arXiv subjects

Eivind Schneider

Publications and source records attributed to Eivind Schneider.

15 recordsLinked to original sources

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

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

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

Recurrent Lorentzian Weyl spaces

We find the local form of all non-closed Lorentzian Weyl manifolds $(M,c,\nabla)$ with recurrent curvature tensor.If the dimension of the manifold is greater than 3, then the conformal structure is flat, and the recurrent Weyl structure is locally determined by a single function. Two local structures are equivalent if and only if the corresponding functions are related by a transformation from $\mathrm{SAff}_1(\mathbb{R}) \times \mathrm{PSL}_2(\mathbb{R}) \times \mathbb{Z}_2$. We find generators for the field of rational scalar differential invariants of this Lie group action. The global structure of the manifold $M$ may be described in terms of a foliation with a transversal projective structure. It is shown that all locally homogeneous structures are locally equivalent, and there is only one simply connected homogeneous non-closed recurrent Lorentzian Weyl manifold. Moreover, there are 5 classes of cohomogeneity-one spaces, and all other spaces are of cohomogeneity-two. If $\dim M=3$, the non-closed recurrent Lorentzian Weyl structures are locally determined by one function of two variables or two functions of one variables, depending on whether its holonomy algebra is 1- or 2-dimensional. In this case, two structures with the same holonomy algebra are locally equivalent if and only if they are related, respectively, by a transformation from an infinite-dimensional Lie pseudogroup or a 4-dimensional subgroup of $\mathrm{Aff}(\mathbb R^3)$. Again we provide generators for the field of rational differential invariants. We find a local expression for the locally homogeneous non-closed recurrent Lorentzian Weyl manifolds of dimension 3, and also of those of cohomogeneity one and two. In the end we give a local description of the non-closed recurrent Lorentzian Weyl manifolds that are also Einstein-Weyl. All of them are 3-dimensional and have a 2-dimensional holonomy algebra.

math.DG↗

arXiv2024

Detecting horizons of symmetric black holes using relative differential invariants

Let $\mathfrak{k}$ be a nontrivial finite-dimensional Lie algebra of vector fields on a manifold M, and consider the family of Lorentzian metrics on M whose Killing algebra contains $\mathfrak{k}$. We show that scalar relative differential invariants, with respect to a Lie algebra of vector fields on M preserving $\mathfrak{k}$, can be used to detect the horizons of several well-known black holes. In particular, using the Lie algebra structure of $\mathfrak{k}$, we construct a general relative differential invariant of order 0 that always vanishes on $\mathfrak{k}$-invariant Killing horizons.

gr-qc↗

arXiv2023

Decomposable $(5,6)$-solutions in eleven-dimensional supergravity

We present decomposable (5,6)-solutions $\widetilde{M}^{1,4} \times M^6$ in eleven-dimensional supergravity by solving the bosonic supergravity equations for a variety of non-trivial flux forms. Many of the bosonic backgrounds presented here are induced by various types of null flux forms on products of certain totally Ricci-isotropic Lorentzian Walker manifolds and Ricci-flat Riemannian manifolds. These constructions provide an analogue of work done by I. Chrysikos and A. Galaev who made similar computations for decomposable (6,5)-solutions. We also present bosonic backgrounds that are products of Lorentzian Einstein manifolds with negative Einstein constant (in the "mostly plus" convention) and Riemannian Kähler-Einstein manifolds with positive Einstein constant. This conclusion generalizes a result of C. N. Pope and P. van Nieuwenhuizen concerning the appearance of six-dimensional Kähler-Einstein manifolds in eleven-dimensional supergravity. In this setting we construct infinitely many non-symmetric decomposable (5, 6)-supergravity backgrounds by using the infinitely many Lorentzian Einstein-Sasakian structures with negative Einstein constant on the 5-sphere, known from the work of C. P. Boyer et al.

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↗

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

Differential invariants of measurements, and their connection to central moments

Due to the principle of minimal information gain, the measurement of points in an affine space $V$ determines a Legendrian submanifold of $V \times V^* \times \mathbb R$. Such Legendrian submanifolds are equipped with additional geometric structures that come from the central moments of the underlying probability distributions and are invariant under the action of the group of affine transformations on $V$. We investigate the action of this group of affine transformations on Legendrian submanifolds of $V \times V^* \times \mathbb R$ by giving a detailed overview of the structure of the algebra of scalar differential invariants. We show how the central moments can be used to construct the scalar differential invariants. In the end, we view the results in the context of equilibrium thermodynamics of gases, where we notice that the heat capacity is one of the differential invariants.

math-ph↗

arXiv2020

Solutions of second-order PDEs with first-order quotients

We describe a way of solving a partial differential equation using the differential invariants of its point symmetries. By first solving its quotient PDE, which is given by the differential syzygies in the algebra of differential invariants, we obtain new differential constraints which are compatible with the PDE under consideration. Adding these constraints to our system makes it overdetermined, and thus easier to solve. We focus on second-order scalar PDEs whose quotients are first-order scalar PDEs. This situation occurs only when the Lie algebra of symmetries of the second-order PDE is infinite-dimensional. We apply this idea to several different PDEs, one of which is the Hunter-Saxton equation.

math.DG↗

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↗

arXiv2018

Projectable Lie algebras of vector fields in 3D

Starting with Lie's classification of finite-dimensional transitive Lie algebras of vector fields on $\mathbb C^2$ we construct Lie algebras of vector fields on the bundle $\mathbb C^2 \times \mathbb C$ by lifting the Lie algebras from the base. There are essentially three types of transitive lifts and we compute all of them for the Lie algebras from Lie's classification. The simplest type of lift is encoded by Lie algebra cohomology.

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↗

arXiv2016

Differential Invariants of Self-Dual conformal structures

We compute the quotient of the self-duality equation for conformal metrics by the action of the diffeomorphism group. We also determine Hilbert polynomial, counting the number of independent scalar differential invariants depending on the jet-order, and the corresponding Poincaré function. We describe the field of rational differential invariants separating generic orbits of the diffeomorphism pseudogroup action, resolving the local recognition problem for self-dual conformal structures.

math.DG↗