Abstract
Classical theory for systems of the first order partial differential equations for a scalar function can be rephrased as the submanifold theory of contact manifolds (geometric first order jet spaces). In the same spirit, we will develop the geometric theory of systems of partial differential equations of second order for a scalar function as the Contact Geometry of Second Order, following E. Cartan.We will formulate the submanifold theory of second order jet spaces as the geometry of PD manifolds (R;D 1 ,D 2 ) of second order. Moreover we will establish the First Reduction Theorem for (R;D 1,D 2) admitting non-trivial Cauchy characteristic systems. By utilizing Parabolic Geometry, we will give, directly or combined with reduction theorems, several classes of systems of partial differential equations of second order, for which the model equation of each class admits the Lie algebra of infinitesimal contact transformations, which is finite dimensional and simple.
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References
T.N. Baily, Parabolic Invariant Theory and Geometry in “The Penrose Transform and Analytic Cohomology in Representation Theory” Contemp. Math. 154, Amer. Math. Soc., (1993).
W. M. Boothby, Homogeneous complex contact manifolds, Proc. Symp. Pure Math., Amer. Math. Soc. 3 (1961), 144-154.
N. Bourbaki, Groupes et algèbles de Lie, Chapitre 4,5 et 6, Hermann, Paris (1968).
R. Bryant, S.S. Chern, R.B. Gardner, H. Goldschmidt, and P. Griffiths, Exterior differential systems, Springer, New York (1986)
E. Cartan, Les systèmes de Pfaff à cinq variables et les équations aux dérivées partielles du second ordre, Ann. Ec. Normale 27 (1910), 109-192.
E. Cartan , Sur les systèmes en involution d'équations aux dérivées partielles du second ordre à une fonction inconnue de trois variables indépendantes, Bull. Soc. Math. France 39 (1911), 352-443.
J. Hwang and K. Yamaguchi, Characterization of hermitian Symmetric Spaces by Fundamental Forms, Duke Math. J. 120 No.3 (2003), 621-634.
J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, New York (1972).
B. Kostant, Lie algebra cohomology and generalized Borel-Weil theorem, Ann. Math. 74 (1961), 329-387.
M. Kuranishi, Lectures on involutive systems of partial differential equations, Pub. Soc. Mat., São Paulo (1967).
T. Sasaki, K. Yamaguchi and M. Yoshida, On the Rigidity of Differential Systems modeled on Hermitian Symmetric Spaces and Disproofs of a Conjecture concerning Modular Interpretations of Configuration Spaces, Advanced Studies in Pure Math. 25 (1997), 318-354.
Y. Se-ashi, On differential invariants of integrable finite type linear differential equations, Hokkaido Math. J. 17 (1988), 151-195.
S. Sternberg Lectures on Differential Geometry, Prentice-Hall, New Jersey (1964).
N. Tanaka On generalized graded Lie algebras and geometric structures I, J. Math. Soc. Japan 19 (1967), 215-254.
N. Tanaka, On differential systems, graded Lie algebras and pseudo-groups, J. Math. Kyoto Univ. 10 (1970), 1-82.
N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. 2 (1976), 131-190.
N. Tanaka, On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J. 8 (1979), 23-84.
K. Yamaguchi Contact geometry of higher order, Japanese J. of Math. 8 (1982), 109-176.
K. Yamaguchi, On involutive systems of second order of codimension 2, Proc. of Japan Acad. 58, Ser A, No.7 (1982), 302-305.
K. Yamaguchi, Geometrization of Jet bundles, Hokkaido Math. J. 12 (1983), 27-40.
K. Yamaguchi, Typical classes in involutive systems of second order, Japanese J. Math. 11 (1985), 265-291.
K. Yamaguchi, Differential systems associated with simple graded Lie algebras, Adv. Studies in Pure Math. 22 (1993), 413-494.
K. Yamaguchi, G2-Geometry of Overdetermined Systems of Second Order, Trends in Math. (Analysis and Geometry in Several Complex Variables) (1999), Birkhäuser, Boston 289-314.
K. Yamaguchi, Geometry of Linear Differential Systems Towards Contact Geometry of Second Order, IMA Volumes in Mathematics and its Applications 144 (Symmetries and Overdetermined Systems of Partial Differential Equations) (2007), 151-203.
K. Yamaguchi and T. Yatsui, Geometry of Higher Order Differential Equations of Finite Type associated with Symmetric Spaces, Advanced Studies in Pure Mathematics 37 (2002), 397-458.
K. Yamaguchi and T. Yatsui, Parabolic Geometries associated with Differential Equations of Finite Type, Progress in Mathematics 252 (From Geometry to Quantum Mechanics: In Honor of Hideki Omori) (2007), 161-209.
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Yamaguchi, K. (2009). Contact Geometry of Second Order I. In: Kruglikov, B., Lychagin, V., Straume, E. (eds) Differential Equations - Geometry, Symmetries and Integrability. Abel Symposia, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00873-3_16
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DOI: https://doi.org/10.1007/978-3-642-00873-3_16
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