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The present paper deals with an intrinsic investigation of the notion of a parallel π-vector field on the pullback bundle of a Finsler manifold (M, L). The effect of the existence of a parallel π-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular β-change, namely the energy β-change ([Formula: see text]with[Formula: see text] being a parallel π-vector field), is established. The relation between the two Barthel connections Γ and [Formula: see text], corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy β-change of the fundamental linear connection in Finsler geometry: The Cartan connection, the Berwald connection, the Chern connection and the Hashiguchi connection. Moreover, the change of their curvature tensors is concluded. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.

In this paper, we present the modified Fermi–Walker parallel transport, the generalized Fermi–Walker parallel transport and related phases according to Frenet and Darboux frame in Minkowski 3-space.

We present a construction transforming a general connection Γ on a fibered manifold Y → M and a classical connection Λ on its base M into a classical connection on the total space Y by means of a vertical parallelism Φ and an auxiliary linear connection Δ. The relations to the theory of gauge-natural operators are discussed.

In the present study, we attend to the canal surfaces with the spine curve [Formula: see text] according to the parallel transport frame in Euclidean [Formula: see text]-space [Formula: see text]. We give an example of these surfaces and obtain some results about curvature conditions in [Formula: see text]. Moreover, the visualizations of projections of canal surfaces are presented. Lastly, we give the necessary and sufficient conditions for canal surfaces to become weak superconformal.

The symmetric product of vector fields on a manifold arises when one studies the controllability of certain classes of mechanical control systems. A novel geometric description of the symmetric product is provided using parallel transport, along the lines of the flow interpretation of the Lie bracket. This geometric interpretation of the symmetric product yields two different applications. First, an intrinsic proof is provided of the fact that the distributions closed under the symmetric product are exactly those distributions invariant under the geodesic flow. Second, some applications in geometric control theory for mechanical systems are clarified.

The aim of this paper is to study from the point of view of linear connections the data [Formula: see text] with M a smooth (n+p)-dimensional real manifold, [Formula: see text] an n-dimensional manifold semi-Riemannian distribution on M, [Formula: see text] the conformal structure generated by g and W a Weyl substructure: a map [Formula: see text] such that W(ḡ) = W(g) - du, ḡ = eug;u ∈ C∞(M). Compatible linear connections are introduced as a natural extension of similar notions from Weyl geometry and such a connection is unique if a symmetry condition is imposed. In the foliated case the local expression of this unique connection is obtained. The notion of Vranceanu connection is introduced for a pair (Weyl structure, distribution) and it is computed for the tangent bundle of Finsler spaces, particularly Riemannian, choosing as distribution the vertical bundle of tangent bundle projection and as one-form the Cartan form.

The present paper deals with an intrinsic generalization of the conformal change and energy β-change on a Finsler manifold (M.L.), namely the energy β-conformal change ([Formula: see text] with [Formula: see text]; [Formula: see text] being a concurrent π-vector field and σ(x) is a function on M). The relation between the two Barthel connections Γ and [Formula: see text], corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy β-conformal change of the fundamental linear connection in Finsler geometry: the Cartan connection, the Berwald connection, the Chern connection and the Hashiguchi connection. Moreover, the change of their curvature tensors is obtained. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.