open access

https://doi.org/10.14943/84163

https://hdl.handle.net/2115/69822

CONTACT GEOMETRY OF SECOND ORDER II

英語

Parabolic Geomeries

2012-09-25

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

1017

1-65

This is the continuation of our previous paper “Contact Geometry of Second Order I” , where we have formulated the contact equivalence of systems of second order partial differential equations for a scalar function as the geometry of PD manifolds of second order. In this paper, we will discuss the Two Step Reduction procedure in Contact Geometry of Second Order. In fact we establish the Second Reduction Theorem for PD manifolds (R;D1,D2) of second order admitting the first order covariant systems ˜N . Utilizing the covariant system ˜N , we construct the intermediate object (W;C:,N), called the IG manifold of corank r, as a submanifold of the Involutive Grassmann bundle Ir(J) over the contact manifold (J,C), where J = R/Ch (D1). We will seek the condition when the equivalence of (R;D1,D2) is reducible to that of (W;C∗,N). Moreover, when Ch (N) is non-trivial, the equivalence of (W;C∗,N) is further reducible to that of (Y ;D∗ N,DN), where Y = W/Ch (N). This theorem gives a sufficient condition for the existence of higher dimensional characteristics of (R;D1,D2). By analyzing the construction parts of the Two Step Reduction procedure, we will show several examples of Parabolic Geometries, which are, through the Second Reduction Theorem, associated with the geometry of PD manifolds of second order.

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