Covariant Tensor Field Research Articles

Riemannian exponential and quantization

This article continues and completes the previous one [18]. First of all, we present two methods of quantization associated with a linear connection given on a differentiable manifold, one of them being the one presented in [18]. The two methods allow quantization of functions that come from covariant tensor fields. The equivalence of both is demonstrated as a consequence of a remarkable property of the Riemannian exponential (Theorem 5.1) that, as far as we know, is new to the literature. In addition, we provide a characterization of the Schrödinger operators as the only ones that by quantization correspond to classical mechanical systems. Finally, it is shown that the extension of the above quantization to functions of a very broad type can be carried out by generalizing the method of [18] in terms of fields of distributions.

Tensor fields associated with integrable systems of chiral

This article describes necessary conditions for chiral-type systems to admit Lax representation with values in simple compact Lie algebras. These conditions state that there exists a covariant constant tensor field with an additional property. It is proposed to construct in an invariant way some covariant tensor fields using the Lax representation of the system under consideration. These fields are constructed by taking linear differential forms with values in the Lie algebra that are constructed using the Lax representation of the system and substituting them into an arbitrary Ad-invariant form on the Lie algebra. The paper proves that such tensor fields are Killing tensor fields or covariant constant fields. The discovered necessary conditions for the existence of the Lax representation are obtained using a special case of such tensor fields associated with the Killing metric of the Lie algebra.

Some Results on Exponential Synchronization of Nonlinear Systems

Based on recent works on transverse exponential stability, we establish some necessary and sufficient conditions for the existence of a (locally) exponential synchronizing control law. We show that the existence of a structured synchronizer is equivalent to the existence of a stabilizer for the individual linearized systems (on the synchronization manifold) by a linear state feedback. This, in turn, is also equivalent to the existence of a symmetric covariant tensor field, which satisfies a control matrix function inequality. Based on this result, we provide the construction of such a synchronizer via backstepping approach. In some particular cases, we show how global exponential synchronization may be obtained.

On singular semi-Riemannian manifolds

On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related operations like the contraction between covariant indices. In this paper, we develop the geometry of singular semi-Riemannian manifolds. First, we introduce an invariant and canonical contraction between covariant indices, applicable even for degenerate metrics. This contraction applies to a special type of tensor fields, which are radical-annihilator in the contracted indices. Then, we use this contraction and the Koszul form to define the covariant derivative for radical-annihilator indices of covariant tensor fields, on a class of singular semi-Riemannian manifolds named radical-stationary. We use this covariant derivative to construct the Riemann curvature, and show that on a class of singular semi-Riemannian manifolds, named semi-regular, the Riemann curvature is smooth. We apply these results to construct a version of Einstein's tensor whose density of weight 2 remains smooth even in the presence of semi-regular singularities. We can thus write a densitized version of Einstein's equation, which is smooth, and which is equivalent to the standard Einstein equation if the metric is non-degenerate.

Spectra and symmetric eigentensors of the Lichnerowicz Laplacian on [formula omitted

We compute the eigenvalues with multiplicities of the Lichnerowicz Laplacian acting on the space of complex symmetric covariant tensor fields on the complex projective space P n ( C ) . The spaces of symmetric eigentensors are explicitly given.

The Lichnerowicz Laplacian on compact symmetric spaces

We show that, on a compact symmetric space, the Lichnerowicz Laplacian acting on the space of covariant tensor fields coincides with the Casimir operator and we deduce that, on a compact semisimple Lie group, the Lichnerowicz Laplacian is the mean of the left invariant Casimir operator and the right invariant Casimir operator. To cite this article: M. Boucetta, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Spectra and symmetric eigentensors of the Lichnerowicz Laplacian on $S^{n}$

We compute the eigenvalues with multiplicities of the Lichnerowicz Laplacian acting on the space of symmetric covariant tensor fields on the Euclidian sphere $S^{n}$. The spaces of symmetric eigentensors are explicitly given.

On the characterization of the kernel of the geodesic X-ray transform

Let Ω ¯ \overline {\Omega } be a compact manifold with boundary. We consider covariant symmetric tensor fields of order two that belong to a Sobolev space H k ( Ω ¯ ) , k ≥ 1 H^{k}(\overline {\Omega }), k \geq 1 . We prove, under the assumption that the metric is simple, that solenoidal tensor fields that belong to the kernel of the geodesic X-ray transform are smooth up to the boundary. As a corollary we obtain that they form a finite-dimensional set in H k H^{k} .

A GENERAL COVARIANT SYMPLECTIC STRUCTURE FOR GRAVITATIONAL, ELECTROMAGNETIC AND DIRAC FIELDS

We derive a general covariant symplectic structure which leads to holonomic and anholonomic formulations of Hamilton equations of motion related to the hydrodynamic picture of mechanics. Such a structure can be achieved starting from generic bilinear Hamiltonians, constructed by covariant vector, bivector or tensor fields. This feature is gage-free and it realizes a deep link toward canonical quantization for interactions like electromagnetism, gravity and Dirac fields.

Inversion of the x-ray transform for 3D symmetric tensor fields with sources on a curve

The x-ray transform I associates with a covariant symmetric tensor field f(x) of order , the line function If(l) by integration over the line l. In this paper, new reconstruction formulae for the solenoidal part of f are obtained. The case when If is known only for the family of lines intersecting a given curve in is considered as well. The formulae obtained contain only a finite number of differentiations, integrations, solving of linear algebraic equations and application the Laplacian powers. The formulae are obtained under the assumption that every plane intersecting the support of f intersects the curve m + 1 times. It is shown that the number of intersection points cannot be decreased. It is shown that the similar considerations are valid for two other admissible families of lines in : lines parallel to a given curve of directions and lines tangential to a given surface.

Poisson structures on tangent bundles

The paper starts with an interpretation of the complete lift of a Poisson structure from a manifold M to its tangent bundle TM by means of the Schouten–Nijenhuis bracket of covariant symmetric tensor fields defined by the cotangent Lie algebroid of M. Then, we discuss Poisson structures of TM which have a graded restriction to the fiberwise polynomial algebra; they must be π-related ( π :TM→M ) with a Poisson structure on M. Furthermore, we define transversal Poisson structures of a foliation, and discuss bivector fields of TM which produce graded brackets on the fiberwise polynomial algebra, and are transversal Poisson structures of the foliation by fibers. Finally, such bivector fields are produced by a process of horizontal lifting of Poisson structures from M to TM via connections.

On generalized hamiltonian systems

The purpose of this paper is to explore an extension of some fundamental properties of the Hamiltonian systems to a more general case. We first extend symplectic group to a generalN-group,GN, and prove that it has certain similar properties. A particular property ofGN is that as a Lie group dim (GN)≥1. Certain properties of its Lie-algebragN are investigated. The results obtained are used to investigate the structure-preserving systems, which generalize the property of symplectic form preserving of Hamiltonian system to a covariant tensor field preserving of certain dynamic systems. The results provide a theoretical benchmark of applying symplectic algorithm to a considerably larger class of structure-preserving systems.

Sub-Riemannian Metrics and Isoperimetric Problems in the Contact Case

In the opposite case, where n is even, (dω) n∆ defines an orientation on ∆, and the orientation on X is given by any Z transversal to ∆; we chose ω(Z) > 0. First, we will discuss normal forms and invariants for such a structure. We obtain a completely reduced normal form for any n. These results will be stated and proved in Sec. 2 of the paper. In the three-dimensional case, a certain number of covariant symmetric tensor fields over ∆ appear; they are invariants of the sub-Riemannian structure. Two of them, denoted by Q2 and V3, are very important, and they have covariance degree 2 and 3, respectively. They are irreducible under the action of SO(2). Generically, Q2 is nonzero outside a smooth curve C ,a ndV3 is nonzero on this curve. They are respectively called the principal and second invariants of the structure. They reflect the most important local properties of the sub-Riemannian metric. 2. Isoperimetric problems. We also consider general isoperimetric problems on a two-dimensional oriented Riemannian manifold (M, g) of the followingtype: we are g iven a two-form η = ψ(Volume), two points q0 ,q 1 ∈ M are fixed, together with a curve ˜

Spaces with contravariant and covariant affine connections and metrics

The theory of spaces with contravariant and covariant affine connections, whose components differ not only in sign, and metrics [(L n ,g) spaces] is worked out within the framework of tensoranalysis over differentiable manifolds and in a volume necessary for further consideration of the kinematics of vector fields and the Lagrangiantheory of tensor fields over (L n ,g) spaces. The possibility of introducing affine connections for contravariant and covariant tensor fields, whose components differ not only in sign, over differentiable manifolds with finite dimensions is discussed. The action of the deviation operator, having an important role for deviation equations in gravitational physics, is considered for the case of contravariant and covariant vector fields over differentiable manifolds with different affine connections (called L n spaces). A deviation identity for contravariant vector fields is obtained. The notions of covariant, contravariant, covariant projective, and contravariant projective metric are introduced in (L n ,g) spaces. The action of the covariant and the Lie differential operator on the different types of metric is found. The concepts of symmetric covariant and contravariant (Riemannian) connection are defined and presented by means of the covariant and contravariant metric and the corresponding torsion tensors. The different types of relative tensor fields (tensor densities) as well as the invariant differential operators acting on them are considered. The invariant volume element and its properties under the action of different differential operators are investigated.

Projective geometry on the bundle of volume forms

The bundle of volume forms on a manifold is examined in terms of the affine connection defined by T. Y. Thomas. The choice of a particular affine connection in the projective class corresponds to the choice of an horizontal distribution on this bundle. The geometric properties of the horizontal distributions are studied. Special lifts of vector fields and covariant tensor fields are examined as well as lifts of metric connections.

Invariance of the Massless Field Equations under Changes of the Metric

It is shown that the Maxwell equations with sources, expressed in terms of the covariant tensor field Fijand the current density four-vector Ji, are invariant under the change of the metric gijby gij′= gij+ liljif liis a principal null direction of Fijand that an analogous result holds in the case of the massless Klein-Gordon equation if liis null and orthogonal to the gradient of the field and in the case of the null dust equations if liis parallel to the dust four-velocity. An elementary proof of the following generalization of the Xanthopoulos theorem is also given: Let (gij, Fij) be an exact solution of the Einstein-Maxwell equations and let libe a principal null direction of Fij, then (gij+ lilj, Fij) is also an exact solution of the Einstein-Maxwell equations if and only if (lilj, 0) satisfies the Einstein-Maxwell equations linearized about the background solution (gij, Fij). Furthermore, analogous theorems, where the source of the gravitational field is a massless Klein-Gordon field or null dust, are presented.

Unified treatment of fermions and bosons

Dirac's matrices can be interpreted as an 8-rank covariant antisymmetric tensor field on an 11-dimensional manifold (space-time ×S7) enforcing a linkage between the Lorentz transformation and “rotations” ofS7, conferring spinorial properties on any quantity having an index in the inner spaceS7.

Some invariant theorems on geometry of Einstein non‐symmetric field theory

This paper generalizes Einstein′s theorem. It is shown that under the transformation urn:x-wiley:01611712:media:ijmm140940:ijmm140940-math-0001 , curvature tensor , Ricci tensor Sik(U), and scalar curvature S(U) are all invariant, where Λ = Λjdxj is a closed 1‐differential form on an n‐dimensional manifold M.It is still shown that for arbitrary U, the transformation that makes curvature tensor (or Ricci tensor Sik(U)) invariant urn:x-wiley:01611712:media:ijmm140940:ijmm140940-math-0004 must be TΛ transformation, where V (its components are ) is a second order differentiable covariant tensor field with vector value.

Retrodictive determinism

With respect to irreversible, non-homeomorphic maps, contravariant and covariant tensor fields have distinctly different natural covariance and transformational behavior. For thermodynamic processes which are non-adiabatie, the fact that the process cannot be represented by a homeomorphic map emphasizes the logical arrow of time, an idea which encompasses a principle of retrodictive determinism for covariant tensor fields.

Natural operations on covariant tensor fields

Natural operations on covariant tensor fields