Exterior Differential Systems

Bryant, Robert L.; Chern, Shiing-Shen; Gardner, Robert B.; Goldschmidt, Hubert; Griffiths, Phillip

doi:10.1007/978-1-4613-9714-4

Cited by 666 publications

(1,129 citation statements)

References 2 publications

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“…Hence for their connection forms, we get Φ * γ = γ , which together with the above implies that (Φ # ) * q * 0γ = q * 0 Φ * γ = q * 0 γ . Together with the fact that (Φ # ) * ξ 0 = λξ 0 , this implies that Φ # is compatible with the partial contact connections constructed in (2) and hence in particular with the contact projective structures they generate.…”

Section: Contactifications Of Conformally Fedosov Structuresmentioning

confidence: 79%

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Parabolic conformally symplectic structures II: parabolic contactification

Čap

Salač

2017

Annali di Matematica

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Parabolic almost conformally symplectic structures were introduced in the first part of this series of articles as a class of geometric structures which have an underlying almost conformally symplectic structure. If this underlying structure is conformally symplectic, then one obtains a PCS-structure. In the current article, we relate PCS-structures to parabolic contact structures. Starting from a parabolic contact structure with a transversal infinitesimal automorphism, we first construct a natural PCS-structure on any local leaf space of the corresponding foliation. Then we develop a parabolic version of contactification to show that any PCS-structure can be locally realized (uniquely up to isomorphism) in this way. In the second part of the paper, these results are extended to an analogous correspondence between contact projective structures and so-called conformally Fedosov structures. The developments in this article provide the technical background for a construction of sequences and complexes of differential operators which are naturally associated to PCS-structures by pushing down BGG sequences on parabolic contact structures. This is the topic of the third part of this series of articles.

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Section: Contactifications Of Conformally Fedosov Structuresmentioning

confidence: 79%

“…Hence by Corollary 2.3 in [2] we can prove that γ # descends to G 0 by showing that it is invariant under the flow of ξ 0 . As discussed in Sect.…”

Section: Distinguished Connections and Contactificationsmentioning

confidence: 89%

Parabolic conformally symplectic structures II: parabolic contactification

Čap

Salač

2017

Annali di Matematica

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“…3 Either we apply the numerical upwind PDEscheme described in subsection 6.1 using the discrete left-invariant vector fields (34), or we apply erosion (32) on the modulus and restore the phase afterwards.…”

Section: Evaluation Of Reassignmentmentioning

confidence: 99%

Left Invariant Evolution Equations on Gabor Transforms

Duits

Führ

Janssen

2011

Computational Imaging and Vision

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By means of the unitary Gabor transform one can relate operators on signals to operators on the space of Gabor transforms. In order to obtain a translation and modulation invariant operator on the space of signals, the corresponding operator on the reproducing kernel space of Gabor transforms must be left invariant, i.e. it should commute with the left regular action of the reduced Heisenberg group H r . By using the left invariant vector fields on H r and the corresponding left-invariant vector fields on on phase space in the generators of our transport and diffusion equations on Gabor transforms we naturally employ the essential group structure on the domain of a Gabor transform. Here we mainly restrict ourselves to non-linear adaptive left-invariant convection (reassignment), while maintaining the original signal.

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“…If R 7 is identified with the imaginary octonians Im(O) and x · y represents octonianic multiplication of x, y ∈ Im(O), then (4) x · y, z = ϕ(x, y, z).…”

Section: Introduction In 1982 Harvey and Lawson Introduced Coassociamentioning

confidence: 99%

“…Exterior differential systems are not the height of fashion these days so I will briefly review the basic definitions. For more information on EDS see the standard text [4]. For a gentler introduction see [8].…”

Section: Introduction In 1982 Harvey and Lawson Introduced Coassociamentioning

confidence: 99%

Coassociative Cones Ruled by 2-planes

Fox

2007

Asian Journal of Mathematics

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Abstract. It is shown that coassociative cones in R 7 that are r-oriented and ruled by 2-planes are equivalent to CR-holomorphic curves in the oriented Grassmanian of 2-planes in R 7 . The geometry of these CR-holomorphic curves is studied and related to holomorphic curves in S 6 . This leads to an equivalence between associative cones on one side and the coassociative cones whose second fundamental form has an O(2) symmetry on the other. It also provides a number of methods for explicitly constructing coassociative 4-folds. One method leads to a family of coassociative 4-folds whose members are neither cones nor are ruled by 2-planes. This family directly generalizes the original family of examples provided by Harvey and Lawson when they introduced coassociative geometry.Key words. Differential geometry, Calibrations, Coassociative AMS subject classifications. Primary 53C381. Introduction. In 1982 Harvey and Lawson introduced coassociative 4-folds in R 7 as an example of a calibrated geometry [6]. Using the perspective of exterior differential systems they proved a local existence theorem. Harvey and Lawson offered a single finite dimensional family of explicit examples (see section 9). In 1985 Mashimo classified the coassociative cones whose links are homogeneous [15]. Then for almost twenty years the literature was quiet on the subject. The silence was broken in 2004 when techniques used for studying special Lagrangian geometry (which is another type of calibrated geometry) began to be applied to coassociative geometry [13], [7], [11]. This renewal of activity was motivated by Joyce's construction of compact manifolds with G 2 holonomy [9] and the appearance of coassociative geometry in M-theory [1].In [13] Lotay studied coassociative 4-folds in R 7 that are ruled by 2-planes. He gave a local existence result and presented a method for using a holomorphic vector field on a naturally associated Riemann surface to deform a coassociative cone that is ruled by 2-planes. His work is an extension of Joyce's work on ruled special Lagrangian 3-folds [10].This article revisits the geometry of coassociative cones that are ruled by 2-planes using the methods introduced by Bryant in [3]. This perspective realizes the Riemann surface that appeared in [13] as a CR-holomorphic curve for a G 2 -invariant CR-structure onG(2, 7), the oriented Grassmanian of 2-planes in R 7 . In fact, Proposition 7.2 asserts that r-oriented 2-ruled coassociative cones are equivalent to CRholomorphic curves inG(2, 7).Studying the geometry of CR-holomorphic curves leads to new methods for constructing coassociative cones that are ruled by 2-planes. These methods arise from the close relationship between CR-holomorphic curves inG(2, 7) and holomorphic curves 1 in S 6 . This relationship can be interpreted as an equivalence between the associative cones and a certain family of coassociative cones. The null-torsion holomorphic curves in S 6 also form the backbone for a new family of coassociative 4-folds that directly generalizes the family introduc...

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Exterior Differential Systems

Abstract: These are notes for a very rapid introduction to the basics of exterior differential systems and their connection with what is now known as Lie theory, together with some typical and not-so-typical applications to illustrate their use.

Cited by 666 publications

References 2 publications

Parabolic conformally symplectic structures II: parabolic contactification

Parabolic conformally symplectic structures II: parabolic contactification

Left Invariant Evolution Equations on Gabor Transforms

Coassociative Cones Ruled by 2-planes

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