English
norskBlar i forfatter "The, Dennis"
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Classification of Simply-Transitive Levi Non-Degenerate Hypersurfaces in C^3
(Journal article; Tidsskriftartikkel; Peer reviewed, 2021-06-24)Holomorphically homogeneous Cauchy–Riemann (CR) real hypersurfaces M<sup>3</sup>⊂C<sup>2</sup> were classified by Élie Cartan in 1932. In the next dimension, we complete the classification of simply-transitive Levi non-degenerate hypersurfaces M<sup>5</sup>⊂C<sup>3</sup> using a novel Lie algebraic approach independent of any earlier classifications of abstract Lie algebras. Central to our approach ... -
Classification of simply-transitive Levi non-degenerate hypersurfaces in C^3
(Journal article; Tidsskriftartikkel; Peer reviewed, 2021-06-24)Holomorphically homogeneous Cauchy–Riemann (CR) real hypersurfaces <i>M<i/><sup>3</sup>⊂C<sup>2</sup> were classified by Élie Cartan in 1932. In the next dimension, we complete the classification of simply-transitive Levi non-degenerate hypersurfaces <i>M<i/><sup>5</sup>⊂C<sup>3</sup> using a novel Lie algebraic approach independent of any earlier classifications of abstract Lie algebras. Central ... -
Exceptionally simple PDE
(Journal article; Tidsskriftartikkel; Peer reviewed, 2017-11-08)We give local descriptions of parabolic contact structures and show how their flat models yield explicit PDE having symmetry algebras isomorphic to all complex simple Lie algebras except <math><msub is="true"><mrow is="true"><mi mathvariant="fraktur" is="true">sl</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub></math>. This yields a remarkably uniform generalization of the Cartan–Engel ... -
Exceptionally simple super-PDE for F (4)
(Journal article; Tidsskriftartikkel; Peer reviewed, 2024-02-01)For the largest exceptional simple Lie superalgebra F(4), having dimension (24|16), we provide two explicit geometric realizations as supersymmetries, namely as the symmetry superalgebra of super-PDE systems of second and third order respectively. -
G(3)-supergeometry and a supersymmetric extension of the Hilbert–Cartan equation
(Journal article; Tidsskriftartikkel; Peer reviewed, 2020-10-23)We realize the simple Lie superalgebra <i>G</i>(3) as supersymmetry of various geometric structures, most importantly super-versions of the Hilbert–Cartan equation (SHC) and Cartan's involutive PDE system that exhibit <i>G</i>(2) symmetry. We provide the symmetries explicitly and compute, via the first Spencer cohomology groups, the Tanaka–Weisfeiler prolongation of the negatively graded Lie ... -
The gap phenomenon in parabolic geometries
(Journal article; Tidsskriftartikkel; Peer reviewed, 2014-09-14)The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type (G,P), we use Tanaka theory to derive a universal upper bound on the submaximal symmetry dimension. We use Kostant’s version ... -
Homogeneous integrable Legendrian contact structures in dimension five
(Journal article; Tidsskriftartikkel; Peer reviewed, 2019-07-04)We consider Legendrian contact structures on odd-dimensional complex analytic manifolds. We are particularly interested in integrable structures, which can be encoded by compatible complete systems of second order PDEs on a scalar function of many independent variables and considered up to point transformations. Using the techniques of parabolic differential geometry, we compute the associated ... -
Homogeneous Levi non-degenerate hypersurfaces in C3
(Journal article; Tidsskriftartikkel; Peer reviewed, 2020-06-09)We classify all (locally) homogeneous Levi non-degenerate real hypersurfaces in C<sup>3</sup> with symmetry algebra of dimension ≥6. -
Homogeneous Levi non-degenerate hypersurfaces in C^3
(Journal article; Tidsskriftartikkel; Peer reviewed, 2020-06-09)We classify all (locally) homogeneous Levi non-degenerate real hypersurfaces in C<sup>3</sup> with symmetry algebra of dimension ≥6. -
Jet-determination of symmetries of parabolic geometries
(Journal article; Tidsskriftartikkel; Peer reviewed, 2017-04-24)We establish 2-jet determinacy for the symmetry algebra of the underlying structure of any (complex or real) parabolic geometry. At non-flat points, we prove that the symmetry algebra is in fact 1-jet determined. Moreover, we prove 1-jet determinacy at any point for a variety of non-flat parabolic geometries—in particular torsion-free, parabolic contact, and several other classes. -
On C-class equations
(Journal article; Tidsskriftartikkel; Peer reviewed, 2023-09-29)The concept of a C-class of differential equations goes back to E. Cartan with the upshot that generic equations in a C-class can be solved without integration. While Cartan’s definition was in terms of differential invariants being first integrals, all results exhibiting C-classes that we are aware of are based on the fact that a canonical Cartan geometry associated to the equations in the class ... -
On uniqueness of submaximally symmetric parabolic geometries
(Journal article; Tidsskriftartikkel, 2021)Among (regular, normal) parabolic geometries of type (G,P), there is a locally unique maximally symmetric structure and it has symmetry dimension dim(G). The symmetry gap problem concerns the determination of the next realizable (submaximal) symmetry dimension. When G is a complex or split-real simple Lie group of rank at least three or when (G,P)=(G2,P2), we establish a local classification result ... -
On uniqueness of submaximally symmetric parabolic geometries
(Journal article; Tidsskriftartikkel; Peer reviewed, 2024-01-24)Among (regular, normal) parabolic geometries of type (<i>G,P</i>), there is a locally unique maximally symmetric structure and it has symmetry dimension dim(<i>G</i>). The symmetry gap problem concerns the determination of the next realizable (submaximal) symmetry dimension. When <i>G</i> is a complex or split-real simple Lie group of rank at least three or when (<i>G,P</i>) = (<i>G<sub>2</sub></i>, ... -
On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class
(Journal article; Tidsskriftartikkel; Peer reviewed, 2023-08-10)The fundamental invariants for vector ODEs of order ≥3 considered up to point transformations consist of generalized Wilczynski invariants and C-class invariants. An ODE of C-class is characterized by the vanishing of the former. For any fixed C-class invariant U , we give a local (point) classification for all submaximally symmetric ODEs of C-class with U≢0 and all remaining C-class invariants ... -
Symmetries of supergeometries related to nonholonomic superdistributions
(Journal article; Tidsskriftartikkel; Peer reviewed, 2022-06-06)We extend Tanaka theory to the context of supergeometry and obtain an upper bound on the supersymmetry dimension of geometric structures related to strongly regular bracket-generating distributions on supermanifolds and their structure reductions. -
Symmetry gaps for higher order ordinary differential equations
(Journal article; Tidsskriftartikkel; Peer reviewed, 2022-07-04)The maximal contact symmetry dimensions for scalar ODEs of order ≥ 4 and vector ODEs of order ≥ 3 are well known. Using a Cartan-geometric approach, we determine for these ODEs the next largest realizable (submaximal) symmetry dimension. Moreover, finer curvature-constrained submaximal symmetry dimensions are also classified. -
Symmetry gaps for higher order ordinary differential equations
(Journal article; Tidsskriftartikkel, 2021)The maximal contact symmetry dimensions for scalar ODEs of order ≥4 and vector ODEs of order ≥3 are well known. Using a Cartan-geometric approach, we determine for these ODE the next largest realizable (submaximal) symmetry dimension. Moreover, finer curvature-constrained submaximal symmetry dimensions are also classified.
