Andreas Cap
Overdetermined Systems, Conformal Differential Geometry, and the BGG Complex
The paper is published:
in M.G. Eastwood, W. Miller (eds.) "Symmetries and Overdetermined Systems of Partial Differential Equations", The IMA Volumes in Mathematics and its Applications 144, Springer (2008), 1-25
- MSC:
- 35N10 Overdetermined systems with variable coefficients (general)
- 53A30 Conformal differential geometry
- 58G05 Differential complexes, See also {35Nxx}; elliptic complexes
- 58G35 Invariance and symmetry properties, See also {35A30}
- 53A40 Other special differential geometries
- 53C15 General geometric structures on manifolds (almost complex, contact, symplectic, almost product structures, etc.)
Abstract: This is an expanded version of a series of two lectures given at the
IMA summer program ``Symmetries and Overdetermined Systems of
Partial Differential Equations''. The main part of the article
describes the Riemannian version of the prolongation procedure for
certain overdetermined system obtained recently in joint work with
T.P.~Branson, M.G.~Eastwood, and A.R.~Gover. First a simple special
case is discussed, then the (Riemannian) procedure is described in
general.
The prolongation procedure was derived from a simplification of the
construction of Bernstein--Gelfand--Gelfand (BGG) sequences of
invariant differential operators for certain geometric structures.
The version of this construction for conformal structures is
described next. Finally, we discuss generalizations of both the
prolongation procedure and the construction of invariant operators
to other geometric structures.
Keywords: overdetermined system, prolongation, invariant differential operator, conformal geometry, parabolic geometry