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Prolongations of linear overdetermined systems on affine and Riemannian manifolds. (English) Zbl 1238.58012
Slovák, Jan (ed.) et al., The proceedings of the 24th winter school “Geometry and physics”, Srní, Czech Republic, January 17–24, 2004. Palermo: Circolo Matemático di Palermo. Supplemento ai Rendiconti del Circolo Matemático di Palermo. Serie II 75, 89-108 (2005).
From the introduction: According to folklore (a precise criterion in the language of exterior differential systems may be found in [R. L. Bryant et al., Exterior differential systems, Publ., Math. Sci. Res. Inst., 18. New York etc.: Springer-Verlag (1991; Zbl 0726.58002)]), a generic overdetermined partial differential equation may be rewritten as a first order ‘closed system’ in which all first partial derivatives of the dependent variables are expressed in terms of the variables themselves. To do this, one must introduce extra dependent variables for unknown derivatives until all derivatives of the original and extra variables can be determined as consequences of the original equation. This is the well-known procedure of ‘prolongation’. Particular prolongations, however, are usually derived ad hoc.
Recent joint work with T. Branson, A. Čap and R. Gover [Int. J. Math. 17, No. 6, 641–664 (2006; Zbl 1101.35060)] shows how to implement this prolongation procedure for a wide class of geometrically defined equations on manifolds with a suitable differential geometric structure. This article presents a special case of our results. Specifically, we consider only linear equations on affine or Riemannian manifolds. By restricting to these special cases, the proofs are considerably simplified. Several further examples are given to complement [Branson et al., loc. cit.].
The ingredients for this work are now known informally as the Bernstein-Gelfand-Gelfand (BGG) machinery This machinery is obtained by interpreting suitable Lie algebra cohomology as providing geometric constructions on manifolds. It has been a common theme at previous Czech Winter Schools.
The advantages of a closed system are considerable. In the linear case, one obtains a bound on the dimension of the solution space (namely the final number of dependent variables) whilst, in the semilinear case, constraints on solutions may be derived by cross differentiation and back substitution.
For the entire collection see [Zbl 1074.53001].

58J10 Differential complexes
35N10 Overdetermined systems of PDEs with variable coefficients
58J70 Invariance and symmetry properties for PDEs on manifolds
17B56 Cohomology of Lie (super)algebras
22E46 Semisimple Lie groups and their representations
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