Conformal theory of curves with tractors

Šilhan, Josef; Žádník, Vojtěch

doi:10.1016/j.jmaa.2018.12.038

Cited by 4 publications

(3 citation statements)

References 15 publications

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“…Let γ : R → c be a curve on (M = K/H, [g]) in some exponential coordinates c : c → M. In this section, we consider more general exponential coordinates than those compatible with decomposition l ⊕ a ⊕ n, because it will simplify some computations. According to [25,38], we assign to nowhere null curve γ, where we always omit writing the argument t for γ and its components in the local coordinates, a conformally invariant curve Σ γ : R → ∧ 3 T as follows. For a chosen g in the conformal class, we denote…”

Section: Conformal Circles Via Conserved Quantitiesmentioning

confidence: 99%

First BGG operators on homogeneous conformal geometries

Gregorovič

Zalabová

2023

Class. Quantum Grav.

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We study first BGG operators and their solutions on homogeneous conformal geometries. We focus on conformal Killing tensors, conformal Killing--Yano forms and twistor spinors in particular. We develop an invariant calculus that allows us to find solutions explicitly using only algebraic computations. We also discuss applications to holonomy reductions and conserved quantities of conformal circles. We demonstrate our result on examples of homogeneous conformal geometries coming mostly from general relativity.

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Section: Conformal Circles Via Conserved Quantitiesmentioning

confidence: 99%

First BGG operators on homogeneous conformal geometries

Gregorovič

Zalabová

2023

Class. Quantum Grav.

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“…In this Section, we consider more general exponential coordinates than those compatible with decomposition l⊕ a ⊕ n, because it will simplify some computations. According to [37,24], we assign to nowhere null curve γ, where we always omit writing the argument t for γ and its components in the local coordinates, a conformally invariant curve Σ γ : R → ∧ 3 T as follows. For a chosen g in the conformal class, we denote…”

Section: Conformal Circles Via Conserved Quantitiesmentioning

confidence: 99%

First BGG operators on homogeneous conformal geometries

Gregorovič¹,

Zalabová²

2022

Preprint

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We study first BGG operators and their solutions on homogeneous conformal geometries. We focus on conformal Killing tensors, conformal Killing-Yano forms and twistor spinors in particular. We develop an invariant calculus that allows us to find solutions explicitly using only algebraic computations. We also discuss applications to holonomy reductions and conserved quantities of conformal circles. We demonstrate our result on examples of homogeneous conformal geometries coming mostly from general relativity.

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“…With a view to various applications, this hypersurface theory was extended in the works [9,67,93,51,98] and this approach has proved to be central in a number of further extensions and applications [2,8,57,58,59,62,61,75]. Rather separately from the general consideration of submanifolds, the distinguished curves in conformal manifolds known as conformal circles or conformal geodesics have been studied classically (see, e.g., [39,85,86,100,101]) and from various modern perspectives recently [44,4,5,96,90,56,35,68,72,14].…”

Section: Introductionmentioning

confidence: 99%

CR Embedded Submanifolds of CR Manifolds

Curry¹,

Gover²

2019

Memoirs of the AMS

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For conformal geometries of Riemannian signature, we provide a comprehensive and explicit treatment of the core local theory for embedded submanifolds of arbitrary dimension. This is based in the conformal tractor calculus and includes a conformally invariant Gauss formula leading to conformal versions of the Gauss, Codazzi, and Ricci equations. It provides the tools for proliferating submanifold conformal invariants, as well for extending to conformally singular Riemannian manifolds the notions of mean curvature and of minimal and CMC submanifolds.A notion of distinguished submanifold is defined by asking the tractor second fundamental form to vanish. We show that for the case of curves this exactly characterises conformal geodesics, also called conformal circles, while for hypersurfaces it is the totally umbilic condition. So, for other codimensions, this unifying notion interpolates between these extremes, and we prove that in all dimensions this coincides with the submanifold being weakly conformally circular, meaning that ambient conformal circles remain in the submanifold. We prove that submanifolds are conformally circular, meaning submanifold conformal circles coincide with ambient conformal circles, if and only also a second conformal invariant also vanishes.Next we provide a very general theory and construction of quantities that are necessarily conserved along distinguished submanifolds. This first integral theory thus vastly generalises the results available for conformal circles in [56]. We prove that any normal solution to an equation from the class of first BGG equations can yield such a conserved quantity, and we show that it is easy to provide explicit formulae for these.Finally we prove that the property of being distinguished is also captured by a type of moving incidence relation. This second characterisation is used to show that, for suitable solutions of conformal Killing-Yano equations, a certain zero locus of the solution is necessarily a distinguished submanifold.

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Conformal theory of curves with tractors

Cited by 4 publications

References 15 publications

First BGG operators on homogeneous conformal geometries

First BGG operators on homogeneous conformal geometries

First BGG operators on homogeneous conformal geometries

CR Embedded Submanifolds of CR Manifolds

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