In this paper, we analyse the question of existence of a natural and projectively equivariant symbol calculus, using the theory of projective Cartan connections. We establish a close relationship between the existence of such a natural symbol calculus and the existence of an sl(m+1,R)equivariant calculus over R in the sense of [15, 1]. Moreover we show that the formulae that hold in the non-cri...

We give a systematic treatment of the treatment of the classical Hannay-Berry phases for mechanical systems in terms of the holonomy of naturally constructed connections on bundles associated to th~ system. We make the costructions using symmetry and reduction and, for moving systems, we use the Cartan connection. These ideas are woven with the idea of Montgomery [1988] on the averaging of conn...

We develop the method of anholonomic frames with associated nonlinear connec-tion (in brief, N–connection) structure and show explicitly how geometries with lo-cal anisotropy (various type of Finsler–Lagrange–Cartan–Hamilton geometry) can bemodeled in the metric–affine spaces. There are formulated the criteria when such gen-eralized Finsler metrics are effectively induced in the...

The present paper deals with an intrinsic investigation of the notion of a concurrent π-vector field on the pullback bundle of a Finsler manifold (M,L). The effect of the existence of a concurrent π-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular β-change, namely the energy β-change (L̃(x, y) = L(x, y) + B(x, y)with B := g(ζ, η); ζ bei...

Randers manifolds are studied in the framework of the pullback bundle formalism, with the aid of intrinsic methods only. After checking a sufficient condition for a Randers manifold to be a Finsler manifold, we provide a systematic description of the Riemann-Finsler metric, the canonical spray, the Barthel endomorphism, the Berwald connection, the Cartan tensors and the Cartan vector field in t...

The present paper deals with an intrinsic investigation of the notion of a concurrent π-vector field on the pullback bundle of a Finsler manifold (M,L). The effect of the existence of a concurrent π-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular β-change, namely the energy β-change (L̃(x, y) = L(x, y) + B(x, y)with B := g(ζ, η); ζ bei...

Abstract We continue our study of the mixed Einstein–Hilbert action as a functional pseudo-Riemannian metric and linear connection. Its geometrical part is total scalar curvature on smooth manifold endowed with distribution or foliation. develop variational formulas for quantities extrinsic geometry metric-affine space use them to derive Euler–Lagrange equations (which in case space-time are an...

We compute the characteristic Cartan connection associated with a system of third order ODEs. Our connection is different from Tanaka normal one, but still is uniquely associated with the system of third order ODEs. This allows us to find all fundamental invariants of a system of third order ODEs and, in particular, determine when a system of third order ODEs is trivializable. As application di...

In 1977, M. Matsumoto and R. Miron [9] constructed an orthonormal frame for an n-dimensional Finsler space, called ‘Miron frame’. The present authors [1, 2, 3, 10, 11] discussed four-dimensional Finsler spaces equipped with such frame. M. Matsumoto [7, 8] proved that in a three-dimensional Berwald space, all the main scalars are h-covariant constants and the h-connection vector vanishes. He als...

We investigate the Cartan formalism in F(R) gravity. gravity has been introduced as a theory to explain cosmologically accelerated expansions by replacing Ricci scalar R Einstein–Hilbert action with function of R. As is well-known, rewritten scalar–tensor using conformal transformation. described based on Riemann–Cartan geometry formulated vierbein-associated local Lorenz symmetry. In formalism...