Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 053, 14 pages      arXiv:1410.7593      https://doi.org/10.3842/SIGMA.2015.053

Constructing Involutive Tableaux with Guillemin Normal Form

Abraham D. Smith
Department of Mathematics, Statistics and Computer Science, University of Wisconsin-Stout, Menomonie, WI 54751-2506, USA

Received December 15, 2014, in final form July 01, 2015; Published online July 09, 2015

Abstract
Involutivity is the algebraic property that guarantees solutions to an analytic and torsion-free exterior differential system or partial differential equation via the Cartan-Kähler theorem. Guillemin normal form establishes that the prolonged symbol of an involutive system admits a commutativity property on certain subspaces of the prolonged tableau. This article examines Guillemin normal form in detail, aiming at a more systematic approach to classifying involutive systems. The main result is an explicit quadratic condition for involutivity of the type suggested but not completed in Chapter IV, § 5 of the book Exterior Differential Systems by Bryant, Chern, Gardner, Goldschmidt, and Griffiths. This condition enhances Guillemin normal form and characterizes involutive tableaux.

Key words: involutivity; tableau; symbol; exterior differential systems.

pdf (413 kb)   tex (20 kb)

References

  1. Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior differential systems, Mathematical Sciences Research Institute Publications, Vol. 18, Springer-Verlag, New York, 1991, available at http://library.msri.org/books/Book18/MSRI-v18-Bryant-Chern-et-al.pdf.
  2. Carlson J., Green M., Griffiths P., Variations of Hodge structure considered as an exterior differential system: old and new results, SIGMA 5 (2009), 087, 40 pages, arXiv:0909.2201.
  3. Guillemin V., Some algebraic results concerning the characteristics of overdetermined partial differential equations, Amer. J. Math. 90 (1968), 270-284.
  4. Guillemin V.W., Quillen D., Sternberg S., The integrability of characteristics, Comm. Pure Appl. Math. 23 (1970), 39-77.
  5. Kruglikov B., Lychagin V., Spencer $\delta$-cohomology, restrictions, characteristics and involutive symbolic PDEs, Acta Appl. Math. 95 (2007), 31-50, math.DG/0503124.
  6. Malgrange B., Cartan involutiveness = Mumford regularity, in Commutative Algebra (Grenoble/Lyon, 2001), Contemp. Math., Vol. 331, Amer. Math. Soc., Providence, RI, 2003, 193-205.
  7. Quillen D.G., Formal properties of over-determined systems of partial differential equations, Ph.D. Thesis, Harvard University, 1964.
  8. Smith A.D., Degeneracy of the characteristic variety, arXiv:1410.6947.
  9. Yang D., Involutive hyperbolic differential systems, Mem. Amer. Math. Soc. 68 (1987), no. 370, xii+93 pages.

Previous article  Next article   Contents of Volume 11 (2015)