Scalar Differential Invariants Research Articles

Differential Invariants of Measurements, and Their Relation 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 . 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 by giving a detailed overview of the structure of the algebra of scalar differential invariants, and we show how the scalar differential invariants can be constructed from the central moments. In the end, we view the results in the context of equilibrium thermodynamics of gases, and notice that the heat capacity is one of the differential invariants.

Differential Invariants of One Parametrical Group of Transformations

The concept of differential invariant, along with the concept of invariant differentiation, is the key in modern geometry [1]-[10]. In the Erlangen program [3] Felix Klein proposed a unified approach to the description of various geometries. According to this program, one of the main problems of geometry is to construct invariants of geometric objects with respect to the action of the group defining this geometry. This approach is largely based on the ideas of Sophus Lee, who introduced continuous geometry groups of transformations, now known as Lie groups, into geometry. In particular, when considering classification problems and equivalence problems in differential geometry, differential invariants with respect to the action of Lie groups should be considered. In this case, the equivalence problem of geometric objects is reduced to finding a complete system of scalar differential invariants. The interpretation of the k- order differential invariant as a function on the space of k- jets of sections of the corresponding bundle made it possible to operate with them efficiently, and using invariant differentiation, new differential invariants can be obtained. Differential invariants with respect to a certain Lie group generate differential equations for which this group is a symmetry group. This allows one to apply the well-known integration methods to such equations, and, in particular, the Li- Bianchi theorem [4]. Depending on the type of geometry, the orders of the first nontrivial differential invariants can be different. For example, in the space R3 equipped with the Euclidean metric, the complete system of differential invariants of a curve is its curvature and torsion, which are second and third order invariants, respectively. Note that scalar differential invariants are the only type of invariants whose components do not change when changing coordinates. For this reason, scalar differential invariants are effectively used in solving equivalence problems. In this paper differential invariants of Lie group of one parametric transformations of the space of two independent and three dependent variables are studied. It is shown method of construction of invariant differential operator. Obtained results applied for finding differential invariants of surfaces.

The equivalence problem for generic four-dimensional metrics with two commuting Killing vectors

We consider the equivalence problem of four-dimensional semi-Riemannian metrics with the $2$-dimensional Abelian Killing algebra. In the generic case we determine a semi-invariant frame and a fundamental set of first-order scalar differential invariants suitable for solution of the equivalence problem. Genericity means that the Killing leaves are not null, the metric is not orthogonally transitive (i.e., the distribution orthogonal to the Killing leaves is non-integrable), and two explicitly constructed scalar invariants $C_\rho$ and $\ell_{\mathcal C}$ are nonzero. All the invariants are designed to have tractable coordinate expressions. Assuming the existence of two functionally independent invariants, we solve the equivalence problem in two ways. As an example, we invariantly characterise the Van den Bergh metric. To understand the non-generic cases, we also find all $\Lambda$-vacuum metrics that are generic in the above sense, except that either $C_\rho$ or $\ell_{\mathcal C}$ is zero. In this way we extend the Kundu class to $\Lambda$-vacuum metrics. The results of the paper can be exploited for invariant characterisation of classes of metrics and for extension of the set of known solutions of the Einstein equations.

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.

On moduli spaces for finite-order jets of linear connections

We describe the ringed-space structure of moduli spaces of jets of linear connections (at a point) as orbit spaces of certain linear representations of the general linear group. Then, we use this fact to prove that the only (scalar) differential invariants associated to linear connections are constant functions, as well as to recover various expressions appearing in the literature regarding the dimensions of generic strata of these moduli spaces.

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 Poincar\'e 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.

Scalar differential invariants of symplectic Monge-Ampère equations

Abstract All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. We also introduce a series of invariant differential forms and vector fields which allow us to construct numerous scalar differential invariants of higher order. The introduced invariants give a solution of the symplectic equivalence of Monge-Ampère equations. As an example we study equations of the form u xy + f(x, y, u x, u y) = 0 and in particular find a simple linearization criterion.

Moduli spaces for jets of Riemannian metrics at a point

We construct the moduli space of r-jets of Riemannian metrics at a point on a smooth manifold. The construction is closely related to the problem of classification of metric jets via scalar differential invariants. The moduli space is proved to be a differentiable space which admits a finite canonical stratification into smooth manifolds. A complete study on the stratification of moduli spaces is carried out for metrics in dimension n = 2 .

ℝ-Conformal invariants of curves

In this paper we describe the structure of the algebra of scalar differential invariants of curves in a plane with Euclidean or Minkowski metric with respect to ℝ-conformal transformations.

Differential Invariants of the 2D Conformal Lie Algebra Action

We consider the Lie algebra that corresponds to the Lie pseudogroup of all conformal transformations on the plane. This conformal Lie algebra is canonically represented as the Lie algebra of holomorphic vector fields in ℝ2≃ℂ. We describe all representations of \(\mathfrak{g}\) via vector fields in J0ℝ2=ℝ3(x,y,u), which project to the canonical representation, and find their algebra of scalar differential invariants.

Differential Invariants of Second Order ODEs, I

This paper is devoted to differential invariants of equations $$y''=a^{3}(x,y)y^{\prime3}+a^{2}(x,y)y^{\prime2}+a^{1}(x,y)y'+a^{0}(x,y).$$ w.r.t. point transformations. The natural bundle of these equations and its bundles of k-jets of sections, k=0,1,2,…, are considered. The action of the pseudogroup of all point transformations on these bundles is investigated. Tensor differential invariants distinguishing orbits of this action on jet bundles of second and third orders are constructed. A complete collection of generators and their differential syzygies is obtained for the algebra of all scalar differential invariants of a wide class of considered equations containing generic equations.

Scalar differential invariants and characteristic classes of homogeneous geometrical structures

Scalar differential invariants and characteristic classes of homogeneous geometrical structures

A Gravitational Field with a Curious Geometrical Property

IT is well known1 that there are fourteen independent absolute scalar differential invariants of the second order associated with the gravitational metric, It can be argued that the vanishing of all the invariants need not imply the vanishing of all the twenty independent components of the curvature tensor Rhijk. On the other hand, it may be pointed out that, when the invariants vanish, there are fourteen differential equations to be satisfied by the ten gµv components ; and it is not at all obvious that a gravitational field with a Riemannian metric exists for which the fourteen invariants vanish. We report here the existence of such a gravitational field described by the metric for which the surviving components of the energy-momentum tensor satisfy the relations, each being equal to The above metric with the conditions (3) may be compared to the gravitational field corresponding to a directed flow of radiation as given by Tolman2. For (2), the conformal curvature tensor vanishes and Tµv has the structure of the electromagnetic energy-momentum tensor ; these circumstances together being responsible for the vanishing of the complete set of invariants.