We generalize reduction theorems for classical connections to operators with values in k-th order natural bundles. Using the 2nd order valued reduction theorems we classify all (0,2)-tensor fields on the cotangent bundle of a manifold with a linear (non-symmetric) connection.

We prove the (first) reduction theorem for general and classical connections, i.e. we prove that any natural operator of a general connection Γ on a fibered manifold and a classical connection Λ on the base manifold can be expressed as a zero order operator of the curvature tensors of Γ and Λ and their appropriate derivatives.

The phase space of general relativistic test particle is defined as the 1-jet space of motions. A Lorentzian metric defines the canonical contact structure on the odd-dimensional phase space. In the paper we study infinitesimal symmetries of the gravitational contact phase structure which are not generated by spacetime infinitesimal symmetries, i.e. they are hidden symmetries. We prove that Killing multivector fields admit hidden symmetries of the gravitational contact phase structure and we give...

Starting by the famous paper by Kirillov, local Lie algebras of functions over smooth manifolds were studied very intensively by mathematicians and physicists. In the present paper we study local Lie algebras of pairs of functions which generate infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds.

We study Lie algebras of generators of infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds. The almost-cosymplectic-contact structure admits on the sheaf of pairs of 1-forms and functions the structure of a Lie algebra. We describe Lie subalgebras in this Lie algebra given by pairs generating infinitesimal symmetries of basic tensor fields given by the almost-cosymplectic-contact structure.

We study relations between functions on the cotangent bundle of a spacetime which are constants of motion for geodesics and functions on the odd-dimensional phase space conserved by the Reeb vector fields of geometrical structures generated by the metric and an electromagnetic field.

We study natural differential operators transforming two tensor fields into a tensor field. First, it is proved that all bilinear operators are of order one, and then we give the full classification of such operators in several concrete situations.

We describe conditions under which a spacetime connection and a scaled Lorentzian metric define natural symplectic and Poisson structures on the tangent bundle of the Einstein spacetime.

Let $M$ be a differentiable manifold with a pseudo-Riemannian metric $g$ and a linear symmetric connection $K$. We classify all natural (in the sense of [KMS]) 0-order vector fields and 2-vector fields on $TM$ generated by $g$ and $K$. We get that all natural vector fields are of the form $$E\left(u\right)=\alpha \left(h\left(u\right)\right)\phantom{\rule{0.166667em}{0ex}}{u}^{H}+\beta \left(h\left(u\right)\right)\phantom{\rule{0.166667em}{0ex}}{u}^{V}\phantom{\rule{0.166667em}{0ex}},$$
where ${u}^{V}$ is the vertical lift of $u\in {T}_{x}M$, ${u}^{H}$ is the horizontal lift of $u$ with respect to $K$, $h\left(u\right)=1/2g(u,u)$ and $\alpha ,\beta $ are smooth real functions defined on $R$. All natural 2-vector fields are of the form $$\Lambda \left(u\right)={\gamma}_{1}\left(h\left(u\right)\right)\phantom{\rule{0.166667em}{0ex}}\Lambda (g,K)+{\gamma}_{2}\left(h\left(u\right)\right)\phantom{\rule{0.166667em}{0ex}}{u}^{H}\wedge {u}^{V}\phantom{\rule{0.166667em}{0ex}},$$
where ${\gamma}_{1}$, ${\gamma}_{2}$ are smooth real functions defined...

This paper is a continuation of [2], dealing with a general, not-necessarily torsion-free, connection. It characterizes all possible systems of generators for vector-field valued operators that depend naturally on a set of vector fields and a linear connection, describes the size of the space of such operators and proves the existence of an ‘ideal’ basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi–Ricci identities without corrections.

All natural operations transforming linear connections on the tangent bundle of a fibred manifold to connections on the 1-jet bundle are classified. It is proved that such operators form a 2-parameter family (with real coefficients).

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