Lagrange–Fedosov nonholonomic manifolds

Abstract

We outline an unified approach to geometrization of Lagrange mechanics, Finsler geometry and geometric methods of constructing exact solutions with generic off-diagonal terms and nonholonomic variables in gravity theories. Such geometries with induced almost symplectic structure are modelled on nonholonomic manifolds provided with nonintegrable distributions defining nonlinear connections. We introduce the concept of Lagrange-Fedosov spaces and Fedosov nonholonomic manifolds provided with almost symplectic connection adapted to the nonlinear connection structure. We investigate the main properties of generalized Fedosov nonholonomic manifolds and analyze exact solutions defining almost symplectic Einstein spaces.

Fedosov quantization of Lagrange–Finsler and Hamilton–Cartan spaces and Einstein gravity lifts on (co) tangent bundles

“…Let us fix any values e i i ′ (x, p) in (1) and associate g ab (x, p) to a * g ab (x, p) (3). This define correspondingly the values * g (20) and * N (15). As a result, we construct an effective Hamilton space, which can be modelled as a canonical almost symplectic structure as we described above.…”

“…The N-lifts of the fundamental tensor fields * g ab (3) and g ab (4) are respectively * g = * g αβ * e α ⊗ * e β = * g ij (x, p)e i ⊗ e j + * g ab (x, p) * e a ⊗ * e b , (20) on T * M, where * g ij is inverse to * g ab , and g = g αβ e α ⊗ e β = g ij (x, y)e i ⊗ e j + g ab (x, y)e a ⊗ e b , on T M, where g ij is stated by g ab following g ij = g n+i n+j .…”

“…where by {α} we label a multi-index and * Λ αβ * θ αβ − i * g αβ , where * θ αβ is the symplectic form (26), with "up" indices, and * g αβ is the inverse to d-tensor (20). This defines a formal Wick algebra * W u associated with the tangent space T u * K 2n , for u ∈ * K 2n , where the local coordinates on * K 2n are parameterized in the form u = {u α } and the local coordinates on T u * K 2n are labelled (u, z) = (u α , z β ), where z β are fiber coordinates.…”

“…[20], it was introduced the concept of Lagrange-Fedosov manifold as nonholonomic manifold with the N-connection and almost symplectic structure defined by a fundamental (in genera, effective) Lagrange function L(x, y). On cotangent bundles, we can consider Proof.…”

“…A rigorous geometric approach to deformation quantization of gravity, gauge theories and geometric mechanics models with constraints and related generalized Lagrange-Finsler theories, see [20,21,22,23,24], shows that the quantization schemes have to be developed for nonholonomic manifolds 1 and tangent and cotangent bundles endowed with nonlinear connection (Nconnection) structure. The natural step in this direction is to apply the methods of the geometry of Hamilton and Cartan spaces and generalizations [31,32] (such spaces are respectively dual to the Lagrange and Finsler spaces and generalizations [33,34]).…”