Anholonomic Frames, Generalized Killing Equations, and Anisotropic Taub NUT Spinning Spaces

By using anholonomic frames in (pseudo) Riemannian spaces we define anisotropic extensions of Euclidean Taub-NUT spaces. With respect to coordinate frames such spaces are described by off-diagonal metrics which could be diagonalized by corresponding anholonomic transforms. We define the conditions when the 5D vacuum Einstein equations have as solutions anisotropic Taub-NUT spaces. The generalized Killing equations for the configuration space of anisotropically spinning particles (anisotropic spinning space) are analyzed. Simple solutions of the homogeneous part of these equations are expressed in terms of some anisotropically modified Killing-Yano tensors. The general results are applied to the case of the four-dimensional locally anisotropic Taub-NUT manifold with Euclidean signature. We emphasize that all constructions are for(pseudo) Riemannian spaces defined by vacuum soltions, with generic anisotropy, of 5D Einstein equations, the solutions being generated by applying the moving frame method.


Introduction
Much attention has been paid to off-diagonal metrics in higher dimensional gravity beginning the Salam, Strathee and Perracci work [1] which showed that including off-diagonal components in higher dimensional metrics is equivalent to including U(1), SU (2) and SU(3) gauge fields. The approach was developed by construction of various locally isotropic solutions of vacuum 5D Einstein equations describing 4D wormholes and/or flux tube gravitationalelectromagnetic configurations (see Refs. [2]).
Recently, the off-diagonal metrics were considered in a new fashion by applying the method of anholonomic frames with associated nonlinear connections [3,4,5] which allowed to construct new classes of solutions of Einstein's equations in three (3D), four (4D) and five (5D) dimensions which had generic local anisotropy, e.g. static black hole and cosmological solutions with ellipsoidal or toroidal symmetry, various soliton-dilaton 2D and 3D configurations in 4D gravity, and wormhole and flux tubes with anisotropic polarizations and/or running constants with different extensions to backgrounds of rotation ellipsoids, elliptic cylinders, bipolar and toroidal symmetry and anisotropy.
Another class of 4D metrics induced from 5D Kaluza-Klein theory is connected with the Euclidean Taub-NUT metric which is involved in many modern studies of physics, for instance, in definition of the gravitational analogue of the Yang-Mills instantons [6] and of Kaluza-Klein monopole [7] related with geodesic geometry of higher dimension (pseudo) Riemannian spaces [8] (see a recent review and original results in [9]).
The construction of monopole and instanton solutions, with deformed symmetries, in modern string theory, extra dimensional gravity and quantum chromodynamics is of fundamental importance in understanding these theories (especially their non-perturbative aspects). Such solutions are difficult to find, and the solutions which are known usually have a high degree of symmetry. In this work we apply the method of anholonomic frames to construct the general form anholonomically constrained Taub NUT metrics in 5D Kaluza-Klein theory. These solutions have local anisotropy which would make their study using holonomic frames difficult. This helps to demonstrate the usefulness of the anholonomic frames method in studying anisotropic solutions. Most physical situations do not possess a high degree of symmetry, and so the anholonomic frames method provides a useful mathematical framework for studying these less symmetric configurations. We emphasize that the anholonomic moving frame method works effectively in construction of anisotropic mass hierarchies with running of constants in modern brane physics [10] (on new directions in extra dimension gravity see Refs. [11]. This allows us to approach a task of of primordial importance of definition of non-perturbative models and finding of exact solutions in higher dimension field theory describing anisotropic monopole/instanton configurations with running constants. The metrics considered for both wormhole and Taub-NUT geometry and physics could be given by 5D line elements with 3D spherical coordinates (r, θ, ϕ), ds 2 5 = −e 2ν(r) dt 2 + ds 2 4 , (1) ds 2 4 = V −1 (r)[dr 2 + a(r)(dθ 2 + sin 2 θ dϕ 2 )] + 16m 2 0 V (r)r 2 0 e 2ψ(r)−2ν(r) [dχ + ω(r)dt + n 0 cos θ dϕ] 2 where the metric coefficients and constants m 0 , r 0 , n 0 have to be correspondingly parametrized in order to select two particular cases: 1. We must put a(r) = r 2 , r 2 0 e 2ψ(r)−2ν(r) = 1, n 0 = 1, ω(r) = 0, ν(r) = 0, m 0 = const, V (r) = 1 + 4m 0 r −1 (2) and to impose on the fifth coordinate the condition 0 ≤ χ < 4π, 4m 0 (χ + ϕ) = −x 5 , if we want to obtain the Taub-NUT metric connected with the gauge field A of a monopole A r = A ϕ = 0, A θ = 4m 0 (1 − cos θ) where r denotes a three-vector r = (r, θ, ϕ); the so called NUT singularity is absent if x 5 is periodic with period 16πm 0 [12].
The solution in [13] satisfied the boundary conditions a(0) = 1, ψ(0) = ν(0) = 0 where it was proved that the free parameters of the metric are varied there are five classes of wormhole /flux tube solutions.
We note that the metric (1) defines solutions of vacuum Einstein equations only for particular parametrizations of type 1 or 2; it is not a vacuum solution for arbitrary values of coefficients.
The purpose of this paper is to construct Taub-NUT like metrics with anisotropic variations of the constant m 0 , when m 0 → m(..., χ), or m 0 → m(..., ϕ). We note that such anisotropic metrics are given by off-diagonal coefficients which define solutions of the 5D vacuum Einstein equations and generalize the constructions from [12,14] to Taub-NUT locally anisotropic gravitational instantons embedded into anisotropic variants of Kaluza-Klein monopoles (the first anisotropic instanton solutions were proposed in in Refs. [15] for the so-called generalized Finsler-Kaluza-Klein spaces and locally anisotropic gauge gravity, here we note that in this paper we shall not concern topics relating generalized Lagrange and Finsler (super) spaces [3]); by using anholonomic frames we can model anisotropic instanton configurations in usual Riemannian spaces. The anisotropic metrics are defined as (pseudo) Riemannian ones which admit a diagonalization with respect to some anholonomic frame bases with associated nonlinear connection structures. Such spacetimes, provided with metrics with generic anisotropy and anholonomic frame structure, are called as anisotropic spaces-time.
With respect to the new extradimensional coordinate ς the component A ϕ of the electromagnetic potential is removed into the component A θ ; this will allow us to treat the coordinates (t, r, θ) as holonomic coordinates but (ϕ, ς) as anholonomic ones. For our further considerations it is convenient to use a conformally transformed (multiplied on the factor V (r)) Taub NUT metric with the 5th coordinate ς which will be used for generalizations in order to obtain new solutions of the vacuum Einstein equations, being anisotropic on coordinates (θ, ϕ, ς). This metric generates a monopole configuration (3). The paper is organized as follow: Section 2 outlines the geometry of anholonomic frames with associated nonlinear connections on (pseudo) Riemannian spaces. The metric ansatz for anisotropic solutions is introduced.
In Section 3, there are analyzed the basic properties of solutions of vacuum Einstein equations with mixed holonomic and anholonomic variables. The method of construction of exact solutions with generic local anisotropy is developed.
In Section 4, we construct three classes of generalized anisotropic Taub NUT metrics, being solutions of the vacuum Einstein equations, which posses anisotropies of parameter m on angular coordinate ϕ, or contains a running constant m(ς) and/or are elliptically polarized on angular coordinate θ.
Section 5 is devoted to a new exact 5D vacuum solution for anisotropic Taub NUT wormholes obtained as a nonlinear superposition of the running on extra dimension coordinate Taub NUT metric and a background metric describing locally isotropic wormhole / flux tube configurations. Section 6 elucidates the problem of definition of integrals of motion for anholonomic spinning of particles in anisotropic spaces. There are introduced Killing, energy and momentum and Runge-Lenz vectors with respect to anholonomic bases with associated nonlinear connection structures defined by anisotropic solutions of vacuum Einstein equations. There are proposed and analyzed the action for anisotropic spinning of particles, defined the Poisson-Dirac brackets on anisotropic spaces. We consider anisotropic Killing equations and discuss the problem of construction their generic solutions and non-generic solutions with anholonomic Killing-Yano tensors.
In Section 7, we approach the problem of definition of Killing-Yano tensors for anisotropic Taub NUT spinning spaces and construct the corresponding Lie algebra with anisotropic variation of constants.
Finally, in Section 8, some conclusion remarks are presented.

Anholonomic Frames and Nonlinear Connections in Riemannian Spaces
In this section we outline the basic formulas on anholonomic frames with mixed holonomic-anholonomic components (variables) and associated nonlinear connection structures in Riemannian spaces.

Metric ansatz
Let us consider a 5D pseudo-Riemannian spacetime of signature (−, +, +, +, +) and denote the local coordinates where (s, p) = (ς, ϕ) , or, inversely, (s, p) = (ϕ, ς) -or more compactly u = (x, y) -where the Greek indices are conventionally split into two subsets x i and y a labeled, respectively, by Latin indices of type i, j, k, ... = 1, 2, 3 and a, b, ... = 4, 5. The local coordinate bases, ∂ α = (∂ i , ∂ a ), and their duals, , are written respectively as and The 5D (pseudo) Riemannian squared linear interval is given by the metric coefficients g αβ (a matrix ansatz definded with respect to the coordinate frame base (9)) in the form where the coefficients are some necessary smoothly class functions of type: w 2,3 = w 2,3 (r, θ, s), n 2,3 = n 2,3 (r, θ, s); one considers dependencies of the coefficients of metric on two so-called isotropic variables (r, θ) and on one anisotropic variable , y 4 = s, (in similar fashions we can alternatively consider dependencies on arbitrary couples of x-coordinates completed with one y-coordinate, for instance, (r, θ) and (r, θ, p)).
The metric (10) with coefficients (11) can be equivalently rewritten in the form with diagonal coefficients if instead the coordinate bases (8) and (9) one introduces the anholonomic frames (anisotropic bases) and where the N-coefficients are parametrized N 4,5 1 = 0, N 4 2,3 = w 2,3 and N 5 2,3 = n 2,3 (they define an associated to some anholonomic frames (16) and (17), nonlinear connection, N-connection, structure, see details in Refs [20,3,16,17,15]). A N-connection induces a global decomposition of the 5D pseudo-Riemannian spacetime into holonomic (horizontal, h) and anholonomic (vertical, v) directions. In a preliminary form the concept of N-connections was applied by E. Cartan in his approach to Finsler geometry [18] and a rigorous definition was given by Barthel [19] (Ref. [20] gives a modern approach to the geometry of N-connections, and to generalized Lagrange and Finsler geometry, see also Ref. [16] for applications of N-connection formalism in supergravity and superstring theory). As a particular case one obtains the linear connections if N a i (x, y) = Γ a bi (x) y a . A quite surprising result is that the N-connection structures can be naturally defined on (pseudo) Riemannian spacetimes [3,16,15] by associating them with some anholonomic frame fields (vielbeins) of type (16) satisfying the relations δ α δ β − δ β δ α = W γ αβ δ γ , with nontrivial anholonomy coefficients where is the nonlinear connection curvature (N-curvature). One says that the N-connection coefficients model a locally anisotropic structure on spacetime ( a locally anisotropic spacetime) when the partial derivative operators and coordinate differentials, (8) and (9), are respectively changed into N-elongated operators (16) and (17).
Conventionally, the N-coefficients decompose the spacetime values (tensors, spinors and connections) into sets of mixed holonomic-anholonomic variables (coordinates) provided respectively with 'holonomic' indices of type i, j, k, ... and with 'anholonomic' indices of type a, b, c, .... Tensors, metrics and linear connections with coefficients defined with respect to anholonomic frames (16) and (17) are distinguished (d) by N-coefficients into holonomic and anholonomic subsets and called, in brief, d-tensors, d-metrics and dconnections.

D-connections, d-torsions and d-curvatures
On (pseudo)-Riemannian spacetimes the associated N-connection structure can be treated as a "pure" anholonomic frame effect which is induced if we are dealing with mixed sets of holonomic-anholonomic basis vectors. When we are transferring our considerations only to coordinate frames (8) and (9) the N-connection coefficients are removed into both off-diagonal and diagonal components of the metric like in (11). In some cases the N-connection (anholonomic) structure is to be stated in a non-dynamical form by definition of some initial (boundary) conditions for the frame structure, following some prescribed symmetries of the gravitational-matter field interactions, or , in another cases, a subset of N-coefficients have to be treated as some dynamical variables defined as to satisfy the Einstein equations.

D-metrics and d-connections:
A metric of type (14), in general, with arbitrary coefficients g ij x k , y a and h ab x k , y a defined with respect to a N-elongated basis (17) is called a dmetric.
A linear connection D δγ δ β = Γ α βγ (x, y) δ α , associated to an operator of covariant derivation D, is compatible with a metric g αβ and N-connection

This defines a canonical linear connection (distinguished by a N-connection)
which is similar to the metric connection introduced by Christoffel symbols in the case of holonomic bases.

D-torsions and d-curvatures:
The anholonomic coefficients w γ αβ and N-elongated derivatives give nontrivial coefficients for the torsion tensor, and for the curvature tensor, We emphasize that the torsion tensor on (pseudo) Riemannian spacetimes is induced by anholonomic frames, whereas its components vanish with respect to holonomic frames. All tensors are distinguished (d) by the N-connection structure into irreducible h-v-components, and are called d-tensors. For instance, the torsion, d-tensor has the following irreducible, nonvanishing, (the d-torsion is computed by substituting the h-v-components of the canonical d-connection (19) and anholonomic coefficients (18) into the formula for the torsion coefficients (20)). The curvature d-tensor has the following ir- (the d-curvature components are computed in a similar fashion by using the formula for curvature coefficients (21)).

Einstein Equations with Anholonomic Variables
In this section we write and analyze the Einstein equations on 5D (pseudo) Riemannian spacetimes provided with anholonomic frame structures and associated N-connections.

Einstein equations with matter sources
The Ricci tensor R βγ = R α β γα has the d-components In general, since 1 P ai = 2 P ia , the Ricci d-tensor is non-symmetric (this could be with respect to anholonomic frames of reference). The scalar curvature of the metric d-connection, where R = g ij R ij and S = h ab S ab .
By substituting (23) and (24) into the 5D Einstein equations where κ and Υ αβ are respectively the coupling constant and the energymomentum tensor. The definition of matter sources with respect to anholonomic frames is considered in Refs. [3].

5D vacuum Einstein equations
In this paper we deal only with vacuum 5D, locally, anisotropic gravitational equations which in invariant h-v-components are written The main 'trick' of the anholonomic frames method for integrating the Einstein equations in general relativity and various (super) string and higher / lower dimension gravitational theories is to find the coefficients N a j such that the block matrices g ij and h ab are diagonalized [3,16]. This greatly simplifies computations. With respect to such anholonomic frames the partial derivatives are N-elongated (locally anisotropic).

Non-trivial Ricci components:
The metric (10) with coefficients (11) (equivalently, the d-metric (14) with coefficients (15)) is assumed to solve the 5D Einstein vacuum equations R αβ = 0, which are distinguished in h-and v-components as where for further applications we gave the formulas with respect to h 4,5 coefficients of metric as well with respect to f 4,5 , see (13), and, for simplicity, the partial derivatives are denoted h • = ∂h/∂x 2 , f ′ = ∂f /∂x 3 and f * = ∂f /∂s. It was possible to construct very general classes of solutions for such equations [3,4,5] describing locally anisotropic soliton, black hole, black tori and wormhole objects.

3.2.2
General properties of anisotropic vacuum solutions: In the vacuum case the equations (27), (28), (29) and (30) form a very simplified system of equations with separations of variables which can be solved consequently for every couples of d-metric coefficients, (g 2 , g 3 ) , (h 4 , h 5 ) , and N-connection coefficients w 2,3 and n 2,3 (see Refs [10] on the main teorems and methods of constructing exact solutions): 1. The equation (27) relates two functions g 2 (r, θ) and g 3 (r, θ) and their partial derivatives on 'isotropic' coordinates r and θ. The solution is trivial if we chose g 2 = 1 and g 3 = r 2 in order to reduce the coefficients from (11), respectively, to those from (7).
If the condition h * 5 = 0 is satisfied, we can write (28), in f -variables (see (12)), as (ln |f * 5 | + f 5 − f 4 ) * = 0, which is solved by arbitrary functions f 5 (r, θ, s) and where The general solution of (28) expressing h 5 via h 4 is for some functions f 5[0,1,2] (r, θ) and ̟(r, θ) stated by boundary conditions and locally anisotropic limits as well from the conditions that the equations (29) and (30) = h 4 (r, θ, s), an arbitrary function if h * 5 = 0, 3. If the functions h 4 (r, θ, s) and h 5 (r, θ, s) were defined, the equations (29) can be solved as independent linear algebraic equations for w 2,3 , w i β + α i = 0, i = 2, 3. For zero matter sources this is a trivial result because in this case the conditions β = 0 and α i = 0 (see the formulas (31), (32) and (33)) are automatically fulfilled. In consequence, the resulting sourceless equations (29) became some trivial equations admitting arbitrary values of functions w i (r, θ, s) ; such functions can be associated to some coordinate transforms for vanishing anholonomy coefficients W 4 αβ = 0, see (18), or to some anholonomy coefficients, in such cases being not contained in the vacuum Einstein equations, which must be stated by some boundary and symmetry conditions.

Taub NUT Metrics with Anisotropies and Running of Constant
The conformally transformed Taub NUT metric (7) can be considered as a locally isotropic background with trivial vanishing local anisotropies. By coordinate transforms of the 5th coordinate and a conformal transform on two holonomic coordinates (here we mention that in two dimensions the coordinate and conformal transforms are equivalent) the isotropic background metric can be transformed into a form parameterizing the usual Taub NUT solution of the vacuum Einstein equations. The aim of this section is to construct and analyze three types of anisotropic generalizations of the Taub NUT solution.

Generalizations of Taub NUT solutions to anisotropies and running of constant
The simplest way to obtain anisotropic Taub NUT like solutions, is to follow the approach developed for generation anisotropic black hole [3,15] and wormhole / flux tube solutions [4] when the constants like mass and charges are considered to be effectively anisotropically polarized by some anholonomic (anisotropic) higher dimension interactions; in our case is to consider that the parameter m 0 from (7) (see also (38)) is not a constant but a renormalized value m 0 → m = m(r, θ, s).

Taub NUT metrics with anisotropic running of constants on s = ς
We generate from the isotropic solution (38) a new one, anisotropic, in this manner: Let us consider the case when h 4 (r, θ) = r 2 sin 2 θ when h * 4 = 0, but h * 5 = 0. Following the solution (36) we parameterize and is just the function (2) but defined by a renormalized value m(r, θ, ς).
So, a parametrization of the ansatz (11), defines a locally anisotropic solution of the vacuum Einstein equations (27)-(30) generalizing the Taub NUT solution (7). We can treat the solution (41) as describing an anisotropic Kaluza-Klein monopole with running constant (on extra dimension coordinate) obtained by embedding the anisotropic Taub NUT gravitational instanton into 5D theory, adding the coordinate in a way as to be compatible with running of constant of effective magnetic configurations (in brief, we shall call such solutions as ς-solutions).
We conclude that the solutions describing gravitational monopoles and instantons [12,14] can be generalized to some anisotropic configurations with running constants.

4.1.2
Taub NUT metric with anisotropy of constants on angle ϕ In a similar fashion we can consider anisotropic (angular) dependencies of constants with s = ϕ (in brief, we call such solutions as ϕ-solutions). The simplest way is to take h * 5 = 0 but h * 4 = 0, i.e to define a solution with where V (r, θ, ϕ) is the function (2)  The d-metric δs 2 = −dt 2 + dr 2 + r 2 dθ 2 + 16m 2 0 exp[̟(r, θ, ϕ)] V 2 (r, θ, ϕ)(dϕ + cos θdθ) 2 (43) +r 2 sin 2 θ dς + n 2 [1] (r, θ)dr ̟(r, θ, ϕ)dϕ + n 3 [1] (r, θ)dθ ̟(r, θ, ϕ) 2 , models a locally anisotropic generalization of the solution (7) for anisotropic dependencies of the constant m on angle ϕ which describe a 5D Kaluza Klein monopole with angular anisotropic constant obtained by embedding the anisotropic Taub NUT gravitational instanton into 5D theory. So, in this subsection, we constructed two classes of generalized, anisotropic, Taub NUT solutions of the 5D vacuum Einstein equations: the first class is for s = ϕ (i.e. anisotropic polarizations) and the second is for s = ς (i.e.with dependence of the constant m on the fifth coordinate). The metric (11), describing these two classes of solutions, can be written with respect to a coordinate frame (8) where the existence of non-diagonal terms is emphasized.

Elliptic polarizations for ς-solutions
The simplest way of definition of such polarizations for ς-solutions (36) and (39) is to consider the off-diagonal metric the function V (r, θ, ς) is the function (2) redefined by an ellipsoidally renor- where there is also a linear dependence on extradimension coordinate ς; for simplicity, we can choose ̟(r, θ) = ̟ 0 = const.

Elliptic polarizations for ϕ-solutions
Such ϕ-solutions with elliptic variations of the constant m on angle θ are distinguished by the metric where V (r, θ, ϕ) generalize the functions (40), (42) for anisotropic dependencies like The constructed solutions of 5D vacuum Einstein equations, (44) and (45) contain some additional elliptic polarizations comparing with respective solutions (41) and (43). Such 5D Kaluza-Klein monopoles induced from 4D Taub NUT instantons behave theirselves as some objects with running on ς, or anisotropic on ϕ, constant m which is also elliptically polarized on the angle θ.

A Superposition of Wormhole and Taub NUT Metrics
By applying the method of anholonomic frames we can construct nonlinear anisotropic superpositions of the Taub NUT metric with some metrics defining wormhole / flux tube configurations: this way one defines anisotropic generalizations of the metric (1) which are called anisotropic Taub NUT wormholes. For simplicity, in this paper, we consider only a ς-solution for the Taub NUT configurations with the wormhole background chosen as to be locally isotropic (see Refs [13], on isotropic wormholes and flux tubes, and [4], on locally anisotropic wormhole solutions). For our purpose the ansatz (11) is generalized by introducing the coefficient N 6 1 = n 1 (r) = ω(r) of the nonlinear connection and, for simplicity, we consider w i = 0 and n 2 = 0.
The data defining a vacuum ς-solution for the (46), including the solution (41) into a wormhole background are given by the coefficients of the d-metric where V (r, ς) and V (r) are respectively those from (40) and (2) and the constant r 2 0 and term exp[2ψ(r)] were introduced as some multiples defining the wormhole / flux tube configuration.
The proprieties of the Taub NUT monopole with running constant m(ς) are the same as those stated for the solution (41), resulting in a similar to (3) magnetic field components = Q 0 /a(r), fields as it is defined respectively by the formulas (4) and (5).
As the free parameters of the wormhole background (see Ref [13]) are varied there are five classes of solutions with the properties: The number wormhole / flux tube classes must be extended to new configurations which arise in the presence of the magnetic field of the Taub NUT monopole, for instance, by considering the 6th class for the pure monopole configuration; we must take into account possible contribution of the monopole magnetic field to the structure of magnetic flux tubes, wormhole-like electromagnetic object and/or magnetic-electric flux tubes by analyzing the total magnetic field H (worm) KK + B (nut) with possible (elliptic and another type) vacuum gravitational polarizations.
Finally, in this section, we remark that in a similar fashion we can construct ϕ-solutions with an anisotropic parameter m(ϕ) describing an anisotropic Taub NUT monopole embedded into a wormhole / flux tube background and to generalized both ς-and ϕ-solutions for various configurations with elliptic polarizations and rotation hypersurface symmetries (like rotation ellipsoids, elliptic cylinders, bipolars and tori) as we constructed exact solutions of 3D-5D Einstein equations describing anisotropic black holes/tori and anisotropic wormholes [3,4,5].

Anisotropically Spinning Particles and Integrals of Motion
In the previous two sections we proved that the Taub NUT and wormhole metrics admit various type of anisotropic generalizations modeled by anholonomic frames with associated N-connection structure. It is of interest the investigation of symmetries of such anisotropic spaces and definition of corresponding invariants of motion of spinning particles. The general rules for developing of corresponding geodesic calculus and definition of generalized anisotropic Killing vectors are: • We have to change the partial derivatives and differentials into Nelongated ones (8) and (9) by redefinition of usual formulas for developing a formalism of differential, integral and variational calculus on anisotropic spaces.
• The metric, linear connection, curvature and Ricci tensors have to be changed into respective d-objects (d-metric (14), d-connection (19) and corresponding curvature and Ricci d-tensors); the d-torsion on (pseudo) Riemannian spaces should be treated as an anholonomic frame effect which vanishes with respect to coordinate bases.
• By using d-metrics and d-connections the differential, integral and variational calculus on Riemannian manifolds is adapted to the anholonomic frame structure with associated N-connection. With respect to N-adapted frames (8) and (9) the geometry is similar to the usual Riemannian one when the anholonomic (constrained dynamics) is coded into the coefficients of N-connection modeling a local anisotropy.
• As a matter of principle, all constructions defined with respect to anholonomic bases can be removed with respect to usual coordinate frames, but in this case the metrics became generically off-diagonal and a number of symmetries (and their constraints) of manifolds are hidden in rather sophisticate structures and relations for redefined holonomic objects.
6.1 Moving of spinless particles in anisotropic Taub NUT spaces

Killing d-vectors:
The geodesic motion of a spinless particle of unit mass moving into a background stated by a d-metric g αβ = (g ij , h ab ) , see (14)) can be derived from the action: where τ is a parameter. The invariance of d-metrics defining anisotropic generalizations of Taub NUT metrics under spatial rotations and ς translations is generated by four Killing d-vectors which are obtained by anholonomic transforms of the usual Killing vectors into corresponding one with elongated partial derivatives ∂ µ → δ µ = (∂ i , δ a ) , where the partial derivative on the new 5th coordinate, are ∂ ∂ς = ∂χ ∂ς ∂ ∂χ , see (6). We write the Killing d-vectors as or, in details, following general considerations we give the formulas for arbitrary transforms of the 5th coordinate, ς → ς (χ, ....) , and elongations of derivatives of type δ/∂θ = ∂/∂θ − n 3 ∂/∂ς.

Energy and momentum d-vector:
We know that in the purely locally isotropic bosonic case such invariances correspond to conservation of angular momentum and "relative electric charge" [9,21]. For anisotropic Taub NUTS we can define similar objects by using anholonomic transforms of the values given with respect to coordinate bases into the corresponding values with anisotropic coefficients and variables given with respect to anholonomic frames: where p = d r/dτ is the "mechanical momentum" which is only part of the momentum canonically conjugate to r (in anisotropization of the momentum formula we do not write the multiple V −1 because the d-metric used for anisotropic constructions has its locally isotropic limit being multiplied on the V as a conformal factor, see (7)). The energy is defined by is also conserved , Π µ = (Π i , Π a ) being the covariant momentum d-vector; this value variates with respect to coordinate frames but behaves itselve as a usual energy with respect to N-adapted frames.

Runge-Lenz d-vector:
There is a conserved vector analogous to the Runge-Lenz vector of the Coulomb problem in the locally isotropic case [9,21,22], which with respect to anholonomic frames with anisotropic variables in spite of the fact of complexity of anholonomic motion in the anisotropic Taub-NUT spaces defined in previous Sections, which implies that the trajectories are anisotropic deformations of conic sections.

Spinning of particles with respect to anholonomic frames
The pseudo-classical limit of the Dirac theory of a spin 1/2 fermion in curved spacetime is described by the supersymmetric extension of the usual relativistic point-particle [23] (the theory of spinors on spaces with generic local anisotropy was developed in Refs. [17], see also some models of locally anisotropic supergravity and superstring theories in Refs. [16]). In this work, the configuration space of spinning particles in anisotropic space(anisotropic spinning space) is an extension of an ordinary Riemannian manifold provided with an anholonomic frame and associated N-connection structure, parametrized by local coordinates {u µ = (x i , y a )}, to a graded manifold parametrized by local coordinates {u µ , ψ µ }, with the first set of variables being Grassmann-even (commuting) and the second set Grassmann-odd (anti-commuting). We emphasize that in anholonomic spaces distinguished by a N-connection structure we must define spinor and Grassmann variables separately on h-subspace (with holonomic variables) and on v-subspace (with anholonomic variables).

Action for anisotropically spinning particles:
The equation of motion of an anisotropically spinning particle on a autoparallel (geodesic) is derived from the action: The corresponding world-line anholonomic Hamiltonian is given by: dτ } is the covariant momentum d-vector.

Poisson-Dirac brackets:
For any integral (we use the term integral instead of the usual one, constant, because on anisotropic spaces we can define any conservations lows with respect to N-adapted anholonomic frames; the invariants of such conservations laws are not constant with respect to coordinate frames) of anholonomic motion J(u, Π, ψ), the bracket with H vanishes, {H, J} = 0, where the Poisson-Dirac brackets for functions of the covariant phase-space variables (u, Π, ψ) is defined In definition of {...} there are used the operators where on anisotropic spaces Γ λ µν is the canonical d-connection (19), R ρσµν is the curvature d-tensor (21) with components (22) and a F is the Grassmann parity of F : a F = (0, 1) for F =(even,odd).

Anisotropic Killing equations:
Expanding J(u, Π, ψ) in a power series on the canonical momentum, J = ∞ n=0 1 n! J (n)µ 1 ...µ n (u, ψ)Π µ 1 . . . Π µ n we conclude that the bracket {H, J} vanishes for arbitrary Π µ if and only if the components of J satisfy the generalized anisotropic Killing equations [23] : where the round brackets (....) denote full symmetrization over the indices enclosed and the covariant derivation ";" is defined by the canonical dconnection (19) and one should be emphasized that every Greek index split into horizontal and vertical groups like µ n = (i n , a n ) which results that this equation will contain both "pure" horizontal or vertical terms as well terms of "mixed" character, like J (n) i 1 ...an . The type of solutions of the generalized anisotropic Killing equations (49) is defined by two classes from the locally isotropic limit [24,25]: the first class of solutions are generic ones, which exists for any spinning particle model and the second class of solutions are non-generic ones, which depend on the specific background space and anisotropy considered.

Generic solutions of Killing equations:
The proper-time translations and supersymmetry are generated by the Hamiltonian and supercharge and belong to the first class. There is also an additional "chiral" symmetry generated by the chiral charge and a dual supersymmetry with generator where d is the dimension of spacetime.

Non-generic solutions and Killing-Yano d-tensors:
The non-generic conserved quantities depend on the explicit form of the metric g µν (u) and, in our case, on N-connection structure. Following Ref. [25], generalizing the constructions for anisotropic spaces, we introduce the Killing-Yano d-tensors as objects generating non-generic N-distinguished supersymmetries. A d-tensor f µ 1 ...µ r is called Killing-Yano of valence r if it is totally antisymmetric and satisfies the equation f µ 1 ...µ r−1 (µ r ;λ) = 0. The method of solution of the system of coupled differential equations (49) is similar to the method developed for locally isotropic spaces [9], that why here we present only the key results which have to be split on h-and vindices if we need explicit formulas for holonomic-anholonomic components.
We start with aJ (n) µ 1 ...µ n solution of the homogeneous equation: This solution is introduced in the r.h.s. of (49) for J (n−1) µ 1 ...µ n−1 and the iteration is carried on to n = 0.
For the bosonic case the first equation shows that J 0 is a trivial constant, the next one is the equation for the Killing d-vectors and so on. In general, the homogeneous equation for a given n defines a Killing d-tensor J (n) µ 1 ...µ n for which J (n) µ 1 ...µ n Π µ 1 . . . Π µ n is a first integral of the geodesic equation [26]. This does not hold for the spinning particles.
Let us consider the case n = 0, whenJ (0) = i 4 f µν ψ µ ψ ν is a solution if f µν is a Killing-Yano d-tensor -covariantly constant, i. e.J (0) is a separately conserved quantity. For n = 1, the natural solution is: where R µ is a Killing d-vector (R (µ;ν) = 0) and f λσ is a Killing-Yano d-tensor d-covariantly constant. Introducing this solution in the r.h.s. of the equation (49) with n = 0, we get where the square bracket denotes the antisymmetrization with norm one.
We define a new integral of anholonomic motion which is peculiar to the spinning case and has its analogous in the locally isotropic limit: We can generate another ψ-dependent solution of the n = 1 by starting from a Killing-Yano d-tensor with r indices,J (1) µ 1 = f µ 1 µ 2 ...µ r ψ µ 2 . . . ψ µ r , or, following the above prescription, we express stating that the integral of motion corresponding to these solutions of the Killing equations is: We conclude that the existence of a Killing-Yano d-tensor with r indices is equivalent to the existence of a supersymmetry for the spinning space with supercharge Q f which anticommutes with Q 0 , such constructions are anholonomic and distinguished by the N-connection structure.

Anisotropic Taub-NUT spinning space
There are four Killing-Yano tensors in the usual, locally isotropic, Taub-NUT geometry [27] which for anisotropic spaces are transformed into corresponding d-tensors for anisotropic Taub NUT spaces, which are d-covariantly constant and the fourth Killing-Yano d-tensor is f Y = 8m(r, θ, s)δς ∧ dr + 4r(r + 2m(r, θ, s))(1 + r 4m(r, θ, s) ) sin θdθ ∧ δϕ having only one non-vanishing component of the field strength f Y rθ;ϕ = 2(1+ r/4m(r, θ, s))r sin θ, where δς = dς + cos θdθ, and the values δx j are Nelongated for the v-components. The corresponding supercharges constructed from the Killing-Yano dtensors are Q i and Q Y . The supercharges Q i together with Q 0 from (50) realize the N=4 supersymmetry algebra [28], in our case distinguished by the N-connection structure [16] : {Q A , Q B } = −2iδ AB H , A, B = 0, . . . , 3 making manifest the link between the existence of the Killing-Yano d-tensors and the hyper-Kähler d-geometry of the anisotropic Taub-NUT manifold. Such distinguished manifolds and geometries are constructed as the usual ones but with respect to N-connection decompositions on holonomic-anholonomic variables.
Starting with these results from the bosonic sector of the Taub-NUT space one can proceed with the spin contributions. The first generalized Killing equation (49) shows that with each Killing vector R µ A there is an associated Killing scalar B A . The expression for the Killing scalar is taken as in Ref. [28]: with that modification that we use a d-covariant derivation which gives that the total angular momentum and "relative electric charge" become in the anisotropic spinning case J Aj = f j λ σ ψ λ ψ σ R Aµ Π µ + i 2 R A[α;β] ψ α ψ β = −4iS j J A , A = 0, . . . , 3; j = 1, 2, 3.
So, there are a sort of Runge-Lenz d-vectors involving only Grassmann components: with the commutation relations: Finally, we note the following Dirac brackets of L i with supercharges: and emphasize that the presented algebraic relations hold true for anisotropic (as on some parameters) dependencies of the constant m = m (r, θ, s) because we are working with respect to anholonomic frames locally adapted to the N-connection structure.

Conclusion Remarks
In this paper, we have extended the method of construction of new exact solutions, with generic local anisotropy, of the Einstein equations by using anholonomic frames with associated nonlinear connection structure (the method was proposed and developed in Refs. [3,16,17,15]) in order to generate vacuum metrics defining locally anisotropic Taub NUT instanton and Kaluza Klein monopoles. Such metrics are off-diagonal with respect to usual coordinate bases and reflect possible interactions of gravitational fields with gauge fields, induced from higher dimension gravity, in a new fashion when the constants interactions could run on the 5th dimension and/or polarized to some anisotropic configurations on angular coordinates.
The first key result of this paper is the proof that the introduced ansatz for the metric and anholonomy coefficients substantially simplifies the structure of resulting vacuum gravitational field equations, the variables being separated and the system of nonlinear partial equations admitting exact solutions. In consequence, a straightforward definition of new classes of anisotropic Taub NUT metrics with the effective constant m varying on the 5th coordinate, as well with anisotropies and elliptic polarizations on angular coordinates, was possible. There were emphasized classes of anisotropic Taub NUT wormhole solutions which can be generalized to various type of rotation hypersurface backgrounds and deformations.
The second key result is connected with the definition of integrals of motion of scalar and spinning particles in curved spacetimes provided with anholonomic frame structure. We proved that the symmetries of such generalized anisotropic Taub NUT spaces are connected with anholonomic Killing vectors and tensors which are subjected to some anholonomic conservation laws. The problem of generation of non-generic anisotropic supersymmetries was solved by introducing Killing-Yano tensors adapted to the anholonomic spacetime structure.
Finally, we note that the results of this paper were extended for 3D solitonic configurations [5].