In this article, we review the current status of Finsler–Lagrange geometry and generalizations. The goal is to aid non–experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kähler geometry and geometric mechanics how they could perform their results in order to be accepted by the community of ”orthodox ” physicists. Although the bulk of former models of Finsler–Lagrange spaces where elaborated on tangent bundles, the surprising result advocated in our works is that such locally anisotropic structures can be modelled equivalently on Riemann–Cartan spaces, even as exact solutions in Einstein and/or string gravity, if nonholonomic distributions and moving frames of references are introduced into consideration.

We propose a new framework for constructing geometric and physical models on nonholonomic manifolds provided both with Clifford – Lie algebroid symmetry and nonlinear connection structure. Explicit parametrizations of generic off–diagonal metrics and linear and nonlinear connections define different types of Finsler, Lagrange and/or Riemann–Cartan spaces. A generalization to spinor fields and Dirac operators on nonholonomic manifolds motivates the theory of Clifford algebroids defined as Clifford bundles, in general, enabled with nonintegrable distributions defining the nonlinear connection. In this work, we elaborate the algebroid spinor differential geometry and formulate the (scalar, Proca, graviton, spinor and gauge) field equations on Lie algebroids. The paper communicates new developments in geometrical formulation of physical theories and this approach is grounded on a number of previous examples when exact solutions with generic off– diagonal metrics and generalized symmetries in modern gravity define nonholonomic spacetime manifolds with uncompactified extra dimensions.

upper R and lower length λ scales (infrared/ultraviolet cutoff). The invariance symmetry leads naturally to the real Clifford algebra Cl(2, 6, R) and complexified Clifford ClC(4) algebra related to Twistors. A unified theory of all Noncommutative branes in Clifford-spaces is developed based on the Moyal-Yang star product deformation quantization whose deformation parameter involves the lower/upper scale (¯hλ/R). Previous work led us to show from first principles why the observed value of the vacuum energy density (cosmological constant) is given by a geometric mean relationship ρ ∼ L −2 P lanck R−2 = L −4 P (LP lanck/R) 2 ∼ 10 −122 M 4 P lanck, and can be obtained when the infrared scale R is set to be of the order of the present value of the Hubble radius. We proceed with an extensive review of Smith’s 8D model based on the Clifford algebra Cl(1, 7) that reproduces at low energies the physics of the Standard Model and Gravity, including the derivation of all the coupling constants, particle masses, mixing angles,....with high precision. Geometric actions are presented like the Clifford-Space extension of Maxwell’s Electrodynamics, and Brandt’s action related to the 8D spacetime tangentbundle involving coordinates and velocities ( Finsler geometries). Finally we outline the reasons why a Clifford-Space Geometric Unification of all forces is a very reasonable avenue to consider and propose an Einstein-Hilbert type action in Clifford-Phase spaces (associated with the 8D Phase space) as a Unified Field theory action candidate that should reproduce the physics of the Standard Model plus Gravity in the low energy limit.

We elaborate an unified geometric approach to classical mechanics, Riemann–Finsler spaces and gravity theories on Lie algebroids provided with nonlinear connection (N–connection) structure. There are investigated the conditions when the fundamental geometric objects like the anchor, metric and linear connection, almost sympletic and related almost complex structures may be canonically defined by a N–connection induced from a regular Lagrangian (or Hamiltonian), in mechanical models, or by generic off–diagonal metric terms and nonholonomic frames, in gravity theories. Such geometric constructions are modelled on nonholonomic manifolds provided with nonintegrable distributions and related chains of exact sequences of submanifolds defining N–connections. We investigate the main properties of the Lagrange, Hamilton, Finsler–Riemann and Einstein–Cartan algebroids and construct and analyze exact solutions describing such objects. Keywords: Lie algebroids, Lagrange, Hamilton and Riemann– Finsler spaces, nonlinear connection, nonholonomic manifold, geometric mechanics and gravity theories 2000 AMS Subject Classification:

The geometry of nonholonomic bundle gerbes provided with nonlinear connection structure and related nonholonomic gerbe modules is elaborated as the theory of Clifford modules on nonholonomic manifolds which positively fail to be spin. We explore an approach to such nonholonomic Dirac operators and derive the related Atiyah–Singer index formulas. There are considered certain applications in modern gravity and geometric mechanics of such Clifford–Lagrange / Finsler gerbes and their realizations as nonholonomic Clifford and Riemann– Cartan modules. Keywords: Nonholonomic gerbes, nonlinear connections, Riemann–Cartan

The geometry of nonholonomic bundle gerbes, provided with nonlinear connection structure, and nonholonomic gerbe modules is elaborated as the theory of Clifford modules on nonholonomic manifolds which positively fail to be spin. We explore an approach to such nonholonomic Dirac operators and derive the related Atiyah–Singer index formulas. There are considered certain applications in modern gravity and geometric mechanics of Clifford–Lagrange / Finsler gerbes and their realizations as nonholonomic Clifford and Riemann–Cartan modules. Keywords: Nonholonomic gerbes, nonlinear connections, Riemann–Cartan and Lagrange–Finsler spaces, nonholonomic spin structure,

In this paper we examine a new class of five dimensional (5D) exact solutions in extra dimension gravity possessing Lie algebroid symmetry. The constructions provide a motivation for the theory of Clifford nonholonomic algebroids elaborated in Ref. [1]. Such Einstein–Dirac spacetimes are parametrized by generic off–diagonal metrics and nonholonomic frames (vielbeins) with associated nonlinear connection structure. They describe self–consistent propagations of (3D) Dirac wave packets in 5D nonholonomically deformed Taub NUT spacetimes and have two physically distinct properties: Fist, the metrics are with polarizations of constants which may serve as indirect signals for the presence of higher dimensions and/or nontrivial torsions and nonholonomic gravitational configurations. Second, such Einstein–Dirac solutions are characterized by new type of symmetries defined as generalizations of the Lie algebra structure constants to nonholonomic Lie algebroid and/or Clifford algebroid structure functions.