Nonlinear Connections and Spinor Geometry

We present an introduction to the geometry of higher order vector and co-vector bundles (including higher order generalizations of the Finsler geometry and Kaluza--Klein gravity) and review the basic results on Clifford and spinor structures on spaces with generic local anisotropy modeled by anholonomic frames with associated nonlinear connection structures. We emphasize strong arguments for application of Finsler like geometries in modern string and gravity theory and noncommutative geometry and noncommutative field theory and gravity.


Introduction
Nowadays, it has been established an interest to non-Riemannian geometries derived in the low energy string theory [9], noncommutative geometry [6,50] and quantum groups [18]. Various types of Finsler like structures can be parametrized by generic off-diagonal metrics, which can not be diagonalized by coordinate transforms but only by anholonomic maps with associated nonlinear connection (in brief, N-connection). Such structures may be defined as exact solutions of gravitational field equations in the Einstein gravity and its generalizations [46,50,47], for instance, in the metric-affine [13] Riemann-Cartan gravity [24]. Finsler like configurations are considered in locally anisotropic thermodynamics and kinetics and related stochastic processes [45] and (super) string theory [36,42,43].
The following natural step in these lines is to elucidate the theory of spinors in effectively derived Finsler geometries and to relate this formalism of Clifford structures to noncommutative Finsler geometry. It should be noted that the rigorous definition of spinors for Finsler spaces and generalizations was not a trivial task because (on such spaces) there are not defined even local groups of authomorphisms. The problem was solved in Refs. [38,39,44] by adapting the geometric constructions with respect to anholonomic frames with associated N-connection structure. The aim of this work is to outline the geometry of generalized Finsler spinors in a form more oriented to applications in modern mathematical physics.
We start with some historical remarks: The spinors studied by mathematicians and physicists are connected with the general theory of Clifford spaces introduced in 1876 [7]. The theory of spinors and Clifford algebras play a major role in contemporary physics and mathematics. The spinors were discovered byÈlie Cartan in 1913 in mathematical form in his researches on representation group theory [5]; he showed that spinors furnish a linear representation of the groups of rotations of a space of arbitrary dimensions. Physicists Pauli [26] and Dirac [10] (in 1927, respectively, for the three-dimensional and four-dimensional space-time) introduced spinors for the representation of the wave functions. In general relativity theory spinors and the Dirac equations on (pseudo) Riemannian spaces were defined in 1929 by H. Weyl [51], V. Fock [11] and E. Schrödinger [29]. The books [27] by R. Penrose and W. Rindler monograph summarize the spinor and twistor methods in spacetime geometry (see additional references [15] on Clifford structures and spinor theory).
Spinor variables were introduced in Finsler geometries by Y. Takano in 1983 [34] where he dismissed anisotropic dependencies not only on vectors on the tangent bundle but on some spinor variables in a spinor bundle on a space-time manifold. Then generalized Finsler geometries, with spinor variables, were developed by T. Ono and Y. Takano in a series of publications during 1990-1993 [25]. The next steps were investigations of anisotropic and deformed geometries with spinor and vector variables and applications in gauge and gravity theories elaborated by P. Stavrinos and his students, S. Koutroubis, P. Manouselis, and V. Balan starting from 1994 [?, 32,33]. In those works the authors assumed that some spinor variables may be introduced in a Finsler-like way but they did not relate the Finlser metric to a Clifford structure and restricted the spinor-gauge Finsler constructions only for antisymmetric spinor metrics on two-spinor fibers with possible generalizations to four dimensional Dirac spinors.
Isotopic spinors, related with SU(2) internal structural groups, were considered in gen-fiber F (cofiber F * ) of dimension m, or as a higher order extended vector/covector bundle (we follow the geometric constructions and definitions of monographs [23,22] which were generalized for vector superbundles in Refs. [42,43]). Such (pseudo) Riemanian spaces and/or vector/covector bundles enabled with compatible fibered and/or anholonomic structures are called anisotropic space-times. If the anholonomic structure with associated nonlinear connection is modeled on higher order vector/covector bundles we use the term of higher order anisotropic space-time. In this section, we usually shall omit proofs which can be found in the mentioned monographs [23,22,43].

(Co) vector and tangent bundles
A locally trivial vector bundle, in brief, v-bundle, E = (E, π, M, Gr, F ) is introduced as a set of spaces and surjective map with the properties that a real vector space F = R m of dimension m (dim F = m, R denotes the real number field) defines the typical fiber, the structural group is chosen to be the group of automorphisms of R m , i. e. Gr = GL (m, R) , and π : E → M is a differentiable surjection of a differentiable manifold E (total space, dim E = n + m) to a differentiable manifold M (base space, dim M = n) .
The local coordinates on E are denoted u α = (x i , y a ) , or in brief u = (x, y) (the Latin indices i, j, k, ... = 1, 2, ..., n define coordinates of geometrical objects with respect to a local frame on base space M; the Latin indices a, b, c, ... =1, 2, ..., m define fiber coordinates of geometrical objects and the Greek indices α, β, γ, ... are considered as cumulative ones for coordinates of objects defined on the total space of a v-bundle).
and matrix K a ′ a (x i ) ∈ GL (m, R) are functions of necessary smoothness class. A local coordinate parametrization of v-bundle E naturally defines a coordinate basis and the reciprocal to (2) coordinate basis which is uniquely defined from the equations d α •∂ β = δ α β , where δ α β is the Kronecher symbol and by "•" we denote the inner (scalar) product in the tangent bundle T E.
A tangent bundle (in brief, t-bundle) (T M, π, M) to a manifold M can be defined as a particular case of a v-bundle when the dimension of the base and fiber spaces (the last one considered as the tangent subspace) are identic, n = m. In this case both type of indices i, k, ... and a, b, ... take the same values 1, 2, ...n. For t-bundles the matrices of fiber coordinates transforms from (1) can be written K i ′ i = ∂x i ′ /∂x i . We shall also use the concept of covector bundle, (in brief, cv-bundles) E = Ȇ , π * , M, Gr, F * , which is introduced as a dual vector bundle for which the typical fiber F * (cofiber) is considered to be the dual vector space (covector space) to the vector space F. The fiber coordinates p a ofȆ are dual to y a in E. The local coordinates on total spaceȆ are denotedȗ = (x, p) = (x i , p a ). The coordinate transform onȆ,ȗ = (x i , p a ) → u ′ = (x i ′ , p a ′ ), are written The coordinate bases on E * are denoted We use "breve" symbols in order to distinguish the geometrical objects on a cv-bundle E * from those on a v-bundle E.
As a particular case with the same dimension of base space and cofiber one obtains the cotangent bundle (T * M, π * , M) , in brief, ct-bundle, being dual to T M. The fibre coordinates p i of T * M are dual to y i in T M. The coordinate transforms (4) on T * M are stated by some matrices K k k ′ (x i ) = ∂x k /∂x k ′ . In our further considerations we shall distinguish the base and cofiber indices.

Higher order (co) vector bundles
The geometry of higher order tangent and cotangent bundles provided with nonlinear connection structure was elaborated in Refs. [22] in order to geometrize the higher order Lagrange and Hamilton mechanics. In this case we have base spaces and fibers of the same dimension. To develop the approach to modern high energy physics (in superstring and Kaluza-Klein theories), we introduced (in Refs [39,44,43,42]) the concept of higher order vector bundle with the fibers consisting from finite 'shells" of vector, or covector, spaces of different dimensions not obligatory coinciding with the base space dimension.
is a vector space decomposed into an invariant oriented direct sum As a particular case we obtain a distinguished vector space, in brief dv-space (a distinguished covector space, in brief dcv-space), if all components of the sum are vector (covector) spaces. We note that we have fixed, for simplicity, an orientation of (co) vector subspaces like in (7).
We define the higher order vector (covector) bundles, in brief, hv-bundles (in brief, hcv-bundles), if the typical fibre is a dv-space (dcv-space) as particular cases of the hvcbundles.

Osculator bundle
The k-osculator bundle is identified with the k-tangent bundle T k M, p (k) , M of a ndimensional manifold M. We denote the local coordinatesũ α = x i , y i (1) , ..., y i (k) , where we have identified y i (1) ≃ y a 1 , ..., y i (k) ≃ y a k , k = z, in order to to have similarity with denotations from [22]. The coordinate transformsũ α ′ →ũ α ′ (ũ α ) preserving the structure of such higher order vector bundles are parametrized where the equalities ) and the coframe isd α = (dx i , dy i (1) , ..., dy i (k) ). These formulas are respectively some particular cases of (8) and (9).

The dual bundle of k-osculator bundle
This higher order vector/covector bundle, denoted as T * k M, p * k , M , is defined as the dual bundle to the k-tangent bundle T k M, p k , M . The local coordinates (parametrized as in the previous paragraph) arẽ The coordinate transforms on T * k M, p * k , M are where the equalities hold for s = 0, ..., k − 2 and y i (0) = x i . The natural coordinate frame on T * k M, p * (k) , M is written in the form , ∂ ∂p i ) and the coframe is writtend α = dx i , dy i (1) , ..., dy i (k−1) , dp i . These formulas are, respespectively, certain particular cases of (8) and (9).

Nonlinear Connections
The concept of nonlinear connection, in brief, N-connection, is fundamental in the geometry of vector bundles and anisotropic spaces (see a detailed study and basic references in [23] and, for supersymmetric and/or spinor bundles in [43,49,42]). A rigorous mathematical definition is possible by using the formalism of exact sequences of vector bundles.

N-connections in vector bundles
Let E = = (E, p, M) be a v-bundle with typical fiber R m and π T : T E → T M being the differential of the map P which is a fibre-preserving morphism of the tangent bundle T E, τ E , E) → E and of tangent bundle (T M, τ, M) → M. The kernel of the vector bundle morphism, denoted as (V E, τ V , E), is called the vertical subbundle over E, which is a vector subbundle of the vector bundle (T E, τ E , E).
A vector X u tangent to a point u ∈ E is locally written as (x, y, We have π T (x, y, X, Y ) = (x, X). Thus the submanifold V E contains the elements which are locally represented as (x, y, 0, Y ).
where T E/V E is the factor bundle.
By definition (2.3) it is defined a morphism of vector bundles C : T E → V E such the superposition of maps C • i is the identity on V E, where i : V E → V E. The kernel of the morphism C is a vector subbundle of (T E, τ E , E) which is the horizontal subbundle, denoted by (HE, τ H , E). Consequently, we can prove that in a v-bundle E a N-connection can be introduced as a distribution for every point u ∈ E defining a global decomposition, as a Whitney sum, into horizontal, HE, and vertical, V E, subbundles of the tangent bundle T E Locally a N-connection in a v-bundle E is given by its coefficients N a i ( u) = N a i (x, y) with respect to bases (2) and (3) We note that a linear connection in a v-bundle E can be considered as a particular case of a N-connection when N a i (x, y) = K a bi (x) y b , where functions K b ai (x) on the base M are called the Christoffel coefficients.

N-connections in covector bundles
A nonlinear connection in a cv-bundleȆ (in brief aŇ-connection) can be introduced in a similar fashion as for v-bundles by reconsidering the corresponding definitions for cvbundles. For instance, it is stated by a Whitney sum, into horizontal, HȆ, and vertical, VȆ, subbundles of the tangent bundle TȆ : Hereafter, for the sake of brevity, we shall omit details on definition of geometrical objects on cv-bundles if they are very similar to those for v-bundles: we shall present only the basic formulas by emphasizing the most important common points and differences.
The same definition is true forŇ-connections in ct-bundles, we have to change in the definition (2.4) the symbolȆ into T * M.
A N-connection in a cv-bundleȆ is given locally by its coefficientsN ia ( u) =N ia (x, p) with respect to bases (2) and (3) We emphasize that if a N-connection is introduced in a v-bundle (cv-bundle), we have to adapt the geometric constructions to the N-connection structure.

N-connections in higher order bundles
The concept of N-connection can be defined for higher order vector / covector bundle in a standard manner like in the usual vector bundles: is a splitting of the left of the exact sequence We can associate sequences of type (12) to every mappings of intermediary subbundles. For simplicity, we present here the Whitney decomposition Locally a N-connectionÑ inẼ is given by its coefficients N a 2 a 1 , N a 1 a 3 , ..., N a 1 a z−1 , N az a 1 , 0, 0, N a 2 a 3 , ..., N a 2 a z−1 , N az a 2 , ..., ..., ..., ..., ..., ..., 0, 0, 0, ..., N a z−2 a z−1 , N az a z−2 , 0, 0, 0, ..., 0, N a z−1 az , which are given with respect to the components of bases (8) and (9).

Anholonomic frames and N-connections
Having defined a N-connection structure in a (vector, covector, or higher order vector / covector) bundle, we can adapt with respect to this structure (by 'N-elongation') the operators of partial derivatives and differentials and to consider decompositions of geometrical objects with respect to adapted bases and cobases.
Anholonomic frames in v-bundles In a v-bundle E provided with a N-connection we can adapt to this structure the geometric constructions by introducing locally adapted basis (N-frame, or N-basis): and its dual N-basis, (N-coframe, or N-cobasis), The anholonomic coefficients, w = {w α βγ (u)}, of N-frames are defined to satisfy the relations A frame bases is holonomic is all anholonomy coefficients vanish (like for usual coordinate bases (3)), or anholonomic if there are nonzero values of w α βγ . The operators (14) and (15) on a v-bundle E enabled with a N-connection can be considered as respective equivalents of the operators of partial derivations and differentials: the existence of a N-connection structure results in 'elongation' of partial derivations on x-variables and in 'elongation' of differentials on y-variables.
An elementt ∈T pr qs , d-tensor field of type p r q s , can be written in local form as We shall respectively use the denotations X Ȇ (or X (M)), Λ p Ȇ or (Λ p (M)) and Anholonomic frames in hvc-bundles The anholnomic frames adapted to a N-connection in hvc-bundleẼ are defined by the set of coefficients (13); having restricted the constructions to a vector (covector) shell we obtain some generalizations of the formulas for corresponding N(orŇ)-connection elongation of partial derivatives defined by (14) (or (17)) and (15) (or (18)). We introduce the adapted partial derivatives (anholonomic N-frames, or N-bases) inẼ by applying the coefficients (13) These formulas can be written in the matrix form: The adapted differentials (anholonomic N-coframes, or N-cobases) inẼ are introduced in the simplest form by using the matrix formalism: The respective dual matrices which defines the formulas for anholonomic N-coframes. The matrix M from (21) is the inverse to N, i. e. satisfies the condition The anholonomic coefficients, w = { w α βγ ( u)}, on hcv-bundleẼ are expressed via coefficients of the matrix N and their partial derivatives following the relations We omit the explicit formulas on shells. A d-tensor formalism can be also developed on the spaceẼ. In this case the indices have to be stipulated for every shell separately, like for v-bundles or cv-bundles.

Distinguished connections and metrics
In general, distinguished objects (d-objects) on a v-bundle E (or cv-bundleȆ) are introduced as geometric objects with various group and coordinate transforms coordinated with the N-connection structure on E (orȆ). For example, a distinguished connection (in brief, d-connection) D on E (orȆ) is defined as a linear connection D on E (orȆ) conserving under a parallelism the global decomposition (10) (or (11)) into horizontal and vertical subbundles of T E (or TȆ). A covariant derivation associated to a d-connection becomes d-covariant. We shall give necessary formulas for cv-bundles in round brackets.

D-connections in v-bundles (cv-bundles)
A N-connection in a v-bundle E (cv-bundleȆ) induces a corresponding decomposition of d-tensors into sums of horizontal and vertical parts, for example, for every d-vector and . In consequence, we can associate to every d-covariant derivation along the d-vector (24), and D (v) XY = D vXY , ∀Y ∈X Ȇ ) for which the following conditions hold: , locally adapted to the Nconnection structure with respect to the frames (14) and (15) ( (17) and (18)), are defined by the equations from which one immediately follows The coefficients of operators of h-and v-covariant derivations, (25)), are introduced as corresponding h-and v-parametrizations of (26) and A set of components (27) and (28) Γ completely defines the local action of a d-connection D in E (D inȆ).
For instance, having taken on E (Ȇ) a d-tensor field of type 1 1 1 1 , where the h-covariant derivative is written ckt ic ja ) and the v-covariant derivative is written For a scalar function

D-connections in hvc-bundles
The theory of connections in higher order anisotropic vector superbundles and vector bundles was elaborated in Refs. [42,44,43]. Here we re-formulate that formalism for the case when some shells of higher order anisotropy could be covector spaces by stating the general rules of covariant derivation compatible with the N-connection structure in hvc-bundleẼ and omit details and cumbersome formulas.
has the next shell decomposition of components (on induction being on the p-th shell, considered as the base space, which in this case a hvc-bundle, we introduce in a usual manner, like a vector or covector fiber, the (p + 1)-th shell) These coefficients determine the rules of a covariant derivationD onẼ.
For example, let us consider a d-tensort of type responding tensor product of components of anholonomic N-frames (20) and (21) t =t The d-covariant derivationD oft is to be performed separately for every shall according the rule (29) if a shell is defined by a vector subspace, or according the rule (30) if the shell is defined by a covector subspace.

D-metrics in v-bundles
We define a metric structure G in the total space E of a v-bundle E = (E, p, M) over a connected and paracompact base M as a symmetric covariant tensor field of type (0, 2), being non degenerate and of constant signature on E.
Nonlinear connection N and metric G structures on E are mutually compatible it there are satisfied the conditions: where ( the matrix h ab is inverse to h ab ). One obtains the following decomposition of metric: where the d-tensor hG(X, Y ) = G(hX, hY ) is of type 0 0 2 0 and the d-tensor vG(X, Y ) = G(vX, vY ) is of type 0 0 0 2 . With respect to the anholonomic basis (14) the d-metric (33) is written where A metric structure of type (33) (equivalently, of type (34)) or a metric on E with components satisfying the constraints (31), (equivalently (32)) defines an adapted to the given N-connection inner (d-scalar) product on the tangent bundle T E.
. With respect to anholonomic frames these conditions are written where by g βγ we denote the coefficients in the block form (34).

D-metrics in cv-and hvc-bundles
The presented considerations on self-consistent definition of N-connection, d-connection and metric structures in v-bundles can re-formulated in a similar fashion for another types of anisotropic space-times, on cv-bundles and on shells of hvc-bundles. For simplicity, we give here only the analogous formulas for the metric d-tensor (34): • On cv-bundleȆ we writȇ whereg ij =G δ i ,δ j andh ab =G ∂ a ,∂ b and the N-coframes are given by formulas (18).
For simplicity, we shall consider that the metricity conditions are satisfied,D γgαβ = 0.
• On osculator bundle T 2 M = Osc 2 M, we have a particular case of (37) wheñ with respect to N-coframes.

Some examples of d-connections
We emphasize that the geometry of connections in a v-bundle E is very reach. If a triple of fundamental geometric objects N a i (u) , Γ α βγ (u) , g αβ (u) is fixed on E, a multi-connection structure (with corresponding different rules of covariant derivation, which are, or not, mutually compatible and with the same, or not, induced d-scalar products in T E) is defined on this v-bundle. We can give a priority to a connection structure following some physical arguments, like the reduction to the Christoffel symbols in the holonomic case, mutual compatibility between metric and N-connection and d-connection structures and so on.
In this subsection we enumerate some of the connections and covariant derivations in vbundle E, cv-bundleȆ and in some hvc-bundles which can present interest in investigation of locally anisotropic gravitational and matter field interactions : which is hv-metric, i.e. there are satisfied the conditions D 3. The canonical d-connection Γ (c) (orΓ (c) ) on a v-bundle (or cv-bundle) is associated to a metric G (orG) of type (34) (or (36)), with coefficients This is a metric d-connection which satisfies conditions In physical applications, we shall use the canonical connection and, for simplicity, we shall omit the index (c). The coefficients (43) are to be extended to higher order if we are dealing with derivations of geometrical objects with "shell" indices. In this case the fiber indices are to be stipulated for every type of shell into consideration. 4. We can consider the N-adapted Christoffel symbols which have the components of d-connection Γ α βγ = L i jk , 0, 0, C a bc , with L i jk and C a bc as in (42) if g αβ is taken in the form (34).

Amost Hermitian anisotropic spaces
The are possible very interesting particular constructions [23,22] on t-bundle T M provided with N-connection which defines a N-adapted frame structure δ α = (δ i ,∂ i ) (for the same formulas (14) and (15) but with identified fiber and base indices). We are using the 'dot' symbol in order to distinguish the horizontal and vertical operators because on t-bundles the indices could take the same values both for the base and fiber objects. This allow us to define an almost complex structure J = {J β α } on T M as follows It is obvious that J is well-defined and J 2 = −I. For d-metrics of type (34), on T M, we can consider the case when g ij (x, y) = h ab (x, y), i. e.
where the index (t) denotes that we have geometrical object defined on tangent space. An almost complex structure J β α is compatible with a d-metric of type (46) and a d-connection D on tangent bundle T M if the conditions is an almost Hermitian structure on T M. One can introduce an almost sympletic 2-form associated to the almost Hermitian structure (G (t) , J), If the 2-form (47), defined by the coefficients g ij , is closed, we obtain an almost Kählerian structure in T M. By straightforward calculation we can prove that a d-connection DΓ = L i jk , L i jk , C i jc , C i jc with the coefficients defined by where L i jk and C e ab → C i jk , on T M are defined by the formulas (42), define a torsionless (see the next section on torsion structures) metric d-connection which satisfy the compatibility conditions (35).
Almost complex structures and almost Kähler models of Finsler, Lagrange, Hamilton and Cartan geometries (of first an higher orders) are investigated in details in Refs. [22,43].

Torsions and Curvatures
We outline the basic definitions and formulas for the torsion and curvature structures in v-bundles and cv-bundles provided with N-connection structure.
3.5.1 N-connection curvature 1. The curvature Ω of a nonlinear connection N in a v-bundle E can be defined in local form as [23]: N a bi being that from (40). 2. For the curvatureΩ, of a nonlinear connectionN in a cv-bundleȆ we introducȇ 3. There were analyzed the curvaturesΩ of different type of nonlinear connectionsÑ in higher order anisotropic bundles were analyzed for higher order tangent/dual tangent bundles and higher order prolongations of generalized Finsler, Lagrange and Hamilton spaces in Refs. [22] and for higher order anisotropic superspaces and spinor bundles in Refs. [43,39,44,42]: For every higher order anisotropy shell, we shall define the coefficients (49) or (50) in dependence of the fact with type of subfiber we are considering (a vector or covector fiber).

d-Torsions in v-and cv-bundles
The torsion T of a d-connection D in v-bundle E (cv-bundleȆ) is defined by the equation One holds the following h-and v-decompositions We consider the projections: and say that, for instance, hT (hX, hY) is the h(hh)-torsion of D , vT (hX, hY) is the v(hh)-torsion of D and so on. The torsion (51) in v-bundle is locally determined by five d-tensor fields, torsions, defined as Using formulas (14), (15), (49) and (51) we can compute [23] in explicit form the components of torsions (52) for a d-connection of type (27) and (28): Formulas similar to (52) and (53) hold for cv-bundles: The formulas for torsion can be generalized for hvc-bundles (on every shell we must write (53) or (55) in dependence of the type of shell, vector or co-vector one, we are dealing).

d-Curvatures in v-and cv-bundles
The curvature R of a d-connection in v-bundle E is defined by the equation One holds the next properties for the h-and v-decompositions of curvature: From (56) and the equation R (X, Y) = −R (Y, X) we get that the curvature of a dconnection D in E is completely determined by the following six d-tensor fields: By a direct computation, using (14), (15), (27), (28) and (57) we get: We note that d-torsions (53) and d-curvatures (58) are computed in explicit form by particular cases of d-connections (41), (43) and (44).
For cv-bundles we havě =∂ dČ bc a. −∂ cČ bd a. +Č bc e.Č ed a. −Č bd e.Č ec .a . The formulas for curvature can be also generalized for hvc-bundles (on every shell we must write (53) or (54) in dependence of the type of shell, vector or co-vector one, we are dealing).

Generalizations of Finsler Geometry
We outline the basic definitions and formulas for Finsler, Lagrange and generalized Lagrange spaces (constructed on tangent bundle) and for Cartan, Hamilton and generalized Hamilton spaces (constructed on cotangent bundle). The original results are given in details in the monographs [23,22], see also developments for suberbundles [42,43].

Finsler Spaces
The Finsler geometry is modeled on tangent bundle T M. 3. The Hessian of F 2 with elements is positively defined on T M .
The function F (x, y) and g ij (x, y) are called respectively the fundamental function and the fundamental (or metric) tensor of the Finsler space F.
One considers "anisotropic" (depending on directions y i ) Christoffel symbols, for simplicity we write g which are used for definition of the Cartan N-connection, This N-connection can be used for definition of an almost complex structure like in (45) and to define on T M a d-metric with g ij (x, y) taken as (61).
In general, we can consider that a Finsler space is provided with a metric g ij = ∂ 2 F 2 /2∂y i ∂y j , but the N-connection and d-connection are be defined in a different manner, even not be determined by F.
The idea of extension was to consider instead of the homogeneous fundamental function F (x, y) in a Finsler space a more general one, a Lagrangian L (x, y), defined as a differ- ij (x, y) = 1 2 is of rank n on M.  L(x, y)) where M is a smooth real n-dimensional manifold provided with regular Lagrangian L(x, y) structure L : T M → R for which g ij (x, y) from (64) has a constant signature over the manifold T M.
The fundamental Lagrange function L(x, y) defines a canonical N-connection as well a d-metric with g ij (x, y) taken as (64). As well we can introduce an almost Kählerian structure and an almost Hermitian model of L n , denoted as H 2n as in the case of Finsler spaces but with a proper fundamental Lagange function and metric tensor g ij . The canonical metric d- cL) jk is to computed by the same formulas (48) and (42)  ij but can be metric, or non-metric with respect to the Lagrange metric.
The next step of generalization is to consider an arbitrary metric g ij (x, y) on T M instead of (64) which is the second derivative of "anisotropic" coordinates y i of a Lagrangian [19]. One can consider different classes of N-and d-connections on T M, which are compatible (metric) or non compatible with (65) for arbitrary g ij (x, y). We can apply all formulas for d-connections, N-curvatures, d-torsions and d-curvatures as in a v-bundle E, but reconsidering them on T M, by changing h ab → g ij (x, y) and N a i → N k i .

Cartan Spaces
The theory of Cartan spaces (see, for instance, [28,16]) was formulated in a new fashion in R. Miron's works [20] by considering them as duals to the Finsler spaces (see details and references in [22]). Roughly, a Cartan space is constructed on a cotangent bundle T * M like a Finsler space on the corresponding tangent bundle T M.  2. K is a positive function, homogeneous on the fibers of the T * M, i. e. K(x, λp) = λF (x, p), λ ∈ R; 3. The Hessian of K 2 with elementš is positively defined on T * M .
The function K(x, y) andǧ ij (x, p) are called respectively the fundamental function and the fundamental (or metric) tensor of the Cartan space C n . We use symbols like "ǧ" as to emphasize that the geometrical objects are defined on a dual space.
One considers "anisotropic" (depending on directions, momenta, p i ) Christoffel symbols, for simplicity, we write the inverse to (66) as g ∂ǧ jk ∂x r , which are used for definition of the canonical N-connection, This N-connection can be used for definition of an almost complex structure like in (45) and to define on T * M a d-metrič withǧ ij (x, p) taken as (66). Using the canonical N-connection (67) and Finsler metric tensor (66) (or, equivalently, the d-metric (68) we can introduce the canonical d-connection The d-connection DΓ Ň (k) has the unique property that it is torsionless and satisfies the metricity conditions both for the horizontal and vertical components, i. e.Ď αǧβγ = 0. The d-curvaturesŘ  K (x, p) .
In general, we can consider that a Cartan space is provided with a metricǧ ij = ∂ 2 K 2 /2∂p i ∂p j , but the N-connection and d-connection could be defined in a different manner, even not be determined by K.

Generalized Hamilton and Hamilton Spaces
The geometry of Hamilton spaces was defined and investigated by R. Miron in Refs. [21] (see details and references in [22]). It was developed on the cotangent bundel as a dual geometry to the geometry of Lagrange spaces. Here we start with the definition of generalized Hamilton spaces and then consider the particular case.
Definition 4.5. A generalized Hamilton space is a pair GH n = (M,ǧ ij (x, p)) where M is a real n-dimensional manifold andǧ ij (x, p) is a contravariant, symmetric, nondegenerate of rank n and of constant signature on T * M .
The valueǧ ij (x, p) is called the fundamental (or metric) tensor of the space GH n . One can define such values for every paracompact manifold M. In general, a N-connection on GH n is not determined byǧ ij . Therefore we can consider arbitrary coefficientsŇ ij (x, p) and define on T * M a d-metric like (36) For Hamilton spaces the canonical N-connection (defined by H and its Hessian) exists, where the Poisson brackets, for arbitrary functions f and g on T * M, act as

In result we can compute the d-torsions and d-curvatures like on cv-bundle or on Cartan spaces. On Hamilton spaces all such objects are defined by the Hamilton function H(x, p)
and indices have to be reconsidered for co-fibers of the co-tangent bundle.

Clifford Bundles and N-Connections
The theory of anisotropic spinors was extended on higher order anisotropic (ha) spaces [44,43,49]. In brief, such spinors will be called ha-spinors which are defined as some Clifford ha-structures defined with respect to a distinguished quadratic form (37) on a hvc-bundle. For simplicity, the bulk of formulas will be given with respect to higher order vector bundles. To rewrite such formulas for hvc-bundles is to consider for the "dual" shells of higher order anisotropy some dual vector spaces and associated dual spinors.

Distinguished Clifford Algebras
The typical fiber of dv-bundle ξ d , π d : , split into horizontal hF and verticals v p F , p = 1, ..., z subspaces, with a bilinear quadratic form G(g, h) induced by a hvc-bundle metric (37). Clifford algebras (see, for example, Refs. [12,27]) formulated for d-vector spaces will be called Clifford dalgebras [38,48]. We shall consider the main properties of Clifford d-algebras. The proof of theorems will be based on the technique developed in Ref. [12,43,49] correspondingly adapted to the distinguished character of spaces in consideration. Let k be a number field (for our purposes k = R or k = C, R and C, are, respectively real and complex number fields) and define F , as a d-vector space on k provided with nondegenerate symmetric quadratic form (metric) G. Let C be an algebra on k (not necessarily commutative) and j : F → C a homomorphism of underlying vector spaces such that j(u) 2 = G(u) · 1 (1 is the unity in algebra C and d-vector u ∈ F ). We are interested in definition of the pair (C, j) satisfying the next universitality conditions. For every k-algebra A and arbitrary homomorphism ϕ : F → A of the underlying d-vector spaces, such that (ϕ(u)) 2 → G (u) · 1, there is a unique homomorphism of algebras ψ : C → A transforming the chain of maps into a commutative diagram.
The algebra solving this problem will be denoted as C (F , A) [equivalently as C (G) or C (F )] and called as Clifford d-algebra associated with pair (F , G) .
Theorem 5.1. The above-presented chain of maps has a unique solution (C, j) up to isomorphism.
Proof: See Refs. [39,43]. Now we re-formulate for d-algebras the Chevalley theorem [14]: Proof. See Refs. [39,43]. From the presented Theorems, we conclude that all operations with Clifford d-algebras can be reduced to calculations for C (hF , g) and C v (p) F , h (p) which are usual Clifford algebras of dimension 2 n and, respectively, 2 mp [12,4].
Of special interest is the case when k = R and F is isomorphic to vector space R p+q,a+b provided with quadratic form In this case, the Clifford algebra, denoted as C p,q , C a,b , is generated by the symbols e Explicit calculations of C p,q and C a,b are possible by the using isomorphisms [12,27] where M s (A) denotes the ring of quadratic matrices of order s with coefficients in ring A.
Here we write the simplest isomorphisms C 1,0 ≃ C, C 0,1 ≃ R ⊕ R and C 2,0 = H, where by H is denoted the body of quaternions. Now, we emphasize that higher order Lagrange and Finsler spaces, denoted H 2n -spaces, admit locally a structure of Clifford algebra on complex vector spaces. Really, by using almost Hermitian structure J β α and considering complex space C n with nondegenarate quadratic form n a=1 |z a | 2 , z a ∈ C 2 induced locally by metric (37) (rewritten in complex coordinates z a = x a + iy a ) we define Clifford algebra and which show that complex Clifford algebras, defined locally for H 2n -spaces, have periodicity 2 on p.
Considerations presented in the proof of theorem 2.2 show that map j : F → C (F ) is monomorphic, so we can identify space F with its image in C (F , G) , denoted as u → u, if u ∈ C (0) (F , G) u ∈ C (1) (F , G) ; then u = u ( respectively, u = −u).
Definition 5.1. The set of elements u ∈ C (G) * , where C (G) * denotes the multiplicative group of invertible elements of C (F , G) satisfying uF u −1 ∈ F , is called the twisted Clifford d-group, denoted as Γ (F ) .
The canonical map j : F → C (F ) can be interpreted as the linear map F → C (F ) 0 satisfying the universal property of Clifford d-algebras. This leads to a homomorphism of algebras, C (F ) → C (F ) t , considered by an anti-involution of C (F ) and denoted as u → t u. More exactly, if u 1 ...u n ∈ F , then t u = u n ...u 1 and t u = t u = (−1) n u n ...u 1 .
Definition 5.2. The spinor norm of arbitrary u ∈ C (F ) is defined as Let us introduce the orthogonal group O (G) ⊂ GL (G) defined by metric G on F and denote sets and Spin (G) = P in (G) ∩ C 0 (F ) . For F ∼ = R n+m we write Spin (n E ) . By straightforward calculations (see similar considerations in Ref. [12]) we can verify the exactness of these sequences: We conclude this subsection by emphasizing that the spinor norm was defined with respect to a quadratic form induced by a metric in dv-bundle E <z> . This approach differs from those presented in Refs. [2] and [25].

Clifford Ha-Bundles
We shall consider two variants of generalization of spinor constructions defined for d-vector spaces to the case of distinguished vector bundle spaces enabled with the structure of Nconnection. The first is to use the extension to the category of vector bundles. The second is to define the Clifford fibration associated with compatible linear d-connection and metric G on a dv-bundle. We shall analyze both variants.

Clifford d-module structure in dv-bundles
Because functor F → C(F ) is smooth we can extend it to the category of vector bundles of type ξ <z> = {π d : Recall that by F we denote the typical fiber of such bundles. For ξ <z> we obtain a bundle of algebras, denoted as C (ξ <z> ) , such that C (ξ <z> ) u = C (F u ) . Multiplication in every fiber defines a continuous map C (ξ <z> ) × C (ξ <z> ) → C (ξ <z> ) . If ξ <z> is a distinguished vector bundle on number field k, C (ξ <z> )-module, the d-module, on ξ <z> is given by the continuous map C (ξ <z> ) × E ξ <z> → ξ <z> with every fiber F u provided with the structure of the C (F u )-module, correlated with its k-module structure, Because F ⊂ C (F ) , we have a fiber to fiber map F × E ξ <z> → ξ <z> , inducing on every fiber the map F u × E ξ <z> (u) → ξ <z> (u) (R-linear on the first factor and k-linear on the second one ). Inversely, every such bilinear map defines on ξ <z> the structure of the C (ξ <z> )-module by virtue of universal properties of Clifford d-algebras. Equivalently, the above-mentioned bilinear map defines a morphism of v-bundles m : ξ <z> → HOM (ξ <z> , ξ <z> ) [HOM (ξ <z> , ξ <z> ) denotes the bundles of homomorphisms] when (m (u)) 2 = G (u) on every point. Vector bundles ξ <z> provided with C (ξ <z> )-structures are objects of the category with morphisms being morphisms of dv-bundles, which induce on every point u ∈ ξ <z> morphisms of C (F u ) −modules. This is a Banach category contained in the category of finitedimensional d-vector spaces on filed k.
Let us denote by H s (E <z> , GL n E (R)) , where n E = n+m 1 +...+m z , the s-dimensional cohomology group of the algebraic sheaf of germs of continuous maps of dv-bundle E <z> with group GL n E (R) the group of automorphisms of R n E (for the language of algebraic topology see, for example, Refs. [12]. We shall also use the group SL n E (R) = {A ⊂ GL n E (R) , det A = 1}. Here we point out that cohomologies H s (M, Gr) characterize the class of a principal bundle π : P → M on M with structural group Gr. Taking into account that we deal with bundles distinguished by an N-connection we introduce into consideration cohomologies H s (E <z> , GL n E (R)) as distinguished classes (d-classes) of bundles E <z> provided with a global N-connection structure.
For a real vector bundle ξ <z> on compact base E <z> we can define the orientation on ξ <z> as an element α d ∈ H 1 (E <z> , GL n E (R)) whose image on map is the d-class of bundle E <z> . Definition 5.3. The spinor structure on ξ <z> is defined as an element β d ∈ H 1 (E <z> , Spin (n E )) whose image in the composition The above definition of spinor structures can be re-formulated in terms of principal bundles. Let ξ <z> be a real vector bundle of rank n+m on a compact base E <z> . If there is a principal bundle P d with structural group SO(n E ) or Spin(n E )], this bundle ξ <z> can be provided with orientation (or spinor) structure. The bundle P d is associated with element α d ∈ H 1 (E <z> , SO(n <z> )) [or β d ∈ H 1 (E <z> , Spin (n E )) .
We remark that a real bundle is oriented if and only if its first Stiefel-Whitney d-class vanishes, where H 1 (E <z> , Z/2) is the first group of Chech cohomology with coefficients in Z/2, Considering the second Stiefel-Whitney class w 2 (ξ <z> ) ∈ H 2 (E <z> , Z/2) it is well known that vector bundle ξ <z> admits the spinor structure if and only if w 2 (ξ <z> ) = 0. Finally, we emphasize that taking into account that base space E <z> is also a v-bundle, p : E <z> → M, we have to make explicit calculations in order to express cohomologies H s (E <z> , GL n+m ) and H s (E <z> , SO (n + m)) through cohomologies which depends on global topological structures of spaces M and E <z> . For general bundle and base spaces this requires a cumbersome cohomological calculus.

Clifford fibration
Another way of defining the spinor structure is to use Clifford fibrations. Consider the principal bundle with the structural group Gr being a subgroup of orthogonal group O (G) , where G is a quadratic nondegenerate form ) defined on the base (also being a bundle space) space E <z> . The fibration associated to principal fibration P (E <z> , Gr) with a typical fiber having Clifford algebra C (G) is, by definition, the Clifford fibration P C (E <z> , Gr) . We can always define a metric on the Clifford fibration if every fiber is isometric to P C (E <z> , G) (this result is proved for arbitrary quadratic forms G on pseudo-Riemannian bases. If, additionally, Gr ⊂ SO (G) a global section can be defined on P C (G) . Let P (E <z> , Gr) be the set of principal bundles with differentiable base E <z> and structural group Gr. If g : Gr → Gr ′ is an homomorphism of Lie groups and P (E <z> , Gr) ⊂ P (E <z> , Gr) (for simplicity in this subsection we shall denote mentioned bundles and sets of bundles as P, P ′ and respectively, P, P ′ ), we can always construct a principal bundle with the property that there is an homomorphism f : P ′ → P of principal bundles which can be projected to the identity map of E <z> and corresponds to isomorphism g : Gr → Gr ′ . If the inverse statement also holds, the bundle P ′ is called as the extension of P associated to g and f is called the extension homomorphism denoted as g.
Now we can define distinguished spinor structures on bundle spaces.
Definition 5.4. Let P ∈ P (E <z> , O (G)) be a principal bundle. A distinguished spinor structure of P, equivalently a ds-structure of E <z> is an extension P of P associated to ho- is the group of orthogonal rotations, generated by metric G, in bundle E <z> .
So, if P is a spinor structure of the space E <z> , then P ∈ P (E <z> , P inG) .
The definition of spinor structures on varieties was given in Ref. [8] one had been proved that a necessary and sufficient condition for a space time to be orientable is to admit a global field of orthonormalized frames. We mention that spinor structures can be also defined on varieties modeled on Banach spaces [1]. As we have shown similar constructions are possible for the cases when space time has the structure of a v-bundle with an N-connection. Definition 5.5. A special distinguished spinor structure, ds-structure, of principal bundle P = P (E <z> , SO (G)) is a principal bundle P = P (E <z> , SpinG) for which a homomorphism of principal bundles p : P → P, projected on the identity map of E <z> and corresponding to representation R : SpinG → SO (G) , is defined.
In the case when the base space variety is oriented, there is a natural bijection between tangent spinor structures with a common base. For special ds-structures we can define, as for any spinor structure, the concepts of spin tensors, spinor connections, and spinor covariant derivations (see Refs. [48,39]).

Almost Complex Spinor Structures
Almost complex structures are an important characteristic of H 2n -spaces and of osculator bundles Osc k=2k 1 (M), where k 1 = 1, 2, ... . For simplicity in this subsection we restrict our analysis to the case of H 2n -spaces. We can rewrite the almost Hermitian metric [23], H 2n -metric in complex form [38]: where y=y(z,z) , and define almost complex spinor structures. For given metric (71) on H 2n -space there is always a principal bundle P U with unitary structural group U(n) which allows us to transform H 2n -space into v-bundle ξ U ≈ P U × U (n) R 2n . This statement will be proved after we introduce complex spinor structures on oriented real vector bundles [12].
Let us consider momentarily k = C and introduce into consideration [instead of the group Spin(n)] the group Spin c × Z/2 U (1) being the factor group of the product Spin(n) × U (1) with the respect to equivalence (y, z) ∼ (−y, −a) , y ∈ Spin(m).
This way we define the short exact sequence where ρ c (y, a) = ρ c (y) . If λ is oriented , real, and rank n, γ-bundle π : E λ → M n , with base M n , the complex spinor structure, spin structure, on λ is given by the principal bundle P with structural group Spin c (m) and isomorphism λ ≈ P × Spin c (n) R n (see (72)). For such bundles the categorial equivalence can be defined as where ǫ c (E c ) = P △ Spin c (n) E c is the category of trivial complex bundles on M n , E λ C (M n ) is the category of complex v-bundles on M n with action of Clifford bundle C (λ) , P △ Spin c (n) and E c is the factor space of the bundle product P × M E c with respect to the equivalence (p, e) ∼ (p g −1 , ge) , p ∈ P, e ∈ E c , where g ∈ Spin c (n) acts on E by via the imbedding Spin (n) ⊂ C 0,n and the natural action U (1) ⊂ C on complex v-bundle ξ c , E c = totξ c , for bundle π c : E c → M n . Now we return to the bundle ξ = E <1> . A real v-bundle (not being a spinor bundle) admits a complex spinor structure if and only if there exist a homomorphism σ : U (n) → Spin c (2n) defining a commutative diagram. The explicit construction of σ for arbitrary γ-bundle is given in Refs. [12] and [4]. Let λ be a complex, rank n, spinor bundle with the homomorphism defined by formula τ (λ, δ) = δ 2 . For P s being the principal bundle with fiber Spin c (n) we introduce the complex linear bundle L (λ c ) = P S × Spin c (n) C defined as the factor space of P S × C on equivalence relation where t ∈ Spin c (n) . This linear bundle is associated to complex spinor structure on λ c . If λ c and λ c ′ are complex spinor bundles, the Whitney sum λ c ⊕ λ c ′ is naturally provided with the structure of the complex spinor bundle. This follows from the holomorphism given by formula where ω is the homomorphism making the chain of maps into a commutative diagram. Here, z, z ′ ∈ U (1) . It is obvious that We conclude this subsection by formulating our main result on complex spinor structures for H 2n -spaces:  (73), H is functor E c → E c ⊗ L (λ c ) and E 0,2n given as correspondence E c → Λ (C n ) ⊗ E c (which is a categorial equivalence), Λ (C n ) is the exterior algebra on C n . W is the real bundle λ c ⊕λ c ′ provided with complex structure.
Proof: See Refs. [38,44,43,49]. Now consider bundle P × Spin c (n) Spin c (2n) as the principal Spin c (2n)-bundle, associated to M ⊕ M being the factor space of the product P × Spin c (2n) on the equivalence relation (p, t, h) ∼ p, µ (t) −1 h . In this case the categorial equivalence (73) can be rewritten as (projections of elements p and e coincides on base M). Every element of ǫ c (E c ) can be represented as P ∆ Spin c (n) E c , i.e., as a factor space P ∆E c on equivalence relation (pt, e) ∼ (p, µ c (t) , e) , when t ∈ Spin c (n) . The complex line bundle L (λ c ) can be interpreted as the factor space of P × Spin c (n) C on equivalence relation (pt, δ) ∼ p, r (t) −1 δ . Putting (p, e) ⊗ (p, δ) (p, δe) we introduce morphism pointing to the fact that we have defined the isomorphism correctly and that it is an isomorphism on every fiber. ✷

Spinors and N-Connection Geometry
The purpose of this section is to show how a corresponding abstract spinor technique entailing notational and calculations advantages can be developed for arbitrary splits of dimensions of a d-vector space F = hF ⊕v 1 F ⊕...⊕v z F , where dim hF = n and dim v p F = m p . For convenience we shall also present some necessary coordinate expressions.

D-Spinor Techniques
The problem of a rigorous definition of spinors on locally anisotropic spaces (d-spinors) was posed and solved [38,39] in the framework of the formalism of Clifford and spinor structures on v-bundles provided with compatible nonlinear and distinguished connections and metric.
We introduced d-spinors as corresponding objects of the Clifford d-algebra C (F , G), defined for a d-vector space F in a standard manner (see, for instance, [12]) and proved that operations with C (F , G) can be reduced to calculations for C (hF , g) , C (v 1 F , h 1 ) , ... and C (v z F , h z ) , which are usual Clifford algebras of respective dimensions 2 n , 2 m 1 , ... and 2 mz (if it is necessary we can use quadratic forms g and h p correspondingly induced on hF and v p F by a metric G (37)). Considering the orthogonal subgroup O(G) ⊂ GL(G) defined by a metric G we can define the d-spinor norm and parametrize d-spinors by ordered pairs of elements of Clifford algebras C (hF , g) and C (v p F , h p ) , p = 1, 2, ...z. We emphasize that the splitting of a Clifford d-algebra associated to a dv-bundle E <z> is a straightforward consequence of the global decomposition defining a N-connection structure in E <z> . In this subsection we shall omit detailed proofs which in most cases are mechanical but rather tedious. We can apply the methods developed in [27,15] in a straightforward manner on h-and v-subbundles in order to verify the correctness of affirmations.

Clifford d-algebra, d-spinors and d-twistors
In order to relate the succeeding constructions with Clifford d-algebras [38] we consider a la-frame decomposition of the metric (37): where the frame d-vectors and constant metric matrices are distinguished as To generate Clifford d-algebras we start with matrix equations where I is the identity matrix, matrices σ < α> (σ-objects) act on a d-vector space F = hF ⊕ v 1 F ⊕ ... ⊕ v z F and theirs components are distinguished as where k = 1, 2, ..., k p = 1, 2, .... The Clifford d-algebra is generated by sums on n + 1 elements of form and sums of m p + 1 elements of form coefficients A 1 , C i j , D i j k , ... and 2 mp+1 coefficients (A 2(p) , C ap bp , D ap bp cp , ...) of the Clifford algebra on F . For simplicity, we shall present the necessary geometric constructions only for h-spin spaces S (h) of dimension N (n) . Considerations for a v-spin space S (v) are similar but with proper characteristics for a dimension N (m) .
In order to define the scalar (spinor) product on S (h) we introduce into consideration this finite sum (because of a finite number of elements σ [ i j... k] ): which can be factorized as and ij km = 2N (n) ǫ km ǫ ij for n = 1(mod4). Antisymmetry of σ i j k... and the construction of the objects (78)-(81) define the properties of ǫ-objects (±) ǫ km and ǫ km which have an eight-fold periodicity on n (see details in [27] and, with respect to locally anisotropic spaces, [38]).
For even values of n it is possible the decomposition of every h-spin space S (h) into irreducible h-spin spaces S (h) and S ′ (h) (one considers splitting of h-indices, for instance, ..) and defines new ǫ-objects ǫ lm = 1 2 (+) ǫ lm + (−) ǫ lm and ǫ lm = We shall omit similar formulas for ǫ-objects with lower indices. In general, the spinor ǫ-objects should be defined for every shell of anisotropy where instead of dimension n we shall consider the dimensions m p , 1 ≤ p ≤ z, of shells.
We define a d-spinor space S (n,m 1 ) as a direct sum of a horizontal and a vertical spinor spaces, for instance, |△ , ... The scalar product on a S (n,m 1 ) is induced by (corresponding to fixed values of n and m 1 ) ǫobjects considered for h-and v 1 -components. We present also an example for S (n,m 1 +...+mz) : Having introduced d-spinors for dimensions (n, m 1 + ... + m z ) we can write out the generalization for ha-spaces of twistor equations [27] by using the distinguished σ-objects (77): where β denotes that we do not consider symmetrization on this index. The general solution of (83) on the d-vector space F looks like as where Ω β and Π ǫ are constant d-spinors. For fixed values of dimensions n and m = m 1 +...m z we mast analyze the reduced and irreducible components of h-and v p -parts of equations (83) and their solutions (84) in order to find the symmetry properties of a d-twistor Z α defined as a pair of d-spinors Z α = (ω α , π ′ β ), where π β ′ = π (0) β ′ ∈ S (n,m 1 ,...,mz) is a constant dual d-spinor. The problem of definition of spinors and twistors on ha-spaces was firstly considered in [48] (see also [35]) in connection with the possibility to extend the equations (84) and theirs solutions (85), by using nearly autoparallel maps, on curved, locally isotropic or anisotropic, spaces. We note that the definition of twistors have been extended to higher order anisotropic spaces with trivial N-and d-connections.

Mutual transforms of d-tensors and d-spinors
The spinor algebra for spaces of higher dimensions can not be considered as a real alternative to the tensor algebra as for locally isotropic spaces of dimensions n = 3, 4 [27]. The same holds true for ha-spaces and we emphasize that it is not quite convenient to perform a spinor calculus for dimensions n, m >> 4. Nevertheless, the concept of spinors is important for every type of spaces, we can deeply understand the fundamental properties of geometical objects on ha-spaces, and we shall consider in this subsection some questions concerning transforms of d-tensor objects into d-spinor ones.

Fundamental d-spinors
We can transform every d-spinor ξ α = (ξ i , ξ a 1 , ..., ξ a z ) into a corresponding d-tensor. For simplicity, we consider this construction only for a h-component ξ i on a h-space being of dimension n. The values with a different number of indices i... j, taken together, defines the h-spinor ξ i to an accuracy to the sign. We emphasize that it is necessary to choose only those h-components of dtensors (90) (or (91)) which are symmetric on pairs of indices αβ (or IJ (or I ′ J ′ )) and the number q of indices i... j satisfies the condition (as a respective consequence of the properties (87) and/ or (88), (89)) n − 2q ≡ 0, 1, 7 (mod 8).
Of special interest is the case when q = 1 2 (n ± 1) (n is odd) (93) or q = 1 2 n (n is even) .
If all expressions (90) and/or (91) are zero for all values of q with the exception of one or two ones defined by the conditions (92), (93) (or (94)), the value ξ i (or ξ I (ξ I ′ )) is called a fundamental h-spinor. Defining in a similar manner the fundamental v-spinors we can introduce fundamental d-spinors as pairs of fundamental h-and v-spinors. Here we remark that a h(v p )-spinor ξ i (ξ ap ) (we can also consider reduced components) is always a fundamental one for n(m) < 7, which is a consequence of (94)).

Differential Geometry of Ha-Spinors
This subsection is devoted to the differential geometry of d-spinors in higher order anisotropic spaces. We shall use denotations of type v <α> = (v i , v <a> ) ∈ σ <α> = (σ i , σ <a> ) and ζ α p = (ζ i p , ζ a p ) ∈ σ αp = (σ ip , σ ap ) for, respectively, elements of modules of d-vector and irreduced d-spinor fields (see details in [38]). D-tensors and d-spinor tensors (irreduced or reduced) will be interpreted as elements of corresponding σ-modules, for instance, .. We can establish a correspondence between the higher order anisotropic adapted to the N-connection metric g αβ (37) and d-spinor metric ǫ αβ (ǫ-objects for both h-and v psubspaces of E <z> ) of a ha-space E <z> by using the relation where which is a consequence of formulas (76)-(82). In brief we can write (95) as if the σ-objects are considered as a fixed structure, whereas ǫ-objects are treated as caring the metric "dynamics " , on higher order anisotropic space. This variant is used, for instance, in the so-called 2-spinor geometry [27] and should be preferred if we have to make explicit the algebraic symmetry properties of d-spinor objects by using metric decomposition (97). An alternative way is to consider as fixed the algebraic structure of ǫ-objects and to use variable components of σ-objects of type (96) for developing a variational d-spinor approach to gravitational and matter field interactions on ha-spaces (the spinor Ashtekar variables [3] are introduced in this manner). We note that a d-spinor metric on the d-spinor space S = (S (h) , S (v 1 ) , ..., S (vz) ) can have symmetric or antisymmetric h (v p ) -components ǫ ij (ǫ a p b p ) , see ǫ-objects. For simplicity, in order to avoid cumbersome calculations connected with eight-fold periodicity on dimensions n and m p of a ha-space E <z> , we shall develop a general d-spinor formalism only by using irreduced spinor spaces S (h) and S (vp) .
Let now analyze the question on uniqueness of action on d-spinors of an operator ▽ <α> satisfying necessary conditions . Denoting by ▽ (1) <α> and ▽ <α> two such d-covariant operators we consider the map Because the action on a scalar f of both operators ▽ (1) α and ▽ α must be identical, i.e.
Connecting the last expression on β and ν and using an orthonormalized d-spinor basis when γ β <γ>β = 0 (a consequence from (102)) we have We also note here that, for instance, for the canonical and Berwald connections and Christoffel d-symbols we can express d-spinor connection (105) through corresponding locally adapted derivations of components of metric and N-connection by introducing corresponding coefficients instead of Γ <α> <γ><β> in (105) and than in (104).
We note that d-spinor tensors X γ δ αβ and Ψ αβγδ are transformed into similar 2-spinor objects on locally isotropic spaces [27] if we consider vanishing of the N-connection structure and a limit to a locally isotropic space.