To the reader familiar with the theory of connections, it may not come as a surprise that some of the ideas which have been introduced may be formulated in the language of that theory. Equations (3.15) for example, suggest that a connection is lurking somewhere. In this section we propose two related formulations of Bäcklund maps in terms of connections. In section 7 we employ these formulations to give a geometric interpretation of the “KdV Lie algebra” discovered by Wahlquist and Estabrook . Similar interpretations, not employing jet bundles, have been suggested by R. Hermann. Our formulations are not as straightforward as we should like, and we suspect that there must be another, more direct, approach to Bäcklund maps in terms of connections yet to be uncovered.
KeywordsVector Field Fibre Bundle Horizontal Distribution Parallel Transport Infinite Order
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