9,269 research outputs found
Iterated Differential Forms IV: C-Spectral Sequence
For the multiple differential algebra of iterated differential forms (see
math.DG/0605113 and math.DG/0609287) on a diffiety (O,C) an analogue of
C-spectral sequence is constructed. The first term of it is naturally
interpreted as the algebra of secondary iterated differential forms on (O,C).
This allows to develop secondary tensor analysis on generic diffieties, some
simplest elements of which are sketched here. The presented here general theory
will be specified to infinite jet spaces and infinitely prolonged PDEs in
subsequent notes.Comment: 8 pages, submitted to Math. Dok
Domains in Infinite Jets: C-Spectral Sequence
Domains in infinite jets present the simplest class of diffieties with
boundary. In this note some basic elements of geometry of these domains are
introduced and an analogue of the C-spectral sequence in this context is
studied. This, in particular, allows cohomological interpretation and analysis
of initial data, boundary conditions, etc, for general partial differential
equations and of transversality conditions in calculus of variations. This kind
applications and extensions to arbitrary diffieties will be considered in
subsequent publications.Comment: 7 pages; no proofs give
Iterated Differential Forms I: Tensors
We interpret tensors on a smooth manifold M as differential forms over a
graded commutative algebra called the algebra of iterated differential forms
over M. This allows us to put standard tensor calculus in a new differentially
closed context and, in particular, enriches it with new natural operations.
Applications will be considered in subsequent notes.Comment: 9 pages, extended version of the published note Dokl. Math. 73, n. 2
(2006) 16
Iterated Differential Forms II: Riemannian Geometry Revisited
A natural extension of Riemannian geometry to a much wider context is
presented on the basis of the iterated differential form formalism developed in
math.DG/0605113 and an application to general relativity is given.Comment: 12 pages, extended version of the published note Dokl. Math. 73, n. 2
(2006) 18
Iterated Differential Forms III: Integral Calculus
Basic elements of integral calculus over algebras of iterated differential
forms, are presented. In particular, defining complexes for modules of integral
forms are described and the corresponding berezinians and complexes of integral
forms are computed. Various applications and the integral calculus over the
algebra will be discussed in subsequent notes.Comment: 7 pages, submitted to Math. Dok
Iterated Differential Forms VI: Differential Equations
We describe the first term of the --spectral
sequence (see math.DG/0610917) of the diffiety (E,C), E being the infinite
prolongation of an l-normal system of partial differential equations, and C the
Cartan distribution on it.Comment: 8 pages, to appear in Dokl. Mat
Iterated Differential Forms V: C-Spectral Sequence on Infinite Jet Spaces
In the preceding note math.DG/0610917 the
--spectral sequence, whose first term is composed of
\emph{secondary iterated differential forms}, was constructed for a generic
diffiety. In this note the zero and first terms of this spectral sequence are
explicitly computed for infinite jet spaces. In particular, this gives an
explicit description of secondary covariant tensors on these spaces and some
basic operations with them. On the basis of these results a description of the
--spectral sequence for infinitely prolonged PDE's
will be given in the subsequent note.Comment: 9 pages, to appear in Math. Dok
Scalar differential invariants of symplectic Monge–Ampère equations
All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère PDEs with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. A series of invariant differential forms and vector fields are also introduced: they allow one to construct numerous scalar differential invariants of higher order. The introduced invariants give a solution to the symplectic equivalence problem for Monge-Ampère equations
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