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Boris Kruglikov

Publications and source records attributed to Boris Kruglikov.

At least 37 records · Page 2Linked to original sources
arXiv2022

Killing Tensors in Koutras-McIntosh Spacetimes

The Koutras-McIntosh family of metrics include conformally flat pp-waves and the Wils metric. It appeared in a paper of 1996 by Koutras-McIntosh as an example of a pure radiation spacetime without scalar curvature invariants or infinitesimal symmetries. Here we demonstrate that these metrics have no "hidden symmetries", by which we mean Killing tensors of low degrees. For the particular case of Wils metrics we show the nonexistence of Killing tensors up to degree 6. The technique we use is the geometric theory of overdetermined PDEs and the Cartan prolongation-projection method. Application of those allows to prove the nonexistence of polynomial in momenta integrals for the equation of geodesics in a mathematical rigorous way. Using the same technique we can completely classify all lower degree Killing tensors and, in particular, prove that for generic pp-waves all Killing tensors of degree 3 and 4 are reducible.

gr-qc↗
arXiv2021

Differential invariants of curves in G(2) flag varieties

We compute the algebra of differential invariants of unparametrized curves in the homogeneous G(2) flag varieties, namely in G(2)/P. This gives a solution to the equivalence problem for such curves. We consider the cases of integral and generic curves and relate the equivalence problems for all three choices of the parabolic subgroup P.

math.DG↗
arXiv2021

Second-order PDEs in 3D with Einstein-Weyl conformal structure

Einstein-Weyl geometry is a triple (D,g,w), where D is a symmetric connection, [g] is a conformal structure and w is a covector such that: (i) connection D preserves the conformal class [g], that is, Dg=wg; (ii) trace-free part of the symmetrised Ricci tensor of D vanishes. Three-dimensional Einstein-Weyl structures arise naturally on solutions of second-order dispersionless integrable PDEs in 3D. In this context, [g] coincides with the characteristic conformal structure and is therefore uniquely determined by the equation. On the contrary, the covector w is a somewhat more mysterious object, recovered from the Einstein-Weyl conditions. We demonstrate that, for generic second-order PDEs (for instance, for all equations not of Monge-Ampere type), the covector w is also expressible in terms of the equation, thus providing an efficient dispersionless integrability test. The knowledge of g and w provides a dispersionless Lax pair by an explicit formula which is apparently new. Some partial classification results of PDEs with Einstein-Weyl characteristic conformal structure are obtained. A rigidity conjecture is proposed according to which for any generic second-order PDE with Einstein-Weyl property, all dependence on the 1-jet variables can be eliminated via a suitable contact transformation.

nlin.SI↗
arXiv2021

Differential invariants of Kundt spacetimes

We find generators for the algebra of rational differential invariants for general and degenerate Kundt spacetimes and relate this to other approaches to the equivalence problem for Lorentzian metrics. Special attention is given to dimensions three and four.

math.DG↗
arXiv2020

Joint Invariants of Linear Symplectic Actions

We review computations of joint invariants on a linear symplectic space, discuss variations for an extension of group and space and relate this to other equivalence problems and approaches, most importantly to differential invariants.

math.DG↗
arXiv2020

Differential Invariants of Linear Symplectic Actions

We consider the equivalence problem for symplectic and conformal symplectic group actions on submanifolds and functions of symplectic and contact linear spaces. This is solved by computing differential invariants via the Lie-Tresse theorem.

math.SG↗
arXiv2020

G(3)-supergeometry and a supersymmetric extension of the Hilbert-Cartan equation

We realize the simple Lie superalgebra G(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 G(2) symmetry. We provide the symmetries explicitly and compute, via the first Spencer cohomology groups, the Tanaka-Weisfeiler prolongation of the negatively graded Lie superalgebras associated with two particular choices of parabolics. We discuss non-holonomic superdistributions with growth vector (2|4,1|2,2|0) deforming the flat model SHC, and prove that the second Spencer cohomology group gives a binary quadratic form, thereby providing a "square-root" of Cartan's classical binary quartic invariant for generic rank 2 distributions in a 5-dimensional space. Finally, we obtain super-extensions of Cartan's classical submaximally symmetric models, compute their symmetries and observe a supersymmetry dimension gap phenomenon.

math.DG↗
arXiv2020

Submaximally Symmetric Quaternion Hermitian Structures

We consider and resolve the gap problem for almost quaternion-Hermitian structures, i.e. we determine the maximal and submaximal symmetry dimensions, both for Lie algebras and Lie groups, in the class of almost quaternion-Hermitian manifolds. We classify all structures with such symmetry dimensions. Geometric properties of the submaximally symmetric spaces are studied, in particular we identify locally conformally quaternion-Kähler structures as well as quaternion-Kähler with torsion.

math.DG↗
arXiv2020

Second-order PDEs in 4D with half-flat conformal structure

We study second-order PDEs in 4D for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge-Ampere property. Since half-flatness of the conformal structure is equivalent to the existence of a nontrivial dispersionless Lax pair, our result explains the observation that all known scalar second-order integrable dispersionless PDEs in dimensions four and higher are of Monge-Ampere type. Some partial classification results of Monge-Ampere equations in 4D with half-flat conformal structure are also obtained.

math.DG↗
arXiv2019

Blow-ups and infinitesimal automorphisms of CR-manifolds

Let $M$ be a real-analytic connected CR-hypersurface of CR-dimension $n>0$ having a point of Levi-nondegeneracy. The following alternative is demonstrated for both the symmetry algebra $s$ and the automorphism group $G$ of $M$. Denote by $d$ the dimension of $s$ or $G$. Then (i) either $d=n^2+4n+3$ and $M$ is spherical everywhere; (ii) or $d\le n^2+2n+2+δ_{2,n}$ and in the case of equality $M$ is spherical of fixed Levi signature in the open dense subset of Levi-nondegenerate points. Explicit examples of CR-hypersurfaces and their infinitesimal and global automorphisms realizing the bound in (ii) are constructed. We provide many other models with large symmetry using the technique of blow-up, in particular we realize all maximal parabolic subalgebras of the pseudo-unitary algebras as a symmetry.

math.CV↗
arXiv2019

Differential invariants of Kundt waves

Kundt waves belong to the class of spacetimes which are not distinguished by their scalar curvature invariants. We address the equivalence problem for the metrics in this class via scalar differential invariants with respect to the equivalence pseudo-group of the problem. We compute and finitely represent the algebra of those on the generic stratum and also specify the behavior for vacuum Kundt waves. The results are then compared to the invariants computed by the Cartan-Karlhede algorithm.

gr-qc↗
arXiv2019

Integrability via geometry: dispersionless differential equations in three and four dimensions

We prove that the existence of a dispersionless Lax pair with spectral parameter for a nondegenerate hyperbolic second order partial differential equation (PDE) is equivalent to the canonical conformal structure defined by the symbol being Einstein-Weyl on any solution in 3D, and self-dual on any solution in 4D. The first main ingredient in the proof is a characteristic property for dispersionless Lax pairs. The second is the projective behaviour of the Lax pair with respect to the spectral parameter. Both are established for nondegenerate determined systems of PDEs of any order. Thus our main result applies more generally to any such PDE system whose characteristic variety is a quadric hypersurface.

math.AP↗
arXiv2018

A criterion for the existence of Killing vectors in 3D

A three-dimensional Riemannian manifold has locally 6, 4, 3, 2, 1 or none independent Killing vectors. We present an explicit algorithm for computing dimension of the infinitesimal isometry algebra. It branches according to the values of curvature invariants. These are relative differential invariants computed via curvature, but they are not scalar polynomial Weyl invariants. We compare our obstructions to the existence of Killing vectors with the known existence criteria due to Singer, Kerr and others.

gr-qc↗
arXiv2018

Poincare function for moduli of differential-geometric structures

The Poincare function is a compact form of counting moduli in local geometric problems. We discuss its property in relation to V.Arnold's conjecture, and derive this conjecture in the case when the pseudogroup acts algebraically and transitively on the base. Then we survey the known counting results for differential invariants and derive new formulae for several other classification problems in geometry and analysis.

math.DG↗
arXiv2018

Differential invariants of Einstein-Weyl structures in 3D

Einstein-Weyl structures on a three-dimensional manifold $M$ is given by a system $E$ of PDEs on sections of a bundle over $M$. This system is invariant under the Lie pseudogroup $G$ of local diffeomorphisms on $M$. Two Einstein-Weyl structures are locally equivalent if there exists a local diffeomorphism taking one to the other. Our goal is to describe the quotient equation $E/G$ whose solutions correspond to nonequivalent Einstein-Weyl structures. The approach uses symmetries of the Manakov-Santini integrable system and the action of the corresponding Lie pseudogroup.

math.DG↗
arXiv2017

Symmetries of second order ODEs

It is demonstrated that point symmetry algebras of general analytic second order ODEs, not necessary of principal type, can have all dimensions between 0 and 8 except for 7. For the symmetry dimension 8 the ODE must be locally trivializable.

math.DG↗
arXiv2017

On the symmetry algebras of 5-dimensional CR-manifolds

We show that for a real-analytic connected holomorphically nondegenerate 5-dimensional CR-hypersurface $M$ and its symmetry algebra $\mathfrak{s}$ one has either: (i) $\dim\mathfrak{s}=15$ and $M$ is spherical (with Levi form of signature either $(2,0)$ or $(1,1)$ everywhere), or (ii) $\dim\mathfrak{s}\le11$ where $\dim\mathfrak{s}=11$ can only occur if on a dense open subset $M$ is spherical with Levi form of signature $(1,1)$. Furthermore, we construct a series of examples of pairwise nonequivalent CR-hypersurfaces with $\dim\mathfrak{s}=11$.

math.CV↗