Boris Kruglikov
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Boris Kruglikov has a total of 42 co-authors. Here are the top co-authors by number of shared publications:
- Evgeny Vladimirovich Ferapontov
Department of Mathematical Sciences, Loughborough University, Loughborough, UK
11 shared publications - Henrik Winther
UiT The Arctic University of Norway
11 shared publications - Eivind Schneider
Faculty of Science, University of Hradec Králové 2 , Rokitanského 62, Hradec Králové 50003, Czech Republic
10 shared publications - Vladimir Sergeevich Matveev
Institut für Mathematik, Friedrich Schiller Universität Jena, 07737 Jena, Germany
9 shared publications - Dennis The
Department of Mathematics and Statistics, UiT The Arctic University of Norway, Tromsø, Norway
5 shared publications - Oleg I. Morozov
Trapeznikov Institute of Control Sciences, Moscow, Russia
5 shared publications
Latest Publications
- Open Access
- 29 September 2026
- Published by Elsevier in Journal of Algebra
- Vol. 713, 917-977
Geometric realizations of the Lie superalgebra D(2,1;a)
- 29 September 2026
- Published by Elsevier in Journal of Algebra
- Vol. 713, 917-977
Open AccessGeometric realizations of the Lie superalgebra D(2,1;a)
- Open Access
On globally invariant Euler--Lagrange equations for curves
Invariant Lagrangians yield invariant Euler-Lagrange equations, and it was discussed in the literature how to compute those using various local methods. The focus of this paper is on global algebraic differential invariants. In this case the computation can be modified in several aspects. We will discuss relations with previous approaches and some foundational aspects. The theory of invariant Euler-Lagrange equations was applied to curves with respect to the motion group in the Euclidean plane and space. We expand those computations to the next dimension four (Minkowski spacetime), which already exhibits computational challenges. We also provide formulas for other examples, namely the projective and conformal (Möbius) groups and relate to some recent applications.
Open AccessOn globally invariant Euler--Lagrange equations for curves
Invariant Lagrangians yield invariant Euler-Lagrange equations, and it was discussed in the literature how to compute those using various local methods. The focus of this paper is on global algebraic differential invariants. In this case the computation can be modified in several aspects. We will discuss relations with previous approaches and some foundational aspects. The theory of invariant Euler-Lagrange equations was applied to curves with respect to the motion group in the Euclidean plane and space. We expand those computations to the next dimension four (Minkowski spacetime), which already exhibits computational challenges. We also provide formulas for other examples, namely the projective and conformal (Möbius) groups and relate to some recent applications.
- Open Access
- 30 October 2025
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 406 (12) , 1-26
Involutive Scroll Structures on Solutions of 4D Dispersionless Integrable Hierarchies
A rational normal scroll structure on an $$(n+1)$$ -dimensional manifold M is defined as a field of rational normal scrolls of degree $$n-1$$ in the projectivised cotangent bundle $${\mathbb {P}} T^*M$$ . We show that geometry of this kind naturally arises on solutions of various 4D dispersionless integrable hierarchies of heavenly type equations. In this context, rational normal scrolls coincide with the characteristic varieties (principal symbols) of the hierarchy. Furthermore, such structures automatically satisfy an additional property of involutivity. Our main result states that involutive scroll structures are themselves governed by a dispersionless integrable hierarchy, namely, the hierarchy of conformal self-duality equations.- 30 October 2025
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 406 (12) , 1-26
Open AccessInvolutive Scroll Structures on Solutions of 4D Dispersionless Integrable Hierarchies
A rational normal scroll structure on an $$(n+1)$$ -dimensional manifold M is defined as a field of rational normal scrolls of degree $$n-1$$ in the projectivised cotangent bundle $${\mathbb {P}} T^*M$$ . We show that geometry of this kind naturally arises on solutions of various 4D dispersionless integrable hierarchies of heavenly type equations. In this context, rational normal scrolls coincide with the characteristic varieties (principal symbols) of the hierarchy. Furthermore, such structures automatically satisfy an additional property of involutivity. Our main result states that involutive scroll structures are themselves governed by a dispersionless integrable hierarchy, namely, the hierarchy of conformal self-duality equations. - 27 October 2025
- Published by IOP Publishing in Nonlinearity
- Vol. 38 (10) , 105022
Conformal geodesics are not variational in higher dimensions
Variationality of the equation of conformal geodesics is an important problem in geometry with applications to general relativity. Recently it was proven that, in three dimensions, this system of equations for un-parametrized curves is the Euler–Lagrange equations of a certain conformally invariant functional, while the parametrized system in three dimensions is not variational. We demonstrate that variationality fails in higher dimensions for both parametrized and un-parametrized conformal geodesics, indicating that variational principle may be the selection principle for the physical dimension.
- 27 October 2025
- Published by IOP Publishing in Nonlinearity
- Vol. 38 (10) , 105022
Conformal geodesics are not variational in higher dimensions
Variationality of the equation of conformal geodesics is an important problem in geometry with applications to general relativity. Recently it was proven that, in three dimensions, this system of equations for un-parametrized curves is the Euler–Lagrange equations of a certain conformally invariant functional, while the parametrized system in three dimensions is not variational. We demonstrate that variationality fails in higher dimensions for both parametrized and un-parametrized conformal geodesics, indicating that variational principle may be the selection principle for the physical dimension.
- Open Access
- 24 October 2025
- Published by Springer Nature in The Journal of Geometric Analysis
- Vol. 35 (12) , 1-61
On 3-Nondegenerate CR Manifolds in Dimension 7 (II): the Intransitive Case
We investigate 3-nondegenerate CR structures in the lowest possible dimension 7 and show that 8 is the maximal dimension for the Lie algebra of symmetries of such structures. The next possible symmetry dimension is 6, and for the automorphism groups the dimension 7 is also realizable. This part (II) is devoted to the case where the symmetry algebra acts intransitively. We use various methods to bound its dimension and demonstrate the existence of infinitely many non-equivalent submaximally symmetric models. Summarizing, we get a stronger form of Beloshapka’s conjecture on the symmetry dimension of hypersurfaces in $$\mathbb C^4$$ .- 24 October 2025
- Published by Springer Nature in The Journal of Geometric Analysis
- Vol. 35 (12) , 1-61
Open AccessOn 3-Nondegenerate CR Manifolds in Dimension 7 (II): the Intransitive Case
We investigate 3-nondegenerate CR structures in the lowest possible dimension 7 and show that 8 is the maximal dimension for the Lie algebra of symmetries of such structures. The next possible symmetry dimension is 6, and for the automorphism groups the dimension 7 is also realizable. This part (II) is devoted to the case where the symmetry algebra acts intransitively. We use various methods to bound its dimension and demonstrate the existence of infinitely many non-equivalent submaximally symmetric models. Summarizing, we get a stronger form of Beloshapka’s conjecture on the symmetry dimension of hypersurfaces in $$\mathbb C^4$$ . - Open Access
- 2 September 2025
- Published by Springer Nature in Analysis and Mathematical Physics
- Vol. 15 (5) , 1-25
Variationality of Conformal Geodesics in dimension 3
Conformal geodesics form an invariantly defined family of unparametrized curves in a conformal manifold generalizing unparametrized geodesics/paths of projective connections. The equation describing them is of third order, and it was an open problem whether they are given by an Euler–Lagrange equation. In dimension 3 (the simplest, but most important from the viewpoint of physical applications) we demonstrate that the equation for unparametrized conformal geodesics is variational.
- 2 September 2025
- Published by Springer Nature in Analysis and Mathematical Physics
- Vol. 15 (5) , 1-25
Open AccessVariationality of Conformal Geodesics in dimension 3
Conformal geodesics form an invariantly defined family of unparametrized curves in a conformal manifold generalizing unparametrized geodesics/paths of projective connections. The equation describing them is of third order, and it was an open problem whether they are given by an Euler–Lagrange equation. In dimension 3 (the simplest, but most important from the viewpoint of physical applications) we demonstrate that the equation for unparametrized conformal geodesics is variational.
- Open Access
- 3 April 2025
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 13
Invariant divisors and equivariant line bundles
Scalar relative invariants play an important role in the theory of group actions on a manifold as their zero sets are invariant hypersurfaces. Relative invariants are central in many applications, where they often are treated locally since an invariant hypersurface may not be a locus of a single function. Our aim is to establish a global theory of relative invariants. For a Lie algebra ${\mathfrak g}$ of holomorphic vector fields on a complex manifold M, any holomorphic ${\mathfrak g}$ -invariant hypersurface is given in terms of a ${\mathfrak g}$ -invariant divisor. This generalizes the classical notion of scalar relative ${\mathfrak g}$ -invariant. Any ${\mathfrak g}$ -invariant divisor gives rise to a ${\mathfrak g}$ -equivariant line bundle, and a large part of this paper is therefore devoted to the investigation of the group $\mathrm {Pic}_{\mathfrak g}(M)$ of ${\mathfrak g}$ -equivariant line bundles. We give a cohomological description of $\mathrm {Pic}_{\mathfrak g}(M)$ in terms of a double complex interpolating the Chevalley-Eilenberg complex for ${\mathfrak g}$ with the Čech complex of the sheaf of holomorphic functions on M. We also obtain results about polynomial divisors on affine bundles and jet bundles. This has applications to the theory of differential invariants. Those were actively studied in relation to invariant differential equations, but the description of multipliers (or weights) of relative differential invariants was an open problem. We derive a characterization of them with our general theory. Examples, including projective geometry of curves and second-order ODEs, not only illustrate the developed machinery but also give another approach and rigorously justify some classical computations. At the end, we briefly discuss generalizations of this theory.- 3 April 2025
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 13
Open AccessInvariant divisors and equivariant line bundles
Scalar relative invariants play an important role in the theory of group actions on a manifold as their zero sets are invariant hypersurfaces. Relative invariants are central in many applications, where they often are treated locally since an invariant hypersurface may not be a locus of a single function. Our aim is to establish a global theory of relative invariants. For a Lie algebra ${\mathfrak g}$ of holomorphic vector fields on a complex manifold M, any holomorphic ${\mathfrak g}$ -invariant hypersurface is given in terms of a ${\mathfrak g}$ -invariant divisor. This generalizes the classical notion of scalar relative ${\mathfrak g}$ -invariant. Any ${\mathfrak g}$ -invariant divisor gives rise to a ${\mathfrak g}$ -equivariant line bundle, and a large part of this paper is therefore devoted to the investigation of the group $\mathrm {Pic}_{\mathfrak g}(M)$ of ${\mathfrak g}$ -equivariant line bundles. We give a cohomological description of $\mathrm {Pic}_{\mathfrak g}(M)$ in terms of a double complex interpolating the Chevalley-Eilenberg complex for ${\mathfrak g}$ with the Čech complex of the sheaf of holomorphic functions on M. We also obtain results about polynomial divisors on affine bundles and jet bundles. This has applications to the theory of differential invariants. Those were actively studied in relation to invariant differential equations, but the description of multipliers (or weights) of relative differential invariants was an open problem. We derive a characterization of them with our general theory. Examples, including projective geometry of curves and second-order ODEs, not only illustrate the developed machinery but also give another approach and rigorously justify some classical computations. At the end, we briefly discuss generalizations of this theory. - Open Access
Involutive scroll structures on solutions of 4D dispersionless integrable hierarchies
A rational normal scroll structure on an $(n+1)$-dimensional manifold $M$ is defined as a field of rational normal scrolls of degree $n-1$ in the projectivised cotangent bundle $\mathbb{P}T^*M$. We show that geometry of this kind naturally arises on solutions of various 4D dispersionless integrable hierarchies of heavenly type equations. In this context, rational normal scrolls coincide with the characteristic varieties (principal symbols) of the hierarchy. Furthermore, such structures automatically satisfy an additional property of involutivity. Our main result states that involutive scroll structures are themselves governed by a dispersionless integrable hierarchy, namely, the hierarchy of conformal self-duality equations.Open AccessInvolutive scroll structures on solutions of 4D dispersionless integrable hierarchies
A rational normal scroll structure on an $(n+1)$-dimensional manifold $M$ is defined as a field of rational normal scrolls of degree $n-1$ in the projectivised cotangent bundle $\mathbb{P}T^*M$. We show that geometry of this kind naturally arises on solutions of various 4D dispersionless integrable hierarchies of heavenly type equations. In this context, rational normal scrolls coincide with the characteristic varieties (principal symbols) of the hierarchy. Furthermore, such structures automatically satisfy an additional property of involutivity. Our main result states that involutive scroll structures are themselves governed by a dispersionless integrable hierarchy, namely, the hierarchy of conformal self-duality equations. - 1 March 2025
- Published by Pleiades Publishing Ltd in Russian Journal of Mathematical Physics
- Vol. 32 (1) , 97-104
On Fractional-Linear Integrals of Geodesics on Surfaces
In this note, we give a criterion for the existence of a fractional-linear integral for a geodesic flow on a Riemannian surface and explain that, modulo the Möbius transformations, the moduli space of such local integrals (if nonempty) is either the two-dimensional projective plane or a finite number of points. We also consider explicit examples and discuss a relation of such rational integrals to Killing vectors. DOI 10.1134/S1061920824601836
- 1 March 2025
- Published by Pleiades Publishing Ltd in Russian Journal of Mathematical Physics
- Vol. 32 (1) , 97-104
On Fractional-Linear Integrals of Geodesics on Surfaces
In this note, we give a criterion for the existence of a fractional-linear integral for a geodesic flow on a Riemannian surface and explain that, modulo the Möbius transformations, the moduli space of such local integrals (if nonempty) is either the two-dimensional projective plane or a finite number of points. We also consider explicit examples and discuss a relation of such rational integrals to Killing vectors. DOI 10.1134/S1061920824601836
- 8 January 2025
- Published by Walter de Gruyter GmbH in Journal für die reine und angewandte Mathematik (Crelles Journal)
On 3-nondegenerate CR manifolds in dimension 7 (I): The transitive case
We investigate 3-nondegenerate CR structures in the lowest possible dimension 7, and one of our goals is to prove Beloshapka’s conjecture on the symmetry dimension bound for hypersurfaces in . We claim that 8 is the maximal symmetry dimension of 3-nondegenerate CR structures in dimension 7, which is achieved on the homogeneous model. This part (I) is devoted to the homogeneous case: we prove that the model is locally the only homogeneous 3-nondegenerate CR structure in dimension 7.- 8 January 2025
- Published by Walter de Gruyter GmbH in Journal für die reine und angewandte Mathematik (Crelles Journal)
On 3-nondegenerate CR manifolds in dimension 7 (I): The transitive case
We investigate 3-nondegenerate CR structures in the lowest possible dimension 7, and one of our goals is to prove Beloshapka’s conjecture on the symmetry dimension bound for hypersurfaces in . We claim that 8 is the maximal symmetry dimension of 3-nondegenerate CR structures in dimension 7, which is achieved on the homogeneous model. This part (I) is devoted to the homogeneous case: we prove that the model is locally the only homogeneous 3-nondegenerate CR structure in dimension 7.