Off-Diagonal Deformations of Kerr Black Holes in Einstein and Modified Massive Gravity and Higher Dimensions

We find general parameterizations for generic off-diagonal spacetime metrics and matter sources in general relativity (GR) and modified gravity theories when the field equations decouple with respect to certain types of nonholonomic frames of reference. This allows us to construct various classes of exact solutions when the coefficients of the fundamental geometric/ physical objects depend on all spacetime coordinates via corresponding classes of generating and integration functions and/or constants. Such (modified) spacetimes display Killing and non-Killing symmetries, describe nonlinear vacuum configurations and effective polarizations of cosmological and interaction constants. Our method can be extended to higher dimensions which simplifies some proofs for embedded and nonholonomically constrained four-dimensional configurations. We reproduce the Kerr solution and show how to deform it nonholonomically into new classes of generic off-diagonal solutions depending on 3-8 spacetime coordinates. Certain examples of exact solutions are analyzed and that are determined by contributions of new type of interactions with sources in massive gravity and/or modified f(R,T) gravity. We conclude that by considering generic off-diagonal nonlinear parametric interactions in GR it is possible to mimic various effects in massive and/or modified gravity, or to distinguish certain classes of"generic"modified gravity solutions which cannot be encoded in GR.


Introduction
The gravitational field equations in general relativity, GR, and modified gravity theories, MGT, are very sophisticate systems of nonlinear partial differential equations (PDEs). Advanced analytic and numerical methods are necessary for constructing exact and approximate solutions of such equations. A number of examples of exact solutions are summarized in the monographs [1,2] where the coefficients of the fundamental geometric/physical objects depend on one and/or two coordinates in four dimensional (4-d) spacetimes and when the diagonalization of the metrics is possible via coordinate transformations. There are well known physically important exact solutions for the Schwarzschild, Kerr, Friedman-Lemaître -Robertson-Worker (FLRW), wormhole spacetimes etc. These classes of solutions are generated by certain ansatzes when the Einstein equations are transformed into certain systems of nonlinear second order ordinary equations (ODE), 2-d solitonic equations etc. Such systems of PDEs display Killing vector symmetries which results in additional parametric symmetries [3,4,5].
The problem of constructing generic off-diagonal exact solutions (which can not be diagonalized via coordinate transformations) with metric coefficients depending on three and/or four coordinates is much more difficult. There are, in general, six independent components of a metric tensor from the ten components in a 4-d (pseudo) Riemannian spacetime 1 . Any such ansatz transforms the Einstein equations into systems of nonlinear coupled PDEs which cannot be integrated in a general analytic form if the constructions are performed in local coordinate frames.
In a series of works [6,5,7,8,9], we have shown that it is possible to decouple the gravitational field equations and perform formal analytic integrations in various theories of gravity with metric and nonlinear, N-, and linear connections structures. To prove the decoupling property in a simplest way we have to consider spacetime fibrations with splitting of dimensions, 2 (or 3) + 2 + 2 + ..., introduce certain adapted frames of reference, consider formal extensions/embeddings of 4-d spacetimes into higher dimensional ones and work with necessary types of linear connections. Such an (auxiliary, in Einstein gravity) adapted connection is also completely defined in a compatible form by the metric structure and contains a nonholonomically induced torsion field. This allows us to decouple the gravitational field equations and generate various classes of exact solutions in generalized/modified gravity theories. After a class of generalized solutions has been constructed in explicit form, we can constrain to zero the induced torsion fields and "extract" solutions in Einstein gravity. We emphasize that it is important to impose the zero-torsion conditions after we found a class of generalized solutions (contrary, we cannot decouple the corresponding systems of PDEs).
It should be noted here that the off-diagonal solutions constructed following the above described anholonomic frame deformation method, AFDM, depend on various classes of generating and integration functions and parameters. The Cauchy problem can be formulated with respect to necessary types of N-adapted frames; it is possible to generate various stable or un-stable solutions with singularities, nontrivial deformed horizons, stochastic behavior, etc ... which depends on the type of nonlinear couplings, prescribed symmetries, asymptotic and boundary conditions, see a number of examples in [10,11,12,13] and references therein. In general, it is not clear what physical importance (if any ?) these classes of such solutions may have. For some well defined conditions, we can speculate about black hole/ellipsoid/wormhole configurations embedded, for instance, into solitonic gravitational backgrounds or to consider small ellipsoidal deformations of certain "primary" spherical/cylindrical configurations.
Our geometric techniques of constructing exact solutions can be applied to four dimensional, 4-d, (pseudo) Riemannian spacetimes with one and two Killing symmetries. For such configurations, the well known Kerr solution can be generated as a particular case. Then these "primary" metrics can be subjected to nonholonomic deformations to "target" off-diagonal exact solutions depending on three, or four, spacetime coordinates.
The first goal of this paper is to show how certain primary physically important solutions depending on two coordinates can be generalized to new classes of exact solutions in Einstein gravity and (higher dimensional) modifications, with zero or nonzero torsion, depending on all possible spacetime coordinates. We consider diagonal and off-diagonal parametrization of primary and target solutions which are different from those in [6,5,7,8] and other works. In this way we generate new classes of Einstein spacetimes and modifications and show that the AFDM encodes various possibilities for generalization.
The second goal is to construct explicit examples of exact solutions as nonholonomic deformations of the Kerr metric determined by nontrivial sources and interactions in massive gravity and/or modified f (R, T ) gravity, see reviews and original results in Refs. [14,15,16,17,18,19,20,21,22,23,24]. For non-Hilbert Lagrangians in gravity theories, the functionals f depend on scalar curvature R (computed, in general, for a linear connection with nontrivial torsion, or for the Levi-Civita one), on various matter and effective matter sources for modified gravity theories etc. We provide a series of exact and/or small parameter-dependent solutions which for small deformations mimic rotoid Kerr -de Sitter like black holes/ellipsoids self-consistently imbedded into generic off-diagonal backgrounds of 4/ 6/ 8 dimensional spacetimes. With respect to nonholonomic frames and via the re-definition of generating and integrations functions and coefficients of the sources, modifications of Einstein gravity are modelled by effective polarized cosmological constants and off-diagonal terms in the new classes of solutions. For certain geometrically well defined conditions, various effects in massive and f -modified gravity can be encoded into vacuum and non-vacuum, configurations (exact solutions) with nontrivial effective cosmological constants in GR. In some sense, we can mimic physically important effects in modified gravity effects (for instance, acceleration of universe, certain dark energy and dark matter locally anisotropic interactions, effective renormalization of quantum gravity models, see Refs. [13,25,26]) via nonlinear generic off-diagonal interactions on effective Einstein spaces. The main question arising from such models and solutions is whether or not we need to modify Einstein's gravitational theory, or to try and solve physically important issues in modern cosmology and quantum gravity by considering only nonlinear and generic off-diagonal interactions based on the general relativity paradigm. There is necessarily additional theoretical and experimental/observational research which is required in order to analyze and solve these problems. Such directions of research cannot be developed if we consider only diagonalizable (and rotating ones, like the Kerr metric) metrics generated by an ansatz with two Killing symmetries.
The plan of the paper is as follows: In section 2 we provide the necessary geometric preliminaries on nonholonomic 2+2+2+... splittings of the spacetime dimensions in GR and MGT. We summarize the key results on the AFDM for constructing generic off-diagonal solutions in gravity theories depending on all spacetime coordinates in dimensions 4,5,..., 8. In section 3 we prove the general decoupling property of the (modified) Einstein equations which allows us to perform formal integrations of corresponding systems of nonlinear PDE. The geometric constructions are performed for the "simplest" case of one Killing symmetry in 4-d and generalized to non-Killing configurations and for higher dimensions. Section 4 is devoted to the theory of nonholonomic deformations of exact solutions in modified gravity theories containing the Kerr solution as a "primary" configurations but with target metrics being constructed as exact solutions in massive gravity and/or f -modified gravity. We show how using the AFDM we can generate as a particular case the Kerr solution. Then we construct solutions with general off-diagonal deformations of the Kerr metrics in 4-d massive gravity, provide examples of (non-Einstein) metrics with nonholonomically induced torsions and study small f -modifications of the Kerr metrics deformed by massive gravity. A separate subsection is devoted to ellipsoidal 4-d deformations of the Kerr metric resulting in a target vacuum rotoid or Kerr-de Sitter configurations. Another subsection is devoted to extra dimensional massive off-diagonal modifications of the Kerr solutions, for the case of 6-d spacetime with nontrivial cosmological constant and for 8-d deformations which may model Finsler-like configurations.
Finally (in section 5), we provide our conclusions and speculate on the physical meaning of the exact solutions constructed using the AFDM for massive modified gravity theories and how such effects can be modelled by nonlinear off-diagonal interactions in Einstein gravity. Some relevant formulae for the coefficients and sketches of the proofs are presented in the Appendix.
2 Nonholonomic Frames with 2+2+.... Splitting In this section, we state the geometric conventions and outline the formalism which are necessary for decoupling and integrating the gravitational field equations in GR and MGTs, see relevant details in [6,5,7,8].
Local frames (bases, e αs ) on s V are denoted in the form where partial derivatives are ∂ βs := ∂/∂u βs , and indices are underlined if it is necessary to emphasize that such values are defined with respect to a coordinate frame. In general, the frames (1) are nonholonomic (equivalently, anholonomic, or non-integrable), e αs e βs − e βs e αs = W γs αsβs e γs , where the anholonomy coefficients W γs αsβs = W γs βsαs (u) vanish for holonomic, i. e. integrable, configurations. The dual frames are e αs = e αs α s ( s u)du α s , which can be defined from the condition e αs ⌋e βs = δ αs βs (the 'hook' operator ⌋ corresponds to the inner derivative and δ αs βs is the Kronecker symbol). The conventional 2 + 2 + ... splitting for a metric is written in the form s g =g αsβs e αs ⊗ e βs = g α s β s du α s ⊗ du β s , s = 0, 1, 2, ..., where coefficients of the metric transform following the rule Similar frame transforms can be considered for all tensor objects. We can not preserve a splitting of dimensions under general frame/coordinate transforms.

Nonholonomic splitting with associated N-connections
To prove the general decoupling property of the Einstein equations and generalizations/ modifications we have to construct a necessary type of nonholonomic 2 + 2 + ... nonholonomic splitting with associated nonlinear connection (N-connection) structure. Such a splitting is introduced using nonholonomic distributions: 3 2 In a similar form, we can split odd dimensions, for instance, dim V = 3 + 2 + ... + 2. Here it should be noted that it is not possible to elaborate any simplified system of notations if we want to integrate in general explicit form certain systems of PDEs related to higher dimensional gravitational theories. It is important to distinguish indices and coordinates corresponding to higher dimensions and nonholonomically constrained variables. 3 In modern gravity, it is largely used the so-called ADM (Arnowit-Deser-Misner) formalism with a 3+1 splitting, see details in [27]. It is not possible to elaborate a technique for a general decoupling of the gravitational field equations and generating off-diagonal solutions if we use only nonholonomic frame bases determined by the "shift" and "lapse" functions. To construct exact solutions is more convenient to work with a correspondingly defined non-integrable 2+2+... splitting [7,8].

A N-connection is stated by a Whitney sum
for a conventional horizontal (h) and vertical (v) "shell by shell" splitting. We shall write boldface letters for spaces and geometric objects enabled/adapted to N-connection structure. This defines a local fibered structure on s V when the coefficients of N-connection, N as is , for s N = N as is ( s u)dx is ⊗ ∂/∂y as , induce a system of N-adapted local bases, with N-elongated partial derivatives, e νs = (e is , e as ), and cobases with N-adapted differentials, e µs = (e is , e as ). On a 4-d V, and on s ≥ 1 shells, e is = dx is , e as = dy as + N as is dx is .
The N-adapted operators (5) and (7) define a subclass of general frame transforms of type (1). The corresponding anholonomy relations [e αs , e βs ] = e αs e βs − e βs e αs = W γs αsβs e γs , are completely defined by the N-connection coefficients and their partial derivatives, W bs isas = ∂ as N bs is and W as jsis = Ω as isjs , where the curvature of the N-connection is Ω as isjs = e js N as is − e is N as js .
2. Any metric structure s g = {g αsβs } on s V can be written as a distinguished metric (d-metric) 4 s g = g isjs ( s u) e is ⊗ e js + g asbs ( s u)e as ⊗ e bs (10) = g ij (x) e i ⊗ e j + g ab (u) e a ⊗ e b + g a 1 b 1 ( 1 u) e a 1 ⊗ e b 1 + .... + g asbs ( s u)e as ⊗ e bs .
In coordinate frames, a metric (2) is parameterized by generic off-diagonal matrices For extra dimensions, such parameterizations are similar to those introduced in the Kaluza-Klein theory when y as , s ≥ 1, are considered as extra dimension coordinates with cylindrical compactification and N es α ( s u) ∼ A es asα (u)y α are for certain (non) Abelian gauge fields A es asα (u). In general, various parameterizations can be used for warped/trapped coordinates in brane gravity and modifications of GR, see examples in [10,11,12,13].

The Levi-Civita and auxiliary N-adapted connections
There is a subclass of linear connections on s V which are adapted to the N-connection splitting (4). By definition, a distinguished connection, d-connection, D = (hD; vD), 1 D =( 1 hD; 1 vD), ..., s D =( s−1 hD; s vD), preserves under parallelism the N-connection structure. 5 The coefficients of a d-connection s D = {D αs } can be computed in N-adapted form with respect to frames (5)-(8) following equations D αs e βs = Γ αs βsγs e γs and covariant derivatives parameterized in the form Such coefficients can be computed with respect to mixed subsets of coordinates and/or N-adapted frames on different shells. It is possible always to consider such frame transforms when all shell frames are N-adapted and To perform computations in N-adapted-shell form we can consider a differential connection 1-form Γ αs βs = Γ αs βsγs e γs and elaborate a differential form calculus with respect to skew symmetric tensor products of Nadapted frames (5)- (8). For instance, the torsion T αs = {T αs βsγs } and curvature R αs βs = {R αs βsγsδs } d-tensors of s D can be computed respectively, T αs := s De αs = de αs + Γ αs βs ∧ e βs (12) R αs βs := s DΓ αs βs = dΓ αs βs − Γ γs βs ∧ Γ αs γs = R αs βsγsδs e γs ∧ e δs , see Refs. [8] for explicit calculation of the coefficients R αs βsγsδs in higher dimensions. For any (pseudo) Riemannian metric s g, we can construct in standard form the Levi-Civita connection (LCconnection), s ∇ = { Γ αs βsγs }, which is completely defined by the metric coefficients following two conditions: This linear connection is metric compatible, s ∇( s g) = 0, and with zero torsion, T αs = 0 (see formulas (12) for s D → s ∇). Such a linear connection is not a d-connection because it does not preserve under general coordinate transforms a N-connection splitting.
To elaborate a covariant differential calculus adapted to decomposition (4) we have to introduce a different type of linear connection. This is the canonical d-connection s D which is completely and uniquely defined by a (pseudo) Riemannian metric s g g ir (e k g jr + e j g kr − e r g jk ) , L α βγ = 1 2 g ατ (e γ g βτ + e β g γτ − e τ g βγ ) , h asds (e cs h bsds + e cs h csds − e ds h bscs ) .
The torsion d-tensor (12) of s D is completely defined by s g (2) for any chosen s N = {N as is } if the above coefficients (14) are introduced "shell by shell" into formulas T αs βsγs = L αs βsγs − L αs γsβs , T αs βsbs = C αs βsbs , T as βsγs = Ω as γsβs .
The N-adapted formulas (14) and (15) show that any coefficient for such objects computed in 4-d can be similarly extended shell by shell by any value s = 1, 2, .... redefining correspondingly the h-and v-indices. Hereafter, we shall present coordinate formulas only for s = 0, omitting label s, i.e. with α = (i, a), or for some arbitrary coefficients α s = (i s , a s ) if that will not result in ambiguities. Because both linear connections s ∇ and s D are defined by the same metric structure, we can compute a canonical distortion relation where the distorting tensor s Z = { Z αs βsγs } is uniquely defined by the same metric s g (2). The values Z αs βsγs are algebraic combinations of T αs βsγs and vanish for zero torsion. For instance, the GR theory in 4-d can be formulated equivalently using the connection ∇ and/or D if the distorting relation (16) is used [5,7]. The nonholonomic variables ( s g (10), s N, s D) are equivalent to standard ones ( s g (2), s ∇). Here we note that s ∇ and s D are not tensor objects and such connections are subjected to different rules of coordinate transforms. It is possible to consider frame transforms with certain s N = {N as is } when the conditions Γ γs αsβs = Γ γs αsβs are satisfied with respect to some N-adapted frames (5)-(8) even, in general, s ∇ = s D and the corresponding curvature tensors R αs βsγsδs = R αs βsγsδs .

The Einstein equations in N-adapted variables
An important motivation to use the linear connection s D is that the Einstein equations written in variables ( s g , s N, s D) decouple with respect to N-adapted frames of reference which gives us the possibility to construct very general classes of solutions, see proofs and examples in [6, 5, 7, 8, 10, 11, 12, 13]. We cannot "see" a general decoupling property for such nonlinear systems of PDE if we work from the very beginning with s ∇, for instance, in coordinate frames or with respect to arbitrary nonholonomic ones: The condition of zero torsion, T αs = 0 states "strong coupling" conditions between various tensor coefficients in the Einstein equations and does not allow to decouple the equations. 6 The main idea of the "anholonomic frame deformation method", AFDM, is to use the data ( s g , s N, s D) in order to decouple certain gravitational and matter field equations, then to solve them in very general off-diagonal form, with possible dependence on all coordinates, and generate exact solutions with nontrivial nonholonomically induced torsion. Such integral varieties of solutions depend on a number of arbitrary generating and integration functions and possible symmetry parameters. This geometric approach can be applied for constructing exact solutions in various modified gravity theories with nonlinear effective Lagrangians and nontrivial torsion. Nevertheless, we can extract "integral subvarieties" of solutions in GR if at the end (after a class of "generalized" solutions was constructed) we impose, additionally, the condition of zero torsion (15). This constrains the set of admissible generating/integration functions but also results in generic off-diagonal solutions depending on all coordinates. We can impose certain symmetry/ asymptotic / boundary / Cauchy conditions in order to determine certain geometrically/physically important off-diagonal configurations. Following additional assumptions, this can be related to small parametric off-diagonal, solitonic or other type, deformations of well known solutions in GR. The goal of this work is to study possible nonholonomic transformations of the Kerr and several wormhole metrics into off-diagonal (4-d or higher dimension) exact solutions.
The Ricci d-tensor Ric = {R αsβs := R τs αsβsτs } of a d-connection s D is introduced via a respective contracting of coefficients of the curvature tensor (13). The explicit formulas for h-/ v-components, are direct recurrent s-modifications of those derived in Refs. [6,5,7,8] (we do not repeat such details in this article). Contracting such values with the inverse d-metric, with coefficients computed for the inverse matrix of s g (10), we define and compute the scalar curvature of s D, with respective h-and v-components of scalar curvature, The Einstein d-tensor s E = {E αsβs } for any data ( s g , s N, s D) can be defined in standard form, It should be noted that s D( s E) = 0 and the d-tensor R αsβs is not symmetric for a general s D. Nevertheless, we can always compute, for instance, s D( s E) as a unique distortion relation determined by (16). This is a consequence of nonholonomic splitting structure (4). It is similar to nonholonomic mechanics when the conservation laws became more sophisticate when we impose certain non-integrable constraints on the dynamical equations. The Einstein equations for a metric g βsγs can be postulated in standard form using the LC-connection s ∇ (with corresponding Ricci tensor, R αsβs , curvature scalar, s R, and Einstein tensor, E αsβs ), where κ is the gravitational constant and T αsβs is the stress-energy tensor for matter fields. In 4-d, there are well-defined geometric/variational and physically motivated procedures of constructing T αsβs . Such values can be similarly (at least geometrically) re-defined with respect to N-adapted frames using distorting relations (16) and introducing extra-dimensions. 7 The gravitational field equations (20) can be rewritten equivalently in N-adapted form for the canonical d-connection s D, L cs asjs = e as (N cs js ), C is jsbs = 0, Ω as jsis = 0, where the sources Υ βsδs are formally defined in GR but for extra dimensions when Υ βsδs → κT βsδs for s D → s ∇. The solutions of (21) are found with nonholonomically induced torsion (12). If the conditions (22) are satisfied, the d-torsion coefficients (15) are zero and we get the LC-connection, i.e. it is possible to "extract" solutions of the standard Einstein equations. The decoupling property can be proved in explicit form working with s D and nonholonomic torsion configurations. Having constructed certain classes of solutions in explicit form, with nonholonomically induced torsions and depending on various sets of integration and generating functions and parameters, we can "extract" solutions for s ∇ imposing at the end additional constraints resulting in zero torsion.

Nonholonomic massive f(R,T) gravity and extra dimensions
We shall consider modified gravity theories constructed on dimension shells derived for the action This generalizes in nonholonomic variables the modified f (R, T ) gravity, see reviews in [14,15,16,17], and the ghost-free massive gravity (by de Rham, Gabadadze and Tolley, dRGT) [18,19,20]. Nontrivial mass terms allow us to solve certain problems of the bimetric theory by Hassan and Rosen, [21,22], with connections to various recent research in black hole physics and modern cosmology [23,24], and allows us to model solutions of (23) in various theories with generalized Finsler branes, stochastic processes, Clifford and phase variables, fractional derivatives etc, see details in Refs. [28,29,25,30,31,32,34]. For instance, y as -coordinates can be treated as "velocity/momentum" variables, to model stochastic and fractional processes, or to be considered as "standard" extra dimensional ones. In this paper, we shall use the units = c = 1 and the Planck mass M P l is defined M 2 P l = 1/8πG via 4-d Newton constant G and similar units will be considered for higher dimensions. We write δ 4+2s u instead of d 4+2s u because N-elongated differentials are used (5) and consider the constant µ g as the mass parameter for gravity (for simplicity, massive gravity theories will be studied for 4-d spacetimes). The geometric and physical meaning of the values contained in this formula will be explained below.
The Lagrangian density m L in action (23) is used for computing the stress-energy tensor of matter. On nonholonomic manifolds/bundles such variations can be considered in N-adapted form, using operators (5) and (6), on inverse metric d-tensor (10). For all shells, we can compute , when the trace is (by definition) s T := g αsβs T αsβs . The functional f ( s R, s T ) modifies the standard Einstein-Hilbert Lagrangian (with a scalar curvature R usually taken for the Levi-Civita connection ∇) to that for the modified f -gravity in various dimensions but with dependence on s R and T. For various applications in modern cosmology, we can assume that for the approximation of perfect fluid matter with the energy density ρ and the pressure p. The four-velocity v αs is subjected to the conditions v αs v αs = 1 and v αs D βs v αs = 0, for m L = −p in a corresponding local N-adapted frame. For simplicity, we can parameterize and denote by 1 F ( s R) A mass term with "gravitational mass" µ g and potential is considered in (23) in addition to the usual f -gravity term (in particular, to the Einstein-Hilbert one). The trace of a shell extended matrix S = (S µsνs ) is denoted by [S] := S νs νs . We understand the square root of such a matrix, √ S = ( √ S νs µs ), to be a matrix for which √ S νs αs √ S αs µs = S νs µs and α 3 and α 4 are free parameters. We use such constants which transform U into standard 4-d one for s = 0. In works [19,20], see additional arguments in [35]), such a nonlinearly extended Fierz-Pauli type potential was shown to result in a theory of massive gravity which is seem to be free from ghost-like degrees of freedom (it takes a special form of total derivative in absence of dynamics). We emphasize that the potential generating matrix S is constructed in a special form which results in a d-tensor with shell decomposition , K νs µs = δ νs µs − √ S νs µs , characterizing metric fluctuations away from a fiducial (flat) 4-d spacetime and possible extra dimensions, or velocity/momentum type variables.
In 4-d, the coefficients with the Minkowski metric η νµ = diag(1, 1, 1, −1), are generated by introducing four scalar Stükelberg fields s ν , which is necessary for restoring the diffeomorphism invariance. Using N-adapted shell extended values g νsαs and e αs we can always transform a tensor S µν into shell distinguished d-tensor S µsνs characterizing nonholonomically constrained fluctuations. This is possible for the values K νs µs , S νs µs , √ S νs µs etc even shell extended s νs transforms as scalar fields under coordinate and frame transforms.
For simplicity, we can consider 4-d variations of the action (23) in N-adapted from for the coefficients of d-metric g να (10). The corresponding generalized/ effective Einstein equations, for the f -modified massive gravity are where the source encodes three terms of different nature, The first component is determined by usual matter fields with energy momentum T βδ tensor but with effective polarization of the gravitational constant ef η = [1 + 2 F/8π]/ 1 F. The second term is for the f -modifications of the energy-momentum tensor, The mass gravity contribution, i.e. the third term in source is computed as a dimensionless effective stressenergy tensor The value K T αβ encodes a bi-metric configurations when the second (fiducial) d-metric f αµ = η νµ e α s ν e µ s µ is determined by the Stükelberg fields s ν . The potential U (26) defines interactions between g µν and f µν via √ S ν µ = √ g νµ f αν and S ν µ := g νµ f αν . We can construct exact solutions in explicit form and study bi-metric gravity models with K T αβ = λ(x k ) g αβ , which can be generated by such configurations of s ν when g µν = ι 2 (x k )f µν with a possible nontrivial conformal factor ι 2 . Such nonholonomic configurations allow us to compute using (27) a diagonal matrices S ν µ := ι −2 δ ν µ . We can express the effective polarized anisotropic constant encoding the contributions of s ν as a functional λ[ι 2 (x k )].
The theories with gravitational field equations (28) are similar to the Einstein one but for a different metric compatible linear connection, D, and with nonlinear "gravitationally polarized" coupling in effective source Υ βδ (29). In next sections, we shall prove that such nonlinear systems of PDE can be integrated in general forms for any N-adapted parameterizations In particular, we can consider for an effective cosmological constant Λ, see details in [5,6,7,8,9]. It should be noted that D δ 1 F |Υ=Λ = 0 in (30) if we prescribe a functional dependence on R = const (we have to chose necessary types of N-coefficients and respective canonical d-connection structure). For certain general distributions of matter fields and effective matter, we can prescribe such values for (32) with T βδ =Ť (x k )g βδ and s R = Λ in (31), then we can write In general, any term may depend on coordinates x i but via re-definition of generating functions they can be transformed into certain effective constants. Prescribing the values Λ,Ť , λ, p and functionals 1 f and 2 f, we describe a nonholonomically constrained matter and effective matter fields dynamics with respect to N-adapted frames.
All above constructions can be extended to extra shells s = 1, 2, ... via formal re-definition of indices for higher dimension. Under very general assumptions, the effective source can be parameterized in the form This formal diagonal form is fixed with respect to N-adapted frames and (see next section) for corresponding re-definition of certain generation functions. Such ( s Λ + s λ)-terms encode via nonholonomic constraints and the canonical d-connection s D various physically important information on modifications of the GR theory by modifications in f -functional and/or massive gravity theories of various dimensions. LC-configurations can be extracted in all such types of theories by imposing additional constraints when D T =0 → ∇.

Decoupling & Integration of (Modified) Einstein Equations
In this section, we show how the gravitational field equations (21) with possible constraints (22), or (20), can be formally integrated in very general forms for generic off-diagonal metrics with coefficients depending on all spacetime coordinates.

Off-diagonal configurations with one Killing symmetries
In the simplest form, the decoupling property can be proven for certain ansatz with at least one Killing symmetry.
h a 1 (u α , y 6 ) e a 1 ⊗ e a 1 + h a 2 (u α 1 , y 8 ) e a 2 ⊗ e b 2 + .... + h as ( u α s−1 , y as )e as ⊗ e as , 8 in our former works, we used a quite different system of notation Such ansatz contains a Killing vector ∂/∂y s−1 because the coordinate y s−1 is not contained in the coefficients of such metrics. With respect to coordinate frames, for instance, in dim s V = 6; s = 1, u α 1 = (x 1 , x 2 , y 3 , y 4 , y 5 , y 6 ), the metrics (35) are written in a form similar to that in Figure 3.1.1. We note that nonholonomic 2+2+... parameterizations of type (2) prescribe certain algebraic symmetries of metrics both with respect to N-adapted and/or coordinate frames. For instance, a splitting 3+3+3+ ... may contain more complex topological configurations but to integrate the Einstein gravitational equations in such cases is not possible for general "non-Killing" ansatz.
In a more general context, a d-metric (35) can be a result of nonholonomic deformations of some "primary" geometric/physical data into certain "target" data, In this work we shall consider that the values labeled by "•" may define, or not, exact solutions in a gravity theory. The metrics with "η" will be constrained always to define a solution of gravitational field equations (21), or (20). For simplicity, we shall use prime ansatz of type where the constants ǫ as take values +1 and/or −1 which depends on the signature of the higher dimensional spacetime and on (g i ,h a ;N a i ). Such an ansatz may define, for instance, a Kerr black hole (or a wormhole) solution trivially imbedded into a 4 + 2s spacetime if the corresponding values of the coefficients are constructed respectively for different type solutions of the gravitational field equations. We choose the target metric ansatz (35) as with so-called gravitational "polarization" functions and extra dimensional N-coefficients, η αs , η a i and η N as is . In order to consider the limits , for ε → 0, depending on a small parameter ε, 0 ≤ ε ≪ 1, we shall introduce "small" polarizations of type η = 1 + εχ(u...) and η N as is = εn as is (u...). It should be noted that if a target d-metric (35) is generated by a nonholonomic deformation with nontrivial η-, or χ,-functions, it contains both "old" geometric/physical information on a prime metric (36) and additional data for a new class of exact solutions.
Using the above formulas, we can compute the Ricci scalar (18) for s D (for simplicity, we consider s = 1), s R = 2( R 1 1 + R 3 3 + R 5 5 ). There are certain N-adapted symmetries of the Einstein d-tensor (19) for the ansatz (35), . In a similar form, we find symmetries for s = 2 : We search for solutions of the nonholonomic Einstein equations (38)-(45) with nontrivial Λ-sources written in the form Similar equations can be written recurrently for arbitrary finite extra dimensions. This constrains us to define such N-adapted frame transformations when the sources Υ βsδs in (21) are parameterized For certain models of extra dimensional gravity, we can write v Re-defining the generating functions (see below) for non-vacuum configurations, we can always introduce such effective sources.

Decoupling of gravitational field equations
Introducing the ansatz (35) for g i (x k ) = ǫ i e ψ(x k ) with nonzero ∂ 4 φ, ∂ 4 h a , ∂ 6 1 φ, ∂ 6 h a 1 , ∂ 8 2 φ, ∂ 8 h a 2 in (38)-(45) with respective sources, we obtain this system of PDEs: (similar equations can be written recurrently for arbitrary finite extra dimensions), where the coefficients are defined respectively and similarly for extra shells. The equations (47)-(54) reflect a very important decoupling property of the (generalized) Einstein equations with respect to the corresponding N-adapted frames. In explicit form, such formulas can be obtained for metrics with at least one Killing symmetry (the constructions can be generalized for non-Killing configurations). Let us explain in brief the decoupling property for 4-d configurations following such steps: 1. The equation (47) is just a 2-d Laplace, or d'Alambert one (depending on prescribed signature), which can be solved for any value Λ(x k ). (48) contains only the partial derivative ∂ 4 and is related to the formula for the coefficient (55) for the values h 3 (x i , y 4 ), h 4 (x i , y 4 ) and φ(x i , y 4 ) and source v Λ(x k , y 4 ). Prescribing any two such functions, we can define (by integrating with respect to y 4 ) the other two such functions.

The equation
3. Using h 3 and φ in the previous point, we can compute the coefficients α i and β, see (56), which allows us to define n i from the algebraic equations (49).

4.
Having computed the coefficient γ (56), the N-connection coefficients w i can be defined after two integrations with respect to y 4 in (50).
The procedure 2-4 can be repeated step by step on the other shells for higher dimensions. We have to add the corresponding dependencies on the extra dimensional coordinates and additional partial derivatives. For instance, the equation (51) and formula (57) with partial derivative ∂ 6 involves the functions h 5 (x i , y a , y 6 ), h 6 (x i , y a , y 6 ) and 1 φ(x i , y a , y 6 ) and source v 1 Λ(u β , y 6 ). We can compute any two such functions integrating with respect to y 6 if the two other ones are prescribed. In a similar form, we follow the steps in points 3 and 4 with 1 α τ , 1 β, 1 γ, see (58), and compute the higher order N-connection coefficients 1 n τ and 1 w τ .

Integration of (modified) Einstein equations by generating functions and effective sources
The system of nonlinear PDEs (47)-(54) can be integrated in general forms for any finite dimension dim s V ≥ 4.

4-d non-vacuum configurations:
The coefficients g i = ǫ i e ψ(x k ) are defined by solutions of the corresponding Laplace/ d'Alambert equation (47).
We can solve (48) and (55) for any ∂ 4 φ = 0, h a = 0 and v Λ = 0 if we re-write the equations as for any nontrivial source v Λ. Inserting the first equation into the second one, we find for Φ := e φ . This formula can be integrated with respect to y 4 , which results in is an integration function and ǫ 3 , ǫ 4 = ±1. To find h 4 we can use the first equation (59) and write These formulas for h a can be simplified if we introduce an "effective" cosmological constant Λ = const = 0 and re-define the generating function Φ →Φ, for which Introducing the integration function 0 h 3 (x k ) and ǫ 3 and ǫ 4 in Φ and, respectively, in v Λ, we can express where Ξ = dy 4 ( v Λ)∂ 4 (Φ 2 ). We can work for convenience with two couples of generating data, (Φ, v Λ) and (Φ,Λ), related by formulas (62).
Using 2+2+... symmetries of off-diagonal parameterizations (36), we can construct exact solutions for arbitrary finite dimension of extra dimensional spacetime s V.
Let us show how we can satisfy the LC-conditions (75) for s = 0. We note that such nonholonomic constraints cannot be solved in explicit form for arbitrary data (Φ, v Λ), or (Φ,Λ), and all types of nonzero integration functions 1 n j (x k ) and 2 n k (x i ) = 0. Nevertheless, certain general classes of solutions can be written in explicit form if via coordinate and frame transformations we can fix 2 n k (x i ) = 0 and 1 n j (x k ) = ∂ j n(x k ) for a function n(x k ). Then we use the property that We can restrict our construction to a subclass of generating data (Φ, In the second step, we solve firstly the condition in (75), for s = 0. Taking the derivative ∂ 4 of w i = ∂ i Φ/∂ 4 Φ (64), we obtain and using (76), we compute were we used the conditions (77) and the property e iΦ = 0. Using the last two formulas, we can obtain Finally, we note that the third condition for s = 0, Following similar considerations for other shells' generating functions ; (78) (similar formulas can be written recurrently for arbitrary extra shells); we can construct quadratic line elements for LC-configurations )du τ 2 (79) In these formulas, the generating functions are functionals of "inverse hat" values, wheň We can compute the values The torsions for such non-vacuum exact solutions (79) generated by the respective data ( sǧ , sŇ , s∇ ) are zero, which is different from the class of exact solutions (72) with nontrivial canonical d-torsions (15) and completely determined by arbitrary data ( s g, s N, s D) with Killing symmetry on ∂ 7 .

Non-Killing configurations
The off-diagonal integral varieties of solutions of gravitational field equations constructed in the previous section possess for any shell s ≥ 0 at least one Killing vector symmetry on ∂/∂y as−1 when the metrics do not depend on coordinate y as−1 in a class of N-adapted frames. There are two general possibilities to generate "non-Killing" configurations: 1) performing a formal embedding into higher dimensional vacuum spacetimes and/or via 2) "vertical" conformal nonholonomic deformations.
We emphasize that the above N-adapted formulas do not depend on y 3 and y 5 .
, for someǍ(x i , y 4 ) and 1Ǎ (x i , y 4 , y 6 ) which are necessary for ∂ i w j = ∂ j w i and ∂ α 1 w β = ∂ β 1 w α . Considering functional derivatives of type (76) and N-coefficients of the type in (82)  Putting together the above formulas, we construct a subclass of metrics of (80) determined by generic off-diagonal metrics as solutions of 6-d vacuum Einstein equations, We note that in this quadratic line element the terms ǫ 3 (dy 3 ) 2 and ǫ 5 (dy 5 ) 2 are used for trivial extensions from 4-d to 6-d. Re-defining the coordinate y 6 → y 3 , we generate vacuum solutions in 4-d gravity with metrics (83) depending on all four coordinates x i , y 3 and y 4 . The anholonomy coefficients (9) are not zero and such metrics cannot be diagonalized by coordinate transformations. This class of 4-d vacuum spacetimes do not possess, in general, Killing symmetries.

"Vertical" conformal nonholonomic deformations
There is another possibility to generate off-diagonal solutions depending on all spacetime coordinates and, in general, with nontrivial sources of the type in (46), see details and proofs in Ref. [8]. By straightforward computations, we can check that any metric with the conformal v-factors subjected to the conditions (similar equations can be written recurrently for arbitrary finite extra dimensions) does not change the Ricci d-tensor (38)-(45). Any class of solutions considered in this section can be generalized to non-Killing configurations using nonholonomic "vertical" conformal transforms. In 4-d, the ansatz (84) can be parameterized with respect to coordinate frames in a form with nontrivial ω 2 (u α ) which is different from that given in Figure 3.1.1, A general metric g αβ (u γ ) can be parameterized in the form (86) if there are any geometrically and physically well-defined frame transformations g αβ = e α α e β β g αβ . For certain given values g αβ and g αβ (in GR, there are 6 + 6 independent components), we have to solve a system of quadratic algebraic equation in order to determine 16 coefficients e α α , up to a fixed coordinate system. We have to fix such nonholonomic 2+2 splitting and partitions on manifolds when the algebraic equations have real nondegenerate solutions.
Finally, we note that we can consider generic off-diagonal coordinate decompositions which are similar to (86) but with dependencies on all coordinates for higher order shells.

Nonholonomic Deformations & the Kerr Metric
In this section, we show how using the AFDM formalism the Kerr solution can be constructed as a particular case when corresponding types of generating and integration functions are prescribed. We provide a series of new classes of solutions when the metrics are nonholonomically deformed into general or ellipsoidal stationary configurations in four dimensional gravity and/or extra dimensions. Explicit examples are studied of generic off-diagonal metrics encoding interactions in massive gravity, f -modifications and nonholonomically induced torsion effects. We find such nonholonomic constraints when modified massive, and zero mass, gravitational effects can be modelled by nonlinear off-diagonal interactions in GR.
The quadratic linear elements (87) (or (88)) with prime datå define solutions of the vacuum Einstein equations parameterized in the form (21) and (22) with zero sources.
Here we note that we have to consider a correspondingly N-adapted system of coordinates instead of the "standard" prolate spherical, or Boyer-Linquist ones because parameterizations with the data (90) are most convenient for a straightforward application of the AFDM. Following such an approach, we can generalize the solutions in order to get dependencies of the coefficients on more than two coordinates, with non-Killing configurations and/or extra dimensions. In some sense, the Kerr vacuum solution in GR consists a "degenerate" case of the 4-d off-diagonal vacuum solutions determined by primary metrics with the data (90) when the diagonal coefficients depend only on two "horizontal" N-adapted coordinates and the off-diagonal terms are induced by rotation frames.

Deformations of Kerr metrics in 4-d massive gravity
Let us consider the coefficients (90) for the Kerr metric as the data for a prime metricg (in general, it may be, or not, an exact solution of the Einstein or other modified gravitational equations, or any fiducial metric). Our goal is to construct nonholonomic deformations, (g,N, vΥ = 0,Υ = 0) → ( g, N, v Υ = λ, Υ = λ), λ = const = 0, see sources (34) for the shell s = 0 and (33). The main condition is that the target metric g positively defines a generic off-diagonal solution of field equations in 4-d massive gravity. The N-adapted deformations of coefficients of the metrics, frames and sources are parameterized in the form where the values η a , w i ,ñ i and ̟ are functions of three coordinates (x k ′ , y 4 ) and η i (x k ) depend only on hcoordinates. The prime datag i ,h a ,ẘ i ,n i are given by coefficients depending only on (x k ).
In terms of η-functions (37) resulting in h * a = 0 and g i = c i e ψ(x k ) , the solutions of type (65) with Λ → λ and 2 n k ′ = 0 (we use "primed" coordinates and prime Kerr data (88) and (90)) can be re-written in the form withΦ 2 /h 4 parameterized using formulas (91). 10 The gravitational polarizations (η i , η a ) and N-coefficients (n i , w i ) are computed following formulas where η A(x k , y 4 ) is introduced via formulas and assumptions similar to (78), for s = 1, and ψ •• + ψ ′′ = 2 µ 2 g λ(x k ′ ). For N-coefficients, the parameterizations are used (64) withΦ = exp[̟(x k ′ , y 4 )] |h 3 ′ |, when h 3 ′h 4 ′ = AC − B 2 and We can take any function η n(x k ) and put λ = const = 0 using the corresponding re-definitions of coordinates and generating functions. The solutions (92) are valid for stationary LC-configurations determined by off-diagonal massive gravity effects on Kerr black holes when the new class of spacetimes have a Killing symmetry in ∂/∂y 3 ′ and a generic dependence on three (from maximally four) coordinates, (x i ′ (r, ϑ), ϕ). Off-diagonal modifications are possible even for very small values of the mass parameter µ g . The solutions depend on the type of generating function ̟(x i ′ , ϕ) we have to fix in order to satisfy certain experimental/observational data in certain fixed systems of reference/coordinates. Various data can be re-parameterized for an effective λ = const = 0. In such variables, we can mimic stationary massive gravity effects by off-diagonal configurations in GR with integration parameters which should be also fixed by imposing additional assumptions on the symmetries of the interactions (for instance, to have an ellipsoid configuration, see section 4.3 and details and discussion on parametric Killing symmetries in Refs. [3,4,5]).

Nonholonomically induced torsion and massive gravity
If we do not impose the LC-conditions (22), a nontrivial source µ Λ = µ 2 g λ from massive gravity induces stationary configuration with nontrivial d-torsion (15). The torsion coefficients are determined by metrics of the type (65) with Λ → λ and parameterizations of coefficients and coordinates distinguishing the prime data for a Kerr metric (90). Such solutions can be written in the form where we use a generating function Φ(x i ′ , ϕ) instead of e ̟ and consider nonzero values of 2 n k (x i ′ ). We can see that nontrivial stationary off-diagonal torsion effects may result in additional effective rotations proportional to µ g if the integration function 2 n k = 0. Considering two different classes of off-diagonal solutions (93) and (92), we can study if a massive gravity theory is described in terms of an induced torsion or characterized by additional nonholonomic constraints as in GR (with zero torsion).
It should be noted that configurations of the type (93) can be constructed in various theories with noncommutative, brane, extra-dimension, warped and trapped brane type variables in string, or Finsler like and/or Hořava-Lifshits theories [6,10,11,13,25] when nonholonomically induced torsion effects play a substantial role. Those classes of solutions were constructed for different sets of interactions constants and, for instance, for propagating Schwarzschild and/or ellipsoid type configurations on Tau NUT backgrounds etc. The off-diagonal deformations and effective polarizations of the coefficients of the metrics correspond to a prime Kerr metric and are related to target configuration in massive gravity.

Small f -modifications of Kerr metrics and massive gravity
Using the AFDM, we can construct off-diagonal solutions for superposition of f -modified and massive gravity interactions. Such nonlinear effects can be distinguished in explicit form if we consider for additional f -deformations, for instance, a "prime" solution for massive gravity/ effectively modelled in GR with source µ Λ = µ 2 g λ(x k ′ ), or re-defined to µΛ = µ 2 gλ = const. Adding a "small" value Λ determined by f -modifications, we work in N-adapted frames with an effective source Υ = Λ + λ (see formulas (33) and (34)). As a result, we construct a class of off-diagonal solutions in modified f -gravity generated from the Kerr black hole solution as a result of two nonholonomic deformations when the target data g = ε g and N = ε N depend on a small parameter ε, 0 < ε ≪ 1. For simplicity, we restrict our considerations for solutions when |ε Λ| ≪ | µΛ |, i.e. consider that f -modifications in N-adapted frames are much smaller than massive gravity effects (in a similar from, we can analyze nonlinear interactions with |ε Λ| ≫ | µΛ |). The corresponding N-adapted transforms are parameterized as leading to a 4-d LC-configuration with d-metric for ∂ i εǍ = ∂ i εǍ + ε∂ i 1Ǎ determined by εΦ =Φ + ε 1Φ following conditions in (82). The values labeled by "•" and " " are taken from (91) (for simplicity, we omit priming of indices). The χ-and w-values ( corresponding to a re-definition of coefficients; for simplicity, we consider εn i = 0) have to be computed to define ε-deformed LC-configurations, see formulas (75) for s = 0, as solutions of the system (46) in the form (47)-(50) for a source Υ = µΛ + ε Λ.
The deformations (94) of the off-diagonal solutions (92) result in a new class of ε-deformed solutions with where there is not summation on index "i" in the last formula andh 3 ′h 4 ′ = AC − B 2 . Such nonholonomic deformations are determined respectively by two generating functionsΦ = e ̟ and 1Φ and two sources µΛ and Λ. Putting all this together, we construct an off-diagonal generalization of the Kerr metric via "main" massive gravity terms and additional ε-parametric f -modifications, We can consider ε-deformations of the type (94) for (93), which allows us to generate new classes of offdiagonal solutions with nonholonomically induced torsion determined both by massive and f -modifications of GR. Such a spacetime cannot be modelled as an effective one with anisotropic polarizations in GR.

Ellipsoidal 4-d deformations of the Kerr metric
We provide some examples how the Kerr primary data (90) is nonholonomically deformed into target generic off-diagonal solutions of vacuum and non-vacuum Einstein equations for the canonical d-connection and/or the Levi-Civita connection.

Vacuum ellipsoidal configurations
Let us construct a class of parametric solutions with such nonholonomic constraints on the coefficients given by (93) which transform the metrics into effective 4-d vacuum LC-configurations of the type (69). This defines a model when f -modifications compensate massive gravity deformations of a Kerr solution, with Υ = µΛ + ε Λ = 0, and result in ellipsoidal off-diagonal configurations in GR, where ε = − µΛ / Λ ≪ 1 can be considered as an eccentricity parameter. We find solutions for ε-deformations into vacuum solutions. The ansatz for the target metrics is of the type for ε Φ = exp(̟ + εχ 3 ′ ), see discussions related to (81) and (82). Because h 4 ′ for (97) can be approximated up to ε 2 to be a functional on ε Φ, we can satisfy for certain classes of generating functions ε Φ = εΦ = εΦ , see (77), the conditions ∂ ϕ w i ′ = e i ′ ln | h 4 |.
If the anholonomy coefficients (9) computed for (97) are not trivial for such w i and n k = 1 n k , the generated solutions cannot be diagonalized via coordinate transformations. The corresponding 4-d spacetimes have one Killing symmetry with respect to ∂/∂y 3 ′ . For small ε, the singularity at Ξ = 0 is "hidden" under ellipsoidal deformed horizons if m 0 ≥ a. Both the event horizon, r + = m(ϕ) + m 2 (ϕ) − a 2 sin 2 ϑ 1/2 , and the Cauchy horizon, r − = m(ϕ) − m 2 (ϕ) − a 2 sin 2 ϑ 1/2 , are ϕ-deformed and are effectively embedded into an off-diagonal background determined by the N-coefficients. In some sense, such configurations determine Kerr-like black hole solutions with additional dependencies on the variable ϕ of certain diagonal and off-diagonal coefficients of the metric. For a = 0, but ε = 0, we get ellipsoidal deformations of the Schwarzschild black holes (see [6] and references therein on the stability and interpretation of such solutions with both commutative and/or noncommutative parameters). Such an interpretation is not possible for "non-small" N -deformations of the Kerr metric. In general, it is not clear what physical importance such target exact solutions may have even if they may be defined to preserve the Levi-Civita configurations. The eccentricity ε = − λ/ Λ ≪ 1 depends both on massive gravity and f -modifications encoded into effective cosmological constants. We proved that via nonholonomic deformations it is possible to transform non-vacuum solutions with an effective locally anisotropically cosmological constant into effective off-diagonal vacuum configurations in GR. If the generating functions are prescribed to possess necessarily certain type of smooth conditions, the solutions are similar to certain Kerr black holes with ellipsoidal ε-deformed horizons and embedded self-consistently into non-trivial off-diagonal vacuum configurations. Polarizations of such vacuums encode massive gravity contributions and f -modifications.

Ellipsoid Kerr -de Sitter configurations
We construct a subclass of solutions (96) with rotoid configurations if we constrain χ 3 appearing in the ε-deformations in (95) to be of the form which is similar to (98). Expressing 1Φ = e ̟ [ Λ/2 µΛ + ζ sin(ω 0 ϕ + ϕ 0 )], forΦ = e ̟ , we generate a class of generic off-diagonal metrics associated with the ellipsoid Kerr -de Sitter configurations Such metrics have a Killing symmetry in ∂/∂y 3 and are completely defined by a generating function ̟(x k ′ , ϕ) and the sources µΛ = µ 2 g λ and Λ. They define ε-deformations of Kerr -de Sitter black holes into ellipsoid configurations with effective (polarized) cosmological constants determined by the constants in massive gravity and f -modifications. If the LC-conditions are satisfied, such metrics can be modelled in GR.

Extra dimension off-diagonal (non) massive modifications of the Kerr solutions
Various classes of generic off-diagonal deformations of the Kerr metric into higher dimensional exact solutions can be constructed. The explicit geometric and physical properties depend on the type of additional generating and integration functions and (non) vacuum configurations and (non) zero sources we consider. Let us analyze a series of 6-d and 8-d solutions encoding possible higher dimensional interactions with effective cosmological constants, nontrivial massive gravity contributions, f -modifications and certain analogies to Finsler gravity models.
Metrics of type (102) can be re-parameterized to define exact solutions in the so-called Einstein-Finsler gravity and fractional derivative modifications constructed on tangent bundles to Lorentz manifolds, see details in Refs. [28,13,25,29] and following different Finsler, or fractional models, [30,31,32,34,35]. For Finsler like theories, we have to consider y 5 , y 6 , y 7 , y 8 as fiber coordinates for a tangent bundle with local coordinates x i ′ , y 3 ′ , ϕ when the 1 v + 2 v coefficients of the metric and other geometric/physical objects can be transformed into standard ones in Finsler geometry via frame and coordinate transformations. In some sense, Finsler-like theories with small corrections to GR are extra-dimensional ones with "velocity/momentum" coordinates and with low "speed/energy" nonlinear corrections.
Finally we note that the class of metrics (102) contains a subclass of the 6d→8d generalization of (101) to those configurations with zero torsion if we choose Φ = e 2̟ and impose on the N-coefficients respective constraints which are necessary for selecting LC-configurations.

Kerr massive deformations and vacuum extra dimensions
In this subsection, we momentarily return to the vacuum ellipsoid solutions (97) and extend the metric to extra dimensions when the source is of type Υ = λ+ ε( Λ + Λ) = 0, µΛ =µ 2 g | λ|, and result in ellipsoidal off-diagonal configurations in GR, where ε = − µΛ /( Λ+ Λ) ≪ 1 can be considered as an eccentricity parameter. We can construct models of off-diagonal extra dimensional interactions when the f -modifications Λ compensate an extra dimensional contribution via the effective constant Λ and which are related to the configurations of massive gravity deformations of a Kerr solution. We select a subclass of solutions for ε-deformations of the vacuum solutions and described by the ansatz for the target metrics The extra-dimensions components of this metric are generated by the functions 1Φ , 2Φ and the N-coefficients similarly to (102)

Extra dimension massive ellipsoid Kerr -de Sitter configurations
Combining the solutions (100) and (102), we construct a class of non-vacuum 8-d solutions with rotoid configurations if we constrain χ 3 in the ε-deformations (for 4-d, see a similar formula (95)) to be of the form We reexpress 1Φ = e ̟ [ ( Λ + Λ)/2 µΛ + ζ sin(ω 0 ϕ + ϕ 0 )], forΦ = e ̟ and (103), and generate a class of generic off-diagonal exra dimensional metrics for ellipsoid Kerr -de Sitter configurations Such non-vacuum solutions can be also modelled for Einstein-Finsler spaces if the extra-dimension coordinates are treated as velocity/momentum ones. The metrics possess a respective Killing symmetry in ∂/∂y 7 . They define ε-deformations of Kerr -de Sitter black holes into ellipsoid configurations with effective cosmological constants determined by the constants in massive gravity, f -modifications and extra dimensional contributions.

Concluding Remarks
In this work, we have elaborated the anholonomic frame deformation method, AFDM, in constructing exact solutions in gravity theories, which we formulated and developed in [5,6,7,8,9], see also references therein. The method is based on a general decoupling property of the gravitational field equations which is possible for certain classes of noholonomic 2 + 2 + ... splitting of the spacetime dimensions. Such solutions are generic off-diagonal, with zero or non-zero torsion structure, and may depend on all (higher dimensions, or 4-d) spacetime coordinates. In the simplest form, the constructions can be performed by using an "auxiliary" metric-compatible connection which is constructed along with the "standard" Levi-Civita connection and from the same metric structure. Both connections are related via a distortion tensor which is completely determined by the coefficients of the metric and the frame splitting. After a class of off-diagonal solutions is constructed in general, we can impose certain conditions on the structure of the nonholonomic frames, when the coefficients of both the auxiliary and standard connections are the same, and we can extract solutions with zero torsion, for instance, in general relativity theory.
In general form, the off-diagonal metrics and nonlinear and linear connections constructed following the AFDM method depend on various classes of generating and integration functions, certain symmetry parameters and on possible nonzero sources and/or (polarized) cosmological constants. This is possible because in our approach the (generalized/modified) Einstein equations are transformed (after choosing the corresponding ansatz for the metrics) into systems of nonlinear partial differential equations which can be integrated in a very general form and depending on certain classes of generating/integration functions. This is different from the case of a diagonal ansatz, for instance, for the Schwarzschild metric when the gravitational field equations transform into a system of nonlinear ordinary differential equations depending on certain integration constants.
We can construct chains of nonholonomic frame deformations in order to transform a given primary metric (it may be an exact solution, or not, in a gravity theory) into other classes of target metrics and which can be fixed to be exact solutions in a "metric compatible" gravity theory. From a formal point of view, the chains' metrics can correspond to spaces with nontrivial topology, have a singular/stochastic/evolution etc behaviour and various types of horizons, symmetries and boundary conditions. In general, it is not possible to formulate some uniqueness property or limiting/ asymptotic conditions. Certain geometric data and physical information of "intermediary" metrics is encoded, step by step, into the target metrics. We can impose certain nonholonomic constraints on such integral varieties in order to relate a new class of target metric solutions to some well-defined primary metrics. However, it is not clear what physical importance these "very general" classes of target metric exact solutions may have.
In a series of works [10,11,12,13] (see details and references in [9]) we studied various examples. When using the AFDM we can construct locally anisotropic black ellipsoid/hole, spinning and/or solitonic spaces etc. Certain configurations seem to be stable [6] and mantain, for instance, the main properties of the Schwarzschild metric but for small rotoid deformations.
The goal of this article was fourfold: 1. to elaborate the AFDM in a form which allows us to construct generic off-diagonal solutions with Killing symmetries and the generalizations to non-Killing configurations using extensions to higher dimensions and so-called "vertical" conformal factors; 2. to study off-diagonal modifications of the Kerr metric under massive gravity and f -modified nonlinear interactions, via higher dimensions, and state the conditions when such configurations can be modelled as effective ones in general gravity, or via nonholonomically induced torsion fields etc; 3. to show how the well-known and physically important exact solution for the Kerr black hole can be constructed, for some special class-types of integration functions, following the AFDM; and 4. to provide certain examples when the solutions in point 2 can be generalized to various vacuum and non-vacuum configurations with ellipsoidal symmetries.
In some cases of rotoidal deformations with small eccentricity parameter, we have been able to prove that the physical properties of the primeval metrics are preserved but with certain effective polarizations of the physical constants and deformation to ellipsoidal configurations. It is possible to construct exact solutions for very general off-diagonal deformations (not depending of small parameters) but the physical properties are not clear if, for instance, additional smooth, symmetry, Cauchy and/or boundary conditions are not imposed. A very important property is that off-diagonal nonlinear gravitational interactions can mimic effective modified gravity theories, with anisotropies and re-scalings, which can find applications in modern cosmology and/or elaborate new models of quantum gravity [13,14,26].
Acknowledgments: The work of TG and SV is partially supported by the Program IDEI, PN-II-ID-PCE-2011-3-0256. SV wrote a part of this article during a recent visit at TH-CERN. He is grateful for important discussions, support and collaboration to S. Basilakos, S. Capozziello, E. Elizalde, N. Mavromatos, D. Singleton and P. C. Stavrinos,

A The Conditions for Zero Torsion
We can consider frame transformations to the N-coefficients and the ansatz (35) when all coefficients of a nonholonomically induced torsion (15) are zero and Γ γs αsβs = Γ γs αsβs . For simplicity, we analyze such conditions for the shell s = 0, i.e. for 4-d spacetimes.
In N-adapted frames, the coefficients of d-torsion (15) are T i jk = L i jk − L i kj = 0, T i ja = C i jb = 0, T a bc = C a bc − C a cb = 0 for any ansatz (35). The nontrivial coefficients are T c aj = L c aj − e a (N c j ) and T a ji = −Ω a ji . The values L a bi = ∂ b N a i + are computed for N 3 i = n i (x k , y 4 ), N 4 i = w i (x k , y 4 ); h bc = diag[h 3 (x k , y 4 ), h 4 (x k , y 4 )]; h ac = diag[(h 3 ) −1 , (h 4 ) −1 ]. We have We obtain such nontrivial values Ω 3 12 = −Ω 3 21 = ∂ 2 n 1 − ∂ 1 n 2 − w 2 ∂ 4 n 1 + w 1 ∂ 4 n 2 , Ω 4 12 = −Ω 4 21 = ∂ 2 w 1 − ∂ 1 w 2 − w 2 ∂ 4 w 1 + w 1 ∂ 4 w 2 . (A.1) Summarizing the above formulas for ∂ 4 n i = 0 and ∂ 2 n 1 − ∂ 1 n 2 = 0, we get the condition of zero torsion for the ansatz in (35) with n k = ∂ k n(x i ), In this form we can define a LC-configuration. The final step is to impose the condition that the coefficients n k do not depend on y 4 . This can be fixed in the form 1 n k (x i ) = ∂ k n(x i ) and 2 n k = 0, i.e. n k = ∂ k n(x i ).
Finally, we note that the LC-conditions can be formulated recurrently, in similar forms, for higher order shells both for zero and non-zero sources.