By Ivan Cherednik

This number of papers at the geometry and topology of submanifolds is taken from conferences held in honour of Professor S.S. Chern. themes coated contain: the category of timelike Bonnet surfaces; parallel natural spinors on pseudo-Riemannian manifolds; harmonic maps; and extra 1. Conservation legislation & Algebraic-Geometric suggestions. 1. neighborhood conservation legislation. 2. Generalized Lax equations. three. Algebraic-geometric ideas of simple equations. four. Algebraic-geometric suggestions of Sin-Gordon, NS, and so forth. -- II. Backlund Transforms and Inverse challenge. 1. Backlund alterations. 2. creation to the scattering concept. three. functions of the inverse challenge procedure

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We also remark that a) itself is a consequence of a certain analogue of the following fundamental theorem of the Galois theory for jets. 4*. Let V be a ring over C with a differentiation d, W an extension of V with a differentiation & which extends d. Suppose that TV is generated by ai,... , am £iy over V and that a i , . . , a m satisfy a system of differential equations of first order with coefRcients in V which is solved with respect to &ai. Then V can be included into the ring T> of formal jets over T>.

In particular, dr exp(fcxJ7o) = 0. One can check that [dr, d] = dr o d — d o dr = 0. 12) with respect to fc-1 with £ on the right we obtain explicit formulae for dr^i in terms of {^i} and their x-derivatives of appropriate orders (cf. 1). We can extend dr uniquely to a differentiation of the ring of differential polynomials of the entries of { $ j } . The action of dT on any polynomial expressions in terms of k, fc-1, d, £ with coefficients from this ring is also well-defined (dr are applied coefficient-wise).

1. 10). 2. , are polynomials in U, Ux, Uxx,.... 6) in terms of $ , using U only). 3*. Suppose that the derivatives ofQ with respect to x of arbitrary order ^ 0 are absolutely integrable from x = —oo to x = oo and U —»• UQ when x —>• oo and x —¥ — oo (for PCF, we need to assume additionally that the limits F ( + o o ) , V(—oo) exist and coincide). 9). 3. Formal j e t s . 1, first we deduce statement b) from statement a). 1. , belong to the ring of polynomials C[U] = C[u\,... , u | , . . , u™] of {u*}).