r and constant on M. The smallest integer s for which this condition holds is called the order of the coframe. A coframe is called fully regular if for all orders s ≥ 0 the structure map T (s) is regular. If a coframe is regular, then the image of the s-th order structure map is a submanifold of K(s) and we speak of the classifying manifold. 56. • The standard coframe θ 1 = dx, θ 2 = dy on R2 is a fully regular coframe of rank 0 (and therefore of order 0).
Contact Structures of Partial Differential Equations by Pieter Thijs Eendebak