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Boris Kruglikov

Publications and source records attributed to Boris Kruglikov.

At least 19 recordsLinked to original sources
arXiv2026

On radicals in differential invariants: global theory vs moving frames

The classical technique of moving frames allows to effectively compute differential invariants of group actions. A standard observation is that the expressions of differential invariants contain radicals and more complicated algebraic functions, however the global Lie-Tresse theorem claims that for algebraic pseudogroup actions the algebra of differential invariants is generated by rational functions in jet variables. Here, we explain this apparent contradiction and illustrate it in a few well-elaborated examples. Constructively, our result gives a method to algorithmically compute global differential invariants.

math.DG↗
arXiv2026

On CR manifolds of CR dimension 1

We classify all maximal symmetry models of CR dimension 1, depending on their Bloom-Graham and Tanaka types, give coordinate realization to some of those models and prove a general extension principle.

math.DG↗
arXiv2026

Differential Invariants of Carrollian Spacetimes

We compute invariants of Carrollian spacetimes, deriving them from the geometry of the screen bundle. For generic Carrollian structures we specify how to generate the entire algebra of differential invariants, with emphasis on dimension 3, which has special physical relevance. Then, in the framework of jet-spaces, we compute the numerology behind these invariants: the Hilbert and Poincaré functions that govern their numbers according to order. Finally, we compute the Spencer cohomology behind the Carrollian geometry that, in particular, contains the spaces of intrinsic torsion and intrinsic curvature, which are fundamental invariants, important in the equivalence problem and symmetry analysis. Thus, we also discuss symmetry sizes of Carrollian spacetimes.

math.DG↗
arXiv2026

Scalar relative differential invariants

Computation of polynomial relative invariants is a classical tool in algebra. Relative differential invariants are central for the equivalence problem of geometric structures. We address the fundamental problem of finite generation of their (differential) algebra and demonstrate both positive and negative results in this respect under various setups. As in the algebraic case, the algebra of polynomial differential invariants is not finitely generated. However we show that after localization on a finite set of relative invariants the differential algebra becomes finitely generated. We also investigate the weights of rational relative differential invariants and bound their order. Several nontrivial examples are considered and further applications are discussed.

math.DG↗
arXiv2026

Prolongation rigidity of sub-free Lie algebras

We prove that if the 0-th Tanaka prolongation $\mathfrak{g}_0=\mathfrak{der}_0(\mathfrak{m})$ of a fundamental graded nilpotent Lie algebra $\mathfrak{m}=\mathfrak{g}_{-s}\oplus\dots\oplus\mathfrak{g}_{-1}$ is irreducible on $\mathfrak{g}_{-1}$, then $\mathfrak{m}$ is prolongation rigid: $\text{pr}_+(\mathfrak{m})=0$. The only exceptions are given by negative gradations of maximal parabolic subalgebras of a simple Lie algebra.

math.DG↗
arXiv2026

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.

math.DG↗
arXiv2025

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.

gr-qc↗
arXiv2025

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 $\mathbb{C}^4$.

math.CV↗
arXiv2025

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.

math.DG↗
arXiv2025

Geometric realizations of the Lie superalgebra D(2,1;a)

For every parabolic subgroup $P$ of a Lie supergroup $G$, the homogeneous superspace $G/P$ carries a $G$-invariant supergeometry. We address the question whether $\mathfrak{g}=\text{Lie}(G)$ is the maximal supersymmetry of this supergeometry in the case of the exceptional Lie superalgebra $D(2,1;a)$. For each choice of parabolic $\mathfrak{p}\subset\mathfrak{g}$, we consider the corresponding negatively graded Lie subalgebra $\mathfrak{m}\subset\mathfrak{g}$, and compute its Tanaka--Weisfeiler prolongations, with reduction of the structure group when required, thus realizing $D(2,1;a)$ via symmetries of supergeometries. This gives 6 inequivalent supergeometries: one of these is a vector superdistribution, two are given by cone fields of supervarieties, and the remaining three are higher order structure reductions (a novel feature). We describe those supergeometries and realize $D(2,1;a)$ supersymmetry explicitly in each case.

math.DG↗
arXiv2025

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.

nlin.SI↗
arXiv2025

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.

math.DG↗
arXiv2024

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 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 will also consider explicit examples and discuss a relation of such rational integrals to Killing vectors.

math.DG↗
arXiv2024

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 $\mathbb{C}^4$. 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.

math.CV↗
arXiv2024

Zero-curvature subconformal structures and dispersionless integrability in dimension five

We extend the recent paradigm "Integrability via Geometry" from dimensions 3 and 4 to higher dimensions, relating dispersionless integrability of partial differential equations to curvature constraints of the background geometry. We observe that in higher dimensions on any solution manifold the symbol defines a vector distribution equipped with a subconformal structure, and the integrability imposes a certain compatibility between them. In dimension 5 we express dispersionless integrability via the vanishing of a certain curvature of this subconformal structure. We also obtain a "master equation" governing all second order dispersionless integrable equations in 5D, and count their functional dimension. It turns out that the obtained background geometry is parabolic of the type $(A_3,P_{13})$. We provide its Cartan theoretic description and compute the harmonic curvature components via the Kostant theorem. Then we relate it to 3D projective and 4D conformal geometries via twistor theory, discuss symmetry reductions and nested Lax sequences, as well as give another interpretation of dispersionless integrability in 5D through Levi-degenerate CR structures~in~7D.

math.DG↗
arXiv2023

ODEs whose symmetry groups are not fiber-preserving

We observe that, up to conjugation, a majority of symmetric higher order ODEs (ordinary differential equations) and ODE systems have only fiber-preserving point symmetries. By exploiting Lie's classification of Lie algebras of vector fields, we describe all the exceptions to this in the case of scalar ODEs and systems of ODEs on a pair of functions. The scalar ODEs whose symmetry algebra is not fiber preserving can be expressed via absolute and relative scalar differential invariants, while a similar description for ODE systems requires us to also invoke conditional differential invariants and vector-valued relative invariants to deal with singular orbits of the action. Investigating prolongations of the actions, we observe some interesting relations between different realizations of Lie algebras. We also note that it may happen that the prolongation of a finite-dimensional Lie algebra acting on a differential equation never becomes free. An example of an underdetermined ODE system for which this phenomenon occurs shows limitations of the method of moving frames.

math.DG↗
arXiv2022

Realization of Lie superalgebras G(3) and F(4) as symmetries of supergeometries

For every parabolic subgroup $P$ of a Lie supergroup $G$ the homogeneous superspace $G/P$ carries a $G$-invariant supergeometry. We address the quesiton whether $\mathfrak{g}=\operatorname{Lie}(G)$ is the maximal symmetry of this supergeometry in the case of exceptional Lie superalgebras $G(3)$ and $F(4)$. Our approach is to consider the negatively graded Lie superalgebras for every choice of parabolic, and to compute the Tanaka-Weisfeiler prolongations, with reduction of the structure group when required (2 resp 3 cases), thus realizing $G(3)$ and $F(4)$ as symmetries of supergeometries. This gives 19 inequivalent $G(3)$-supergeometries and 55 inequivalent $F(4)$-supergeometries, in majority of cases (17 resp 52 cases) those being encoded as vector superdistributions. We describe those supergeometries and realize supersymmetry explicitly in some cases.

math.DG↗