Quasi-stationary solutions in gravity theories with modified dispersion relations and Finsler–Lagrange–Hamilton Geometry

Abstract

Modified gravity theories, MGTs, with modified (nonlinear) dispersion relations, MDRs, encode via indicator functionals possible modifications and effects of quantum gravity; in string/brane, noncommutative, and/or nonassociative gravity theories etc. MDRs can be with global and/or local Lorentz invariance violations, LIVs, determined by quantum fluctuations, random, kinetic, statistical, and/or thermodynamical processes, etc. Such MGTs with MDRs and corresponding models of locally anisotropic spacetime and curved phase spaces can be geometrized in an axiomatic form for theories constructed on (co) tangent bundles with base spacetime Lorentz manifolds. In certain canonical nonholonomic variables, the geometric/physical objects are defined equivalently as in generalized Einstein–Finsler and/or Lagrange–Hamilton spaces. In such Finsler like MGTs, the coefficients of metrics and connections depend both on local Lorentz spacetime coordinates and, additionally, on (co) fiber velocity and/or momentum-type variables. The main goal of this work is to elaborate on a nonholonomic diadic “shell by shell” formulation of MGTs with MDRs, with a conventional $(2+2)+(2+2)$ splitting of total phase space dimensions, when the (dual) modified Einstein–Hamilton equations can be decoupled in general forms. We show how this geometric formalism allows us to construct various classes of exact and parametric solutions determined by generating and integration functions and effective sources depending, in principle, on all phase space coordinates. There are derived certain most general and important formulas for nonlinear quadratic elements and studied the main geometric and physical properties of quasi-stationary generic off-diagonal and diagonalizable phase spaces. This work provides a self-consistent geometric and analytic method for constructing in our further partner papers different types of black hole solutions for theories with MDRs and LIVs, and elaborating various applications in modern cosmology and astrophysics.

Appendices

N-adapted dyadic coefficient formulas

We provide some important N-adapted formulas and examples of proofs, see similar methods in [1, 2] and references therein.

1.1 Proof of Lemma 2.1 for canonical N-connection coefficients

There are canonical N-elongated bases and dual bases with nonholonomic $(4+4)$ splitting on $\mathbf {TV}$ enabled with N-connection structure (4) which are constructed by definition:

$$\begin{aligned} \begin{aligned} {\mathbf {e}}_{\alpha }&=\left( {\mathbf {e}}_{i}=\frac{\partial }{\partial x^{i}} -N_{i}^{a}(x,v)\frac{\partial }{\partial v^{a}},e_{b}=\frac{\partial }{ \partial v^{b}}\right) , \text{ on } \ T\mathbf {TV;} \\ {\mathbf {e}}^{\alpha }&=(e^{i}=\mathrm{d}x^{i},{\mathbf {e}} ^{a}=\mathrm{d}v^{a}+N_{i}^{a}(x,v)\mathrm{d}x^{i}), \text{ on } T^{*}\mathbf {TV}. \end{aligned} \end{aligned}$$

(A.1)

Here, we note that a local basis ${\mathbf {e}}_{\alpha }$ is nonholonomic if the commutators:

$$\begin{aligned} {\mathbf {e}}_{[\alpha }{\mathbf {e}}_{\beta ]}:={\mathbf {e}}_{\alpha }{\mathbf {e}} _{\beta }-{\mathbf {e}}_{\beta }{\mathbf {e}}_{\alpha }=C_{\alpha \beta }^{\gamma }(u){\mathbf {e}}_{\gamma } \end{aligned}$$

(A.2)

contain non-trivial anholonomy coefficients $C_{\alpha \beta : }^{\gamma }=\{C_{ia}^{b}=\partial _{a}N_{i}^{b},C_{ji}^{a}={\mathbf {e}}_{j}N_{i}^{a}- {\mathbf {e}}_{i}N_{j}^{a}\}.$

Similarly, we define N-elongated dual bases, cobase, for cotangent bundles:

$$\begin{aligned} \begin{aligned} \ ^{\shortmid }{\mathbf {e}}_{\alpha }&=\left( \ ^{\shortmid }{\mathbf {e}}_{i}=\frac{ \partial }{\partial x^{i}}-\ ^{\shortmid }N_{ia}(x,p)\frac{\partial }{ \partial p_{a}},\ ^{\shortmid }e^{b}=\frac{\partial }{\partial p_{b}}\right) , \text{ on } \ T{\mathbf {T}}^{*}\mathbf {V;} \\ \ \ ^{\shortmid }{\mathbf {e}}^{\alpha }&=\left( \ ^{\shortmid }e^{i}=\mathrm{d}x^{i},\ ^{\shortmid }{\mathbf {e}}_{a}=\mathrm{d}p_{a}+\ ^{\shortmid }N_{ia}(x,p)\mathrm{d}x^{i}\right) , \text{ on } T^{*}{\mathbf {T}}^{*}\mathbf {V.} \end{aligned} \end{aligned}$$

(A.3)

For “shell by shell” decompositions, we consider N-adapted bases:

$$\begin{aligned} {\mathbf {e}}_{\alpha _{s}}=\left( \ {\mathbf {e}}_{i_{s}}=\frac{\partial }{\partial x^{i_{s}}}-N_{i_{s}}^{a_{s}}\frac{\partial }{\partial v^{a_{s}}},e_{b_{s}}= \frac{\partial }{\partial v^{b_{s}}}\right) \text{ on } \ \, _{s}T\mathbf {TV}, \end{aligned}$$

where (respectively, for $s=1,2,3,4$):

$$\begin{aligned} {\mathbf {e}}_{\alpha _{1}}= & {} e_{i_{1}}=\frac{\partial }{\partial x^{i_{1}}},\ {\mathbf {e}}_{\alpha _{2}}=\left( \ {\mathbf {e}}_{i_{2}}=\frac{\partial }{\partial x^{i_{2}}}-N_{i_{2}}^{a_{2}}\frac{\partial }{\partial x^{a_{2}}},\ e_{b_{2}}= \frac{\partial }{\partial x^{b_{2}}}\right) , \text{ where } i=(i_{1},i_{2}), \nonumber \\ \ {\mathbf {e}}_{\alpha _{3}}= & {} \left( \ {\mathbf {e}}_{i_{3}}=\frac{\partial }{ \partial x^{i_{3}}}-N_{i_{3}}^{a_{3}}\frac{\partial }{\partial v^{a_{3}}},\ e_{b_{3}}=\frac{\partial }{\partial v^{b_{3}}}\right) ,\\ \ {\mathbf {e}}_{\alpha _{4}}= & {} \left( \ {\mathbf {e}}_{i_{4}}=\frac{\partial }{\partial x^{i_{4}}} -N_{i_{4}}^{a_{4}}\frac{\partial }{\partial v^{a_{4}}},e_{b_{4}}=\frac{ \partial }{\partial v^{b_{4}}}\right) . \nonumber \end{aligned}$$

(A.4)

Similarly, there are defined N-elongated bases:

$$\begin{aligned} \ ^{\shortmid }{\mathbf {e}}_{\alpha _{s}}=\left( \ \ ^{\shortmid }{\mathbf {e}} _{i_{s}}=\ \frac{\partial }{\partial x^{i_{s}}}-\ ^{\shortmid }N_{\ i_{s}a_{s}}\frac{\partial }{\partial p_{a_{s}}},\ \ ^{\shortmid }e^{b_{s}}= \frac{\partial }{\partial p_{b_{s}}}\right) \text{ on } \ \, _{s}T{\mathbf {T}}^{*} \mathbf {V,} \end{aligned}$$

when

$$\begin{aligned} \ ^{\shortmid }{\mathbf {e}}_{\alpha _{1}}= & {} \ \ ^{\shortmid }e_{i_{1}}=\frac{ \partial }{\partial x^{i_{1}}},\ ^{\shortmid }{\mathbf {e}}_{\alpha _{2}} =\left( \ ^{\shortmid }{\mathbf {e}}_{i_{2}}=\frac{\partial }{\partial x^{i_{2}}}-\ ^{\shortmid }N_{i_{2}}^{a_{2}}\frac{\partial }{\partial x^{a_{2}}},\ \ ^{\shortmid }e_{b_{2}}=\frac{\partial }{\partial x^{b_{2}}}\right) ,\nonumber \\ \ \ ^{\shortmid }{\mathbf {e}}_{\alpha _{3}}= & {} \left( \ \ ^{\shortmid }{\mathbf {e}} _{i_{3}}=\ \frac{\partial }{\partial x^{i_{3}}}-\ ^{\shortmid }N_{i_{3}a_{3}} \frac{\partial }{\partial p_{a_{3}}},\ \ ^{\shortmid }e^{b_{3}}=\frac{ \partial }{\partial p_{b_{3}}}\right) ,\\ \ \ ^{\shortmid }{\mathbf {e}}_{\alpha _{4}}= & {} \left( \ \ ^{\shortmid }{\mathbf {e}}_{i_{4}}=\ \frac{\partial }{\partial x^{i_{4}}}-\ ^{\shortmid }N_{\ i_{4}a_{4}}\frac{\partial }{\partial p_{a_{4}}},\ \ ^{\shortmid }e^{b_{4}}=\frac{\partial }{\partial p_{b_{4}}}\right) .\nonumber \end{aligned}$$

(A.5)

In dual form, we introduce (we omit explicit shell parameterizations):

$$\begin{aligned} \begin{aligned} {\mathbf {e}}^{\alpha _{s}}&=(e^{i_{s}}=\mathrm{d}x^{i_{s}},{\mathbf {e}} ^{a_{s}}=\mathrm{d}v^{a_{s}}+N_{j_{s}}^{a_{s}}\mathrm{d}x^{j_{s}}), \text{ on } \ \, _{s}T^{*} \mathbf {TV;} \\ \ ^{\shortmid }{\mathbf {e}}^{\alpha _{s}}&=(\ ^{\shortmid }e^{i}=\mathrm{d}x^{i},\ ^{\shortmid }{\mathbf {e}}_{a}=\mathrm{d}p_{a}+\ ^{\shortmid }N_{ia}(x,p)\mathrm{d}x^{i}) \text{ on } \ \, _{s}T^{*}{\mathbf {T}}^{*}{\mathbf {V}}. \end{aligned} \end{aligned}$$

(A.6)

Finally, we note that a N-connection on $\mathbf {TV,}$ or ${\mathbf {T}}^{*} \mathbf {V,}$ is characterized by such coefficients of N-connection curvature (called also Neijenhuis tensors):

$$\begin{aligned} \begin{aligned} \Omega _{ij}^{a}&=\frac{\partial N_{i}^{a}}{\partial x^{j}}-\frac{\partial N_{j}^{a}}{\partial x^{i}}+N_{i}^{b}\frac{\partial N_{j}^{a}}{\partial y^{b}} -N_{j}^{b}\frac{\partial N_{i}^{a}}{\partial y^{b}}, \text{ or } \ \\ \mathbf { \ ^{\shortmid }}\Omega _{ija}&=\frac{\partial \mathbf {\ ^{\shortmid }}N_{ia}}{ \partial x^{j}}-\frac{\partial \mathbf {\ ^{\shortmid }}N_{ja}}{\partial x^{i} }+\ \mathbf {^{\shortmid }}N_{ib}\frac{\partial \mathbf {\ ^{\shortmid }}N_{ja} }{\partial p_{b}}-\mathbf {\ ^{\shortmid }}N_{jb}\frac{\partial \mathbf {\ ^{\shortmid }}N_{ia}}{\partial p_{b}}. \end{aligned} \end{aligned}$$

(A.7)

Similar formulas can be written for dyadic decompositions considering shell indices and respective coordinates.

1.2 Off-diagonal local coefficients for d-metrics

Introducing formulas of type (B.9) and (B.10), respectively, into (6), (7), and (8), (9) and regrouping with respect to local coordinate bases, one proves the following.

Corollary A.1

(Equivalent re-writing of d-metrics and s-metrics as off-diagonal metrics) With respect to local coordinate frames, any d-metric structures on $\mathbf {TV,}$${\mathbf {T}}^{*}{\mathbf {V}}$ and s-metric on ${\mathbf {T}} ^{*}{\mathbf {V}}$:

$$\begin{aligned} {\mathbf {g}}= & {} {\mathbf {g}}_{\alpha \beta }(x,y){\mathbf {e}}^{\alpha }\mathbf { \otimes e}^{\beta }=g_{\underline{\alpha }\underline{\beta }}(x,y)\mathrm{d}u^{ \underline{\alpha }}\mathbf {\otimes }\mathrm{d}u^{\underline{\beta }} \text{ and/or } \\ \ ^{\shortmid }{\mathbf {g}}= & {} \ ^{\shortmid }{\mathbf {g}}_{\alpha \beta }(x,p)\ ^{\shortmid }{\mathbf {e}}^{\alpha }\mathbf {\otimes \ ^{\shortmid }e}^{\beta }=\ ^{\shortmid }g_{\underline{\alpha }\underline{\beta }}(x,p)d\ ^{\shortmid }u^{\underline{\alpha }}\mathbf {\otimes }d\ ^{\shortmid }u^{ \underline{\beta }}, \end{aligned}$$

can be parameterized via frame transforms, ${\mathbf {g}}_{\alpha \beta }=e_{\ \alpha }^{\underline{\alpha }}e_{\ \beta }^{\underline{\beta }}g_{\underline{ \alpha }\underline{\beta }},$$\ ^{\shortmid }{\mathbf {g}}_{\alpha \beta }=\ ^{\shortmid }e_{\ \alpha }^{\underline{\alpha }}\ ^{\shortmid }e_{\ \beta }^{ \underline{\beta }}\ ^{\shortmid }g_{\underline{\alpha }\underline{\beta }},$ in respective off-diagonal forms:

$$\begin{aligned} \begin{aligned} g_{\underline{\alpha }\underline{\beta }}&=\left[ \begin{array}{cc} g_{ij}(x)+g_{ab}(x,y)N_{i}^{a}(x,y)N_{j}^{b}(x,y) &{} g_{ae}(x,y)N_{j}^{e}(x,y) \\ g_{be}(x,y)N_{i}^{e}(x,y) &{} g_{ab}(x,y) \end{array} \right] , \\ \ ^{\shortmid }g_{\underline{\alpha }\underline{\beta }}&=\left[ \begin{array}{cc} \ ^{\shortmid }g_{ij}(x)+\ ^{\shortmid }g^{ab}(x,p)\ ^{\shortmid }N_{ia}(x,p)\ ^{\shortmid }N_{jb}(x,p) &{} \ ^{\shortmid }g^{ae}\ ^{\shortmid }N_{je}(x,p) \\ \ ^{\shortmid }g^{be}\ ^{\shortmid }N_{ie}(x,p) &{} \ ^{\shortmid }g^{ab}(x,p)\ \end{array} \right] \end{aligned} \end{aligned}$$

(A.8)

and, for nonholonomic $(2+2)+(2+2)$ splitting on $\, _{s}{\mathbf {T}}^{*} {\mathbf {V}}$ determined by respective data (5) and with coordinate parameterizations in footnote 8, using frame transforms $\ ^{\shortmid }{\mathbf {g}}_{\alpha _{s}\beta _{s}}=\ ^{\shortmid }e_{\ \alpha _{s}}^{\underline{\alpha }}\ ^{\shortmid }e_{\ \beta _{s}}^{\underline{\beta }}\ ^{\shortmid }g_{\underline{\alpha }\underline{\beta }}$ the coefficients of (9) are parameterized as follows:

$$\begin{aligned} \, _{s}^{\shortmid }{\mathbf {g}}= & {} \ ^{\shortmid }{\mathbf {g}}_{\alpha _{s}\beta _{s}}(x,y,\, _{s}p)\ ^{\shortmid }{\mathbf {e}}^{\alpha _{s}}\mathbf {\otimes \ ^{\shortmid }e}^{\beta _{s}}=\ ^{\shortmid }{\mathbf {g}} _{i_{s}j_{s}}(x^{k_{s}})\ ^{\shortmid }{\mathbf {e}}^{i_{s}}\mathbf {\otimes \ ^{\shortmid }e}^{j_{s}}+\ ^{\shortmid }{\mathbf {g}} ^{a_{s}b_{s}}(x^{k_{s}},p_{c_{s}})\ ^{\shortmid }{\mathbf {e}}_{a_{s}}\mathbf { \otimes \ ^{\shortmid }e}_{b_{s}} \nonumber \\= & {} \ ^{\shortmid }{\mathbf {g}}_{i_{1}j_{1}}(x^{k_{1}})\ \mathrm{d}x^{i_{1}}\mathbf { \otimes \ }\mathrm{d}x^{j_{1}}+\ ^{\shortmid }{\mathbf {g}} _{a_{2}b_{2}}(x^{k_{1}},x^{c_{2}})\ ^{\shortmid }{\mathbf {e}}^{a_{2}}\mathbf { \otimes \ ^{\shortmid }e}^{b_{2}} \nonumber \\&+\ ^{\shortmid }{\mathbf {g}}^{a_{3}b_{3}}(x^{k_{1}},x^{c_{2}},p_{c_{3}})\ ^{\shortmid }{\mathbf {e}}_{a_{3}}\mathbf {\otimes \ ^{\shortmid }e}_{b_{3}}+\ ^{\shortmid }{\mathbf {g}} ^{a_{4}b_{4}}(x^{k_{1}},x^{c_{2}},p_{c_{3}},p_{c_{4}})\ ^{\shortmid }\mathbf { e}_{a_{4}}\mathbf {\otimes \ ^{\shortmid }e}_{b_{4}}, \nonumber \\&\text{ where } \nonumber \\ \ ^{\shortmid }{\mathbf {e}}^{i_{1}}= & {} \mathrm{d}x^{i_{1}}, \text{ for } i_{1}=1,2 \nonumber \\ \ ^{\shortmid }{\mathbf {e}}^{a_{2}}= & {} \mathrm{d}x^{a_{2}}+\ ^{\shortmid }N_{j_{1}}^{a_{2}}(x^{k_{1}},x^{c_{2}})\ \mathrm{d}x^{i_{1}}, \text{ for } j_{1}=1,2 \text{ and } a_{2},c_{2}=3,4; \nonumber \\ \ ^{\shortmid }{\mathbf {e}}_{a_{3}}= & {} \mathrm{d}p_{a_{3}}+\ ^{\shortmid }N_{j_{2}\ a_{3}}(x^{k_{1}},x^{c_{2}},p_{c_{3}})\ \mathrm{d}x^{e_{2}}; \text{ for } j_{2}=1,2,3,4;e_{2}=3,4;a_{3},c_{3}=5,6; \nonumber \\ \ ^{\shortmid }{\mathbf {e}}_{a_{4}}= & {} \mathrm{d}p_{a_{4}}+\ ^{\shortmid }N_{j_{3}\ a_{4}}(x^{k_{1}},x^{k_{2}},p_{c_{3}},p_{c_{4}})\ \mathrm{d}p_{e_{3}}; \text{ for } j_{3}=1,2,3,4,5,6;a_{4},c_{4}=7,8;\nonumber \\ \end{aligned}$$

(A.9)

and, in generic off-diagonal local coordinate form:

$$\begin{aligned} \ ^{\shortmid }{\mathbf {g}}=\, _{s}^{\shortmid }{\mathbf {g}}=\, _{s}^{\shortmid }g_{\underline{\alpha }\underline{\beta }}(x,p)d\ ^{\shortmid }u^{\underline{ \alpha }}\mathbf {\otimes }d\ ^{\shortmid }u^{\underline{\beta }}, \end{aligned}$$

the coefficients can be expressed in matrix forms:

$$\begin{aligned} \, _{s}^{\shortmid }g_{\underline{\alpha }\underline{\beta }}=\left[ \begin{array}{cccc} \begin{array}{c} [\ ^{\shortmid }g_{i_{1}j_{1}}+ \\ \ ^{\shortmid }g_{a_{2}b_{2}}\ ^{\shortmid }N_{i_{1}}^{a_{2}}\ ^{\shortmid }N_{j_{1}}^{a_{2}}+ \\ \ ^{\shortmid }g^{c_{3}f_{3}}\ ^{\shortmid }N_{i_{1}c_{3}}\ ^{\shortmid }N_{j_{1}f_{3}}+ \\ \ ^{\shortmid }g^{c_{4}f_{4}}\ ^{\shortmid }N_{i_{1}c_{4}}\ ^{\shortmid }N_{j_{1}f_{4}}] \end{array} &{} \ ^{\shortmid }g_{a_{2}b_{2}}\ ^{\shortmid }N_{j_{1}}^{b_{2}} &{} \ \ ^{\shortmid }g^{a_{3}e_{3}}\ ^{\shortmid }N_{j_{1}e_{3}} &{} \ ^{\shortmid }g^{a_{4}e_{4}}\ ^{\shortmid }N_{j_{1}e_{4}} \\ \ ^{\shortmid }g_{a_{2}b_{2}}\ ^{\shortmid }N_{i_{1}}^{b_{2}} &{} \begin{array}{c} [\ ^{\shortmid }g_{a_{2}b_{2}}+ \\ \ ^{\shortmid }g^{c_{3}f_{3}}\ ^{\shortmid }N_{a_{2}c_{3}}\ ^{\shortmid }N_{b_{2}f_{3}}+ \\ \ ^{\shortmid }g^{c_{4}f_{4}}\ ^{\shortmid }N_{a_{2}c_{4}}\ ^{\shortmid }N_{b_{2}f_{4}}] \end{array} &{} \ \ ^{\shortmid }g^{a_{3}e_{3}}\ ^{\shortmid }N_{a_{2}e_{3}} &{} \ ^{\shortmid }g^{a_{4}e_{4}}\ ^{\shortmid }N_{a_{2}e_{4}} \\ \ ^{\shortmid }g^{b_{3}e_{3}}\ ^{\shortmid }N_{i_{1}e_{3}} &{} \ ^{\shortmid }g^{a_{3}e_{3}}\ ^{\shortmid }N_{i_{2}e_{3}} &{} [\ ^{\shortmid }g^{a_{3}b_{3}}+\ ^{\shortmid }g^{c_{4}f_{4}}\ ^{\shortmid }N_{\ c_{4}}^{a_{3}}\ ^{\shortmid }N_{\ f_{4}}^{b_{3}}] &{} \ ^{\shortmid }g^{a_{4}e_{4}}\ ^{\shortmid }N_{\ e_{4}}^{a_{3}} \\ \ ^{\shortmid }g^{b_{4}e_{4}}\ ^{\shortmid }N_{i_{1}e_{4}} &{} \ ^{\shortmid }g^{b_{4}e_{4}}\ ^{\shortmid }N_{i_{2}e_{4}} &{} \ ^{\shortmid }g^{b_{4}e_{4}}\ ^{\shortmid }N_{\ e_{4}}^{a_{3}} &{} \ ^{\shortmid }g^{a_{4}b_{4}} \end{array} \right] . \nonumber \\ \end{aligned}$$

(A.10)

Parameterizations of type (A.8) are considered, for instance, in the Kaluza–Klein theory for associated vector bundles when gauge field interactions are modeled for extra-dimension theories. These types of d- and/or s-metrics are generic off-diagonal if the corresponding N-adapted structure is not integrable. For MDR generalizations of the Einstein gravity, we can consider that the h-metrics $g_{ij}(x)=\ ^{\shortmid }g_{ij}(x)$ are determined by a solution of standard Einstein equations, but the terms with N-coefficients are determined by solutions of certain generalized gravitational field equations on nonholonomic phase spaces with a corresponding shell-by-shell splitting. In general, such nonholonomic dyadic solutions are not compactified on velocity/momentum-like coordinates, $v^{a}$/$p_{a}$ like in the standard Kaluza–Klein models.

Finally, we note that a s-metric written in a form (A.9) describes a 8-d phase space with curved both the base manifold and typical fiber subspaces which is different from (3) defined for extensions of a base metric ${\mathbf {g}}_{ij}(x^{k})$ on a Lorentz manifold ${\mathbf {V}}$ in GR to $\ ^{\shortmid }{\mathbf {g}}_{\alpha \beta }=[{\mathbf {g}} _{ij}(x^{k}),\eta ^{ab}]=[{\mathbf {g}}_{ij}(x^{k}),\eta ^{a_{3}b_{3}},\eta ^{a_{4}b_{4}}]$ on $T^{*}{\mathbf {V}}$ or $\, _{s}T^{*}{\mathbf {V}}$ with a flat typical fiber characterized by $\eta ^{ab}=[\eta ^{a_{3}b_{3}}=diag(1,1),\eta ^{a_{4}b_{4}}=diag(1,-1)]$.

1.3 Curvatures of d-connections with dyadic structure

By explicit computations for ${\mathbf {X}}={\mathbf {e}}_{\alpha },{\mathbf {Y}}= {\mathbf {e}}_{\beta },{\mathbf {D}}=\{\varvec{\Gamma }_{\ \alpha \beta }^{\gamma }\},$$\ ^{\shortmid }{\mathbf {X}}=\ ^{\shortmid }{\mathbf {e}}_{\alpha },$$\ ^{\shortmid }{\mathbf {Y}}=\ ^{\shortmid }{\mathbf {e}}_{\beta },\ ^{\shortmid } {\mathbf {D}}=\{\ ^{\shortmid }\varvec{\Gamma }_{\ \alpha \beta }^{\gamma }\},$ and $\, _{s}^{\shortmid }{\mathbf {X}}=\ ^{\shortmid }{\mathbf {e}}_{\alpha _{s}},$$\, _{s}^{\shortmid }{\mathbf {Y}}=\ ^{\shortmid }{\mathbf {e}}_{\beta _{s}},\, _{s}^{\shortmid }{\mathbf {D}}=\{\ ^{\shortmid }\varvec{\Gamma }_{\ \alpha _{s}\beta _{s}}^{\gamma _{s}}\}$ introduced, respectively, in (13 ) and (14), we prove the following.

Corollary A.2

For a d-connection ${\mathbf {D}}$ or $\ ^{\shortmid } \mathbf {D,}$ there are computed corresponding N-adapted coefficients: d-curvature, ${\mathcal {R}}=\mathbf {\{R}_{\ \beta \gamma \delta }^{\alpha }=(R_{\ hjk}^{i},R_{\ bjk}^{a},P_{\ hja}^{i},P_{\ bja}^{c},S_{\ hba}^{i},S_{\ bea}^{c})\},$ for:

$$\begin{aligned} R_{\ hjk}^{i}= & {} {\mathbf {e}}_{k}L_{\ hj}^{i}-{\mathbf {e}}_{j}L_{\ hk}^{i}+L_{\ hj}^{m}L_{\ mk}^{i}-L_{\ hk}^{m}L_{\ mj}^{i}-C_{\ ha}^{i}\Omega _{\ kj}^{a}, \nonumber \\ R_{\ bjk}^{a}= & {} {\mathbf {e}}_{k}\acute{L}_{\ bj}^{a}-{\mathbf {e}}_{j}\acute{L} _{\ bk}^{a}+\acute{L}_{\ bj}^{c}\acute{L}_{\ ck}^{a}-\acute{L}_{\ bk}^{c} \acute{L}_{\ cj}^{a}-C_{\ bc}^{a}\Omega _{\ kj}^{c}, \nonumber \\ P_{\ jka}^{i}= & {} e_{a}L_{\ jk}^{i}-D_{k}\acute{C}_{\ ja}^{i}+\acute{C}_{\ jb}^{i}T_{\ ka}^{b},\ P_{\ bka}^{c}=e_{a}\acute{L}_{\ bk}^{c}-D_{k}C_{\ ba}^{c}+C_{\ bd}^{c}T_{\ ka}^{c}, \nonumber \\ S_{\ jbc}^{i}= & {} e_{c}\acute{C}_{\ jb}^{i}-e_{b}\acute{C}_{\ jc}^{i}+\acute{C }_{\ jb}^{h}\acute{C}_{\ hc}^{i}-\acute{C}_{\ jc}^{h}\acute{C}_{\ hb}^{i}, \nonumber \\ \ S_{\ bcd}^{a}= & {} e_{d}C_{\ bc}^{a}-e_{c}C_{\ bd}^{a}+C_{\ bc}^{e}C_{\ ed}^{a}-C_{\ bd}^{e}C_{\ ec}^{a}, \end{aligned}$$

(A.11)

or $\ ^{\shortmid }{\mathcal {R}}=\mathbf {\{\ ^{\shortmid }R}_{\ \beta \gamma \delta }^{\alpha }=(\ ^{\shortmid }R_{\ hjk}^{i},\ ^{\shortmid }R_{a\ jk}^{\ b},\ ^{\shortmid }P_{\ hj}^{i\ \ \ a},\ ^{\shortmid }P_{c\ j}^{\ b\ a},\ ^{\shortmid }S_{\ hba}^{i},\ ^{\shortmid }S_{\ bea}^{c})\},$ for:

$$\begin{aligned} \ ^{\shortmid }R_{\ hjk}^{i}= & {} \ ^{\shortmid }{\mathbf {e}}_{k}\ ^{\shortmid }L_{\ hj}^{i}-\ ^{\shortmid }{\mathbf {e}}_{j}\ ^{\shortmid }L_{\ hk}^{i}+\ ^{\shortmid }L_{\ hj}^{m}\ ^{\shortmid }L_{\ mk}^{i}-\ ^{\shortmid }L_{\ hk}^{m}\ ^{\shortmid }L_{\ mj}^{i}-\ ^{\shortmid }C_{\ h}^{i\ a}\ ^{\shortmid }\Omega _{akj}, \\ \ ^{\shortmid }R_{a\ jk}^{\ b}= & {} \ ^{\shortmid }{\mathbf {e}}_{k}\ ^{\shortmid }\acute{L}_{a\ j}^{\ b}-\ ^{\shortmid }{\mathbf {e}}_{j}\ ^{\shortmid }\acute{L} _{a\ k}^{\ b}+\ ^{\shortmid }\acute{L}_{c\ j}^{\ b}\ ^{\shortmid }\acute{L} _{a\ k}^{\ c}-\ ^{\shortmid }\acute{L}_{c\ k}^{\ b}\ ^{\shortmid }\acute{L} _{a\ j}^{\ c}-\ ^{\shortmid }C_{a\ }^{\ bc}\ ^{\shortmid }\Omega _{ckj}, \\ \ ^{\shortmid }P_{\ jk}^{i\ \ \ a}= & {} \ ^{\shortmid }e^{a}\ ^{\shortmid }L_{\ jk}^{i}-\ ^{\shortmid }D_{k}\ ^{\shortmid }\acute{C}_{\ j}^{i\ a}+\ ^{\shortmid }\acute{C}_{\ j}^{i\ b}\ ^{\shortmid }T_{bk}^{\ \ \ a},\\ \ \ ^{\shortmid }P_{c\ k}^{\ b\ a}= & {} \ ^{\shortmid }e^{a}\ ^{\shortmid }\acute{L} _{c\ k}^{\ b}-\ ^{\shortmid }D_{k}\ ^{\shortmid }C_{c\ }^{\ ba}+\ ^{\shortmid }C_{\ bd}^{c}\ ^{\shortmid }T_{\ ka}^{c}, \\ \ ^{\shortmid }S_{\ j}^{i\ bc}= & {} \ ^{\shortmid }e^{c}\ ^{\shortmid }\acute{C }_{\ j}^{i\ b}-\ ^{\shortmid }e^{b}\ ^{\shortmid }\acute{C}_{\ j}^{i\ c}+\ ^{\shortmid }\acute{C}_{\ j}^{h\ b}\ ^{\shortmid }\acute{C}_{\ h}^{i\ c}-\ ^{\shortmid }\acute{C}_{\ j}^{h\ c}\ ^{\shortmid }\acute{C}_{\ h}^{i\ b}, \\ \ ^{\shortmid }S_{a\ }^{\ bcd}= & {} \ ^{\shortmid }e^{d}\ ^{\shortmid }C_{a\ }^{\ bc}-\ ^{\shortmid }e^{c}\ ^{\shortmid }C_{a\ }^{\ bd}+\ ^{\shortmid }C_{a\ }^{\ bc}\ ^{\shortmid }C_{b\ }^{\ ed}-\ ^{\shortmid }C_{e}^{\ bd}\ ^{\shortmid }C_{a}^{\ ec}; \end{aligned}$$

d-torsion, $\ {\mathcal {T}}=\{{\mathbf {T}}_{\ \alpha \beta }^{\gamma }=(T_{\ jk}^{i},T_{\ ja}^{i},T_{\ ji}^{a},T_{\ bi}^{a},T_{\ bc}^{a})\},$ for:

$$\begin{aligned} T_{\ jk}^{i}=L_{jk}^{i}-L_{kj}^{i},T_{\ jb}^{i}=C_{jb}^{i},T_{\ ji}^{a}=-\Omega _{\ ji}^{a},\ T_{aj}^{c}=L_{aj}^{c}-e_{a}(N_{j}^{c}),T_{\ bc}^{a}=C_{bc}^{a}-C_{cb}^{a}, \nonumber \\ \end{aligned}$$

(A.12)

or $\ ^{\shortmid }{\mathcal {T}}=\{\ ^{\shortmid }{\mathbf {T}}_{\ \alpha \beta }^{\gamma }=(\ ^{\shortmid }T_{\ jk}^{i},\ ^{\shortmid }T_{\ j}^{i\ a},\ ^{\shortmid }T_{aji},\ ^{\shortmid }T_{a\ i}^{\ b},\ ^{\shortmid }T_{a\ }^{\ bc})\},$ for:

$$\begin{aligned} \ ^{\shortmid }T_{\ jk}^{i}= & {} \ ^{\shortmid }L_{jk}^{i}-\ ^{\shortmid }L_{kj}^{i},\ ^{\shortmid }T_{\ j}^{i\ a}=\ ^{\shortmid }C_{j}^{ia},\ ^{\shortmid }T_{aji}=-\ ^{\shortmid }\Omega _{aji},\\ \ \ ^{\shortmid }T_{c\ j}^{\ a}= & {} \ ^{\shortmid }L_{c\ j}^{\ a}-\ ^{\shortmid }e^{a}(\ ^{\shortmid }N_{cj}),\ ^{\shortmid }T_{a\ }^{\ bc}=\ ^{\shortmid }C_{a}^{\ bc}-\ ^{\shortmid }C_{a}^{\ cb}; \end{aligned}$$

d-nonmetricity, $\ {\mathcal {Q}}=\mathbf {\{Q}_{\gamma \alpha \beta }=\left( Q_{kij},Q_{kab},Q_{cij},Q_{cab}\right) \},$ for:

$$\begin{aligned} Q_{kij}=D_{k}g_{ij},Q_{kab}=D_{k}g_{ab},Q_{cij}=D_{c}g_{ij},Q_{cab}=D_{c}g_{ab} \end{aligned}$$

(A.13)

or $\ ^{\shortmid }{\mathcal {Q}}=\mathbf {\{\ ^{\shortmid }Q}_{\gamma \alpha \beta }=\left( \ ^{\shortmid }Q_{kij},\ ^{\shortmid }Q_{kab},\ ^{\shortmid }Q_{cij},\ ^{\shortmid }Q_{cab}\right) \},$ for

$$\begin{aligned} \ ^{\shortmid }Q_{kij}=\ ^{\shortmid }D_{k}\ ^{\shortmid }g_{ij},\ ^{\shortmid }Q_{k}^{\ ab}=\ ^{\shortmid }D_{k}\ ^{\shortmid }g^{ab},\ ^{\shortmid }Q_{\ ij}^{c}=\ ^{\shortmid }D^{c}\ ^{\shortmid }g_{ij},\ ^{\shortmid }Q^{cab}=\ ^{\shortmid }D^{c}\ ^{\shortmid }g^{ab}. \end{aligned}$$

N-adapted formulas for $\, _{s}{\mathbf {T}}^{*}{\mathbf {V}}$ are written, respectively:

$\, _{s}^{\shortmid }{\mathcal {R}}=\mathbf {\{\ ^{\shortmid }R}_{\ \beta _{s}\gamma _{s}\delta _{s}}^{\alpha _{s}}=(\ ^{\shortmid }R_{\ h_{s}j_{s}k_{s}}^{i_{s}},\ ^{\shortmid }R_{a_{s}\ j_{s}k_{s}}^{\ b_{s}},\ ^{\shortmid }P_{\ h_{s}j_{s}}^{i_{s}\ \ \ a_{s}},\ ^{\shortmid }P_{c_{s}\ j_{s}}^{\ b_{s}\ a_{s}},\ ^{\shortmid }S_{\ h_{s}b_{s}a_{s}}^{i_{s}},\ ^{\shortmid }S_{\ b_{s}e_{s}a_{s}}^{c_{s}})\},$ for:

$$\begin{aligned} \ ^{\shortmid }R_{\ h_{s}j_{s}k_{s}}^{i_{s}}= & {} \ ^{\shortmid }{\mathbf {e}} _{k_{s}}\ ^{\shortmid }L_{\ h_{s}j_{s}}^{i_{s}}-\ ^{\shortmid }{\mathbf {e}} _{j_{s}}\ ^{\shortmid }L_{\ h_{s}k_{s}}^{i_{s}}+\ ^{\shortmid }L_{\ h_{s}j_{s}}^{m_{s}}\ ^{\shortmid }L_{\ m_{s}k_{s}}^{i_{s}}-\ ^{\shortmid }L_{\ h_{s}k_{s}}^{m_{s}}\ ^{\shortmid }L_{\ m_{s}j_{s}}^{i_{s}}-\ ^{\shortmid }C_{\ h_{s}}^{i_{s}\ a_{s}}\ ^{\shortmid }\Omega _{a_{s}k_{s}j_{s}}, \\ \ ^{\shortmid }R_{a_{s}\ j_{s}k_{s}}^{\ b_{s}}= & {} \ ^{\shortmid }{\mathbf {e}} _{k_{s}}\ ^{\shortmid }\acute{L}_{a_{s}\ j_{s}}^{\ b_{s}}-\ ^{\shortmid } {\mathbf {e}}_{j_{s}}\ ^{\shortmid }\acute{L}_{a_{s}\ k_{s}}^{\ b_{s}}+\ ^{\shortmid }\acute{L}_{c_{s}\ j_{s}}^{\ b_{s}}\ ^{\shortmid }\acute{L} _{a_{s}\ k_{s}}^{\ c_{s}}-\ ^{\shortmid }\acute{L}_{c_{s}\ k_{s}}^{\ b_{s}}\ ^{\shortmid }\acute{L}_{a_{s}\ j_{s}}^{\ c_{s}}-\ ^{\shortmid }C_{a_{s}\ }^{\ b_{s}c_{s}}\ ^{\shortmid }\Omega _{c_{s}k_{s}j_{s}}, \\ \ ^{\shortmid }P_{\ j_{s}k_{s}}^{i_{s}\ \ \ a_{s}}= & {} \ ^{\shortmid }e^{a_{s}}\ ^{\shortmid }L_{\ j_{s}k_{s}}^{i_{s}}-\ ^{\shortmid }D_{k_{s}}\ ^{\shortmid }\acute{C}_{\ j_{s}}^{i_{s}\ a_{s}}+\ ^{\shortmid }\acute{C}_{\ j_{s}}^{i_{s}\ b_{s}}\ ^{\shortmid }T_{b_{s}k_{s}}^{\ \ \ a_{s}}, \\ \ \ ^{\shortmid }P_{c_{s}\ k_{s}}^{\ b_{s}\ a_{s}}= & {} \ ^{\shortmid }e^{a_{s}}\ ^{\shortmid }\acute{L}_{c_{s}\ k_{s}}^{\ b_{s}}-\ ^{\shortmid }D_{k_{s}}\ ^{\shortmid }C_{c_{s}\ }^{\ b_{s}a_{s}}+\ ^{\shortmid }C_{\ b_{s}d_{s}}^{c_{s}}\ ^{\shortmid }T_{\ k_{s}a_{s}}^{c_{s}}, \\ \ ^{\shortmid }S_{\ j_{s}}^{i_{s}\ b_{s}c_{s}}= & {} \ ^{\shortmid }e^{c_{s}}\ ^{\shortmid }\acute{C}_{\ j_{s}}^{i_{s}\ b_{s}}-\ ^{\shortmid }e^{b_{s}}\ ^{\shortmid }\acute{C}_{\ j_{s}}^{i_{s}\ c_{s}}+\ ^{\shortmid }\acute{C}_{\ j_{s}}^{h_{s}\ b_{s}}\ ^{\shortmid }\acute{C}_{\ h_{s}}^{i_{s}\ c_{s}}-\ ^{\shortmid }\acute{C}_{\ j_{s}}^{h_{s}\ c_{s}}\ ^{\shortmid }\acute{C}_{\ h_{s}}^{i_{s}\ b_{s}}, \\ \ ^{\shortmid }S_{a_{s}\ }^{\ b_{s}c_{s}d_{s}}= & {} \ ^{\shortmid }e^{d_{s}}\ ^{\shortmid }C_{a_{s}\ }^{\ b_{s}c_{s}}-\ ^{\shortmid }e^{c_{s}}\ ^{\shortmid }C_{a_{s}\ }^{\ b_{s}d_{s}}+\ ^{\shortmid }C_{a_{s}\ }^{\ b_{s}c_{s}}\ ^{\shortmid }C_{b_{s}\ }^{\ e_{s}d_{s}}-\ ^{\shortmid }C_{e_{s}}^{\ b_{s}d_{s}}\ ^{\shortmid }C_{a_{s}}^{\ e_{s}c_{s}}; \end{aligned}$$

s-torsion $\, _{s}^{\shortmid }{\mathcal {T}}=\{\ ^{\shortmid }{\mathbf {T}}_{\ \alpha _{s}\beta _{s}}^{\gamma _{s}}=(\ ^{\shortmid }T_{\ j_{s}k_{s}}^{i_{s}},\ ^{\shortmid }T_{\ j_{s}}^{i_{s}\ a_{s}},\ ^{\shortmid }T_{a_{s}j_{s}i_{s}},\ ^{\shortmid }T_{a_{s}\ i_{s}}^{\ b_{s}},\ ^{\shortmid }T_{a_{s}\ }^{\ b_{s}c_{s}})\},$ for:

$$\begin{aligned} \ ^{\shortmid }T_{\ j_{s}k_{s}}^{i_{s}}= & {} \ ^{\shortmid }L_{j_{s}k_{s}}^{i_{s}}-\ ^{\shortmid }L_{k_{s}j_{s}}^{i_{s}},\ ^{\shortmid }T_{\ j_{s}}^{i_{s}\ a_{s}}=\ ^{\shortmid }C_{j_{s}}^{i_{s}a_{s}},\ ^{\shortmid }T_{a_{s}j_{s}i_{s}}=-\ ^{\shortmid }\Omega _{a_{s}j_{s}i_{s}}, \\ \ \ ^{\shortmid }T_{c_{s}\ j_{s}}^{\ a_{s}}= & {} \ ^{\shortmid }L_{c_{s}\ j_{s}}^{\ a_{s}}-\ ^{\shortmid }e^{a_{s}}(\ ^{\shortmid }N_{c_{s}j_{s}}),\ ^{\shortmid }T_{a_{s}\ }^{\ b_{s}c_{s}}=\ ^{\shortmid }C_{a_{s}}^{\ b_{s}c_{s}}-\ ^{\shortmid }C_{a_{s}}^{\ c_{s}b_{s}}; \end{aligned}$$

s-nonmetricity $\, _{s}^{\shortmid }{\mathcal {Q}}=\mathbf {\{\ ^{\shortmid }Q} _{\gamma _{s}\alpha _{s}\beta _{s}}=\left( \ ^{\shortmid }Q_{k_{s}i_{s}j_{s}},\ ^{\shortmid }Q_{k_{s}a_{s}b_{s}},\ ^{\shortmid }Q_{c_{s}i_{s}j_{s}},\ ^{\shortmid }Q_{c_{s}a_{s}b_{s}}\right) \},$ for:

$$\begin{aligned} \ ^{\shortmid }Q_{k_{s}i_{s}j_{s}}=\ ^{\shortmid }D_{k_{s}}\ ^{\shortmid }g_{i_{s}j_{s}},\ ^{\shortmid }Q_{k_{s}}^{\ a_{s}b_{s}}=\ ^{\shortmid }D_{k_{s}}\ ^{\shortmid }g^{a_{s}b_{s}},\ ^{\shortmid }Q_{\ i_{s}j_{s}}^{c_{s}}=\ ^{\shortmid }D^{c_{s}}\ ^{\shortmid }g_{i_{s}j_{s}},\ ^{\shortmid }Q^{c_{s}a_{s}b_{s}}=\ ^{\shortmid }D^{c_{s}}\ ^{\shortmid }g^{a_{s}b_{s}}. \end{aligned}$$

Similar formulas as in this Remarks can be proven for $\, _{s}\mathbf {TV}$ (the coefficients are written without label “$\ ^{\shortmid }$” and $a,b,c\ldots$ symbols go up, or down, comparing with corresponding low, or up, ones, and inversely).

1.4 The coefficients of canonical d-connections and dyadic splitting

Such d-connections are very important for elaborating MGTs on (co)tangent bundles, because they allow a very general decoupling and integration of generalized Einstein and matter field equations. By explicit computations in N-adapted frames, we can prove that necessary conditions for defining and constructing, respectively, $\widehat{{\mathbf {D}}}$ (19) and $\ ^{\shortmid }\widehat{{\mathbf {D}}}$ (20), are satisfied the following:

Corollary A.3

The N-adapted coefficients of canonical Lagrange and Hamilton d-connections are computed, respectively:

$$\begin{aligned} \text{ on } T\mathbf {TV},\ \widehat{{\mathbf {D}}}= & {} \left\{ \widehat{\varvec{\Gamma } }_{\ \alpha \beta }^{\gamma }=\left( {\widehat{L}}_{jk}^{i},{\widehat{L}}_{bk}^{a}, {\widehat{C}}_{jc}^{i},{\widehat{C}}_{bc}^{a}\right) \right\} , \text{ for } [{\mathbf {g}} _{\alpha \beta }=(g_{jr},g_{ab})\mathbf {,N=\{}N_{i}^{a}\mathbf {\}]}, \nonumber \\ {\widehat{L}}_{jk}^{i}= & {} \frac{1}{2}g^{ir}\left( {\mathbf {e}}_{k}g_{jr}+\mathbf { e}_{j}g_{kr}-{\mathbf {e}}_{r}g_{jk}\right) ,\nonumber \\ \ {\widehat{L}} _{bk}^{a}= & {} e_{b}(N_{k}^{a})+\frac{1}{2}g^{ac}(e_{k}g_{bc}-g_{dc}\ e_{b}N_{k}^{d}-g_{db}\ e_{c}N_{k}^{d}), \nonumber \\ {\widehat{C}}_{jc}^{i}= & {} \frac{1}{2}g^{ik}e_{c}g_{jk},\ {\widehat{C}}_{bc}^{a}= \frac{1}{2}g^{ad}\left( e_{c}g_{bd}+e_{b}g_{cd}-e_{d}g_{bc}\right) \end{aligned}$$

(A.14)

$$\begin{aligned} \text{ and, } \text{ on } T{\mathbf {T}}^{*}{\mathbf {V}},\ \ ^{\shortmid }\widehat{ {\mathbf {D}}}= & {} \left\{ \ ^{\shortmid }\widehat{\varvec{\Gamma }}_{\ \alpha \beta }^{\gamma }=\left( \ ^{\shortmid }{\widehat{L}}_{jk}^{i},\ ^{\shortmid }{\widehat{L}} _{a\ k}^{\ b},\ ^{\shortmid }{\widehat{C}}_{\ j}^{i\ c},\ ^{\shortmid } {\widehat{C}}_{\ j}^{i\ c}\right) \right\} ,\nonumber \\&\text{ for } [\ ^{\shortmid }{\mathbf {g}} _{\alpha \beta }=(\ ^{\shortmid }g_{jr},\ ^{\shortmid }g^{ab})\mathbf {,\ ^{\shortmid }N=\{}\ ^{\shortmid }N_{ai}\mathbf {\}]}, \nonumber \\ \ ^{\shortmid }{\widehat{L}}_{jk}^{i}= & {} \frac{1}{2}\ ^{\shortmid }g^{ir}(\ ^{\shortmid }{\mathbf {e}}_{k}\ ^{\shortmid }g_{jr}+\ ^{\shortmid }{\mathbf {e}} _{j}\ ^{\shortmid }g_{kr}-\ ^{\shortmid }{\mathbf {e}}_{r}\ ^{\shortmid }g_{jk}),\ \nonumber \\ \ ^{\shortmid }{\widehat{L}}_{a\ k}^{\ b}= & {} \ ^{\shortmid }e^{b}(\ ^{\shortmid }N_{ak})+\frac{1}{2}\ ^{\shortmid }g_{ac}(\ ^{\shortmid }e_{k}\ ^{\shortmid }g^{bc}-\ ^{\shortmid }g^{dc}\ \ ^{\shortmid }e^{b}\ ^{\shortmid }N_{dk}-\ ^{\shortmid }g^{db}\ \ ^{\shortmid }e^{c}\ ^{\shortmid }N_{dk}), \nonumber \\ \ ^{\shortmid }{\widehat{C}}_{\ j}^{i\ c}= & {} \frac{1}{2}\ ^{\shortmid }g^{ik}\ ^{\shortmid }e^{c}\ ^{\shortmid }g_{jk},\ \ ^{\shortmid }{\widehat{C}}_{\ a}^{b\ c}=\frac{1}{2}\ ^{\shortmid }g_{ad}(\ ^{\shortmid }e^{c}\ ^{\shortmid }g^{bd}+\ ^{\shortmid }e^{b}\ ^{\shortmid }g^{cd}-\ ^{\shortmid }e^{d}\ ^{\shortmid }g^{bc}). \nonumber \\ \end{aligned}$$

(A.15)

We use formulas (A.14) for the shells $s=1,2$ (with $i_{1},j_{1},\ldots =1,2$ and $a_{2},b_{2},\ldots =3,4)$ of the cotangent Lorentz bundle:

$$\begin{aligned} \text{ on } \ \, _{2}T{\mathbf {T}}^{*}{\mathbf {V}},\ \, _{2}^{\shortmid } \widehat{{\mathbf {D}}}= & {} \left\{ \widehat{\varvec{\Gamma }}_{\ \alpha _{2}\beta _{2}}^{\gamma _{2}}=\left( {\widehat{L}}_{j_{1}k_{1}}^{i_{1}},{\widehat{L}} _{b_{2}k_{1}}^{a_{2}},{\widehat{C}}_{j_{1}c_{2}}^{i_{1}},{\widehat{C}} _{b_{2}c_{2}}^{a_{2}}\right) \right\} ,\nonumber \\ \text{ for } [{\mathbf {g}}_{\alpha _{1}\beta _{1}}= & {} (g_{j_{1}r_{1}},g_{a_{2}b_{2}})\mathbf {,}\ \, _{2}^{\shortmid }\mathbf { N=\{}N_{i_{1}}^{a_{2}}\mathbf {\}]}, \nonumber \\ {\widehat{L}}_{j_{1}k_{1}}^{i_{1}}= & {} \frac{1}{2}g^{i_{1}r_{1}}\left( \mathbf {e }_{k_{1}}g_{j_{1}r_{1}}+{\mathbf {e}}_{j_{1}}g_{k_{1}r_{1}}-{\mathbf {e}} _{r_{1}}g_{j_{1}k_{1}}\right) ,\ \nonumber \\ {\widehat{L}}_{b_{2}k_{1}}^{a_{2}}= & {} e_{b_{2}}(N_{k_{1}}^{a_{2}})+\frac{1}{2} g^{a_{2}c_{2}}(e_{k_{1}}g_{b_{2}c_{2}}-g_{d_{2}c_{2}}\ e_{b_{2}}N_{k_{1}}^{d_{2}}-g_{d_{2}b_{2}}\ e_{c_{2}}N_{k_{1}}^{d_{2}}), \nonumber \\ {\widehat{C}}_{j_{1}c_{2}}^{i_{1}}= & {} \frac{1}{2} g^{i_{1}k_{1}}e_{c_{2}}g_{j_{1}k_{1}},\ {\widehat{C}}_{b_{2}c_{2}}^{a_{2}}= \frac{1}{2}g^{a_{2}d_{2}}\left( e_{c_{2}}g_{b_{2}d_{2}}+e_{b_{2}}g_{c_{2}d_{2}}-e_{d_{2}}g_{b_{2}c_{2}} \right) , \nonumber \\ \end{aligned}$$

(A.16)

and formulas (A.15) for shells $s=3,4$ ($i_{3},j_{3},\ldots =1,2,3,4$ and $a_{3},b_{3}=5,6;i_{4},j_{4},\ldots =1,2,3,4,5,6$ and $a_{4},b_{4}=7,8)$ and:

$$\begin{aligned} \text{ on } \, _{s}T{\mathbf {T}}^{*}{\mathbf {V}},\ \, _{s}^{\shortmid }\widehat{ {\mathbf {D}}}= & {} \left\{ \ ^{\shortmid }\widehat{\varvec{\Gamma }}_{\ \alpha _{s}\beta _{s}}^{\gamma _{s}}=\left( \ ^{\shortmid }{\widehat{L}} _{j_{s}k_{s}}^{i_{s}},\ ^{\shortmid }{\widehat{L}}_{a_{s}\ k_{s}}^{\ b_{s}},\ ^{\shortmid }{\widehat{C}}_{\ j_{s}}^{i_{s}\ c_{s}},\ ^{\shortmid }{\widehat{C}} _{\ j_{s}}^{i_{s}\ c_{s}}\right) \right\} , \text{ where, } \nonumber \\&\text{ for } [\ ^{\shortmid }{\mathbf {g}}_{\alpha _{s}\beta _{s}}=(\ ^{\shortmid }g_{j_{s}r_{s}},\ ^{\shortmid }g^{a_{s}b_{s}})\mathbf {,\, _{s}^{\shortmid }N=\{}\ ^{\shortmid }N_{a_{s}i_{s}}\mathbf {\}]} \nonumber \\ \ ^{\shortmid }{\widehat{L}}_{j_{s}k_{s}}^{i_{s}}= & {} \frac{1}{2}\ ^{\shortmid }g^{i_{s}r_{s}}(\ ^{\shortmid }{\mathbf {e}}_{k_{s}}\ ^{\shortmid }g_{j_{s}r_{s}}+\ ^{\shortmid }{\mathbf {e}}_{j_{s}}\ ^{\shortmid }g_{k_{s}r_{s}}-\ ^{\shortmid }{\mathbf {e}}_{r_{s}}\ ^{\shortmid }g_{j_{s}k_{s}}),\ \nonumber \\ \ ^{\shortmid }{\widehat{L}}_{a_{s}\ k_{s}}^{\ b_{s}}= & {} \ ^{\shortmid }e^{b_{s}}(\ ^{\shortmid }N_{a_{s}k_{s}})+\frac{1}{2}\ ^{\shortmid }g_{a_{s}c_{s}}(\ ^{\shortmid }e_{k_{s}}\ ^{\shortmid }g^{b_{s}c_{s}}\nonumber \\&-\ ^{\shortmid }g^{d_{s}c_{s}}\ \ ^{\shortmid }e^{b_{s}}\ ^{\shortmid }N_{d_{s}k_{s}}-\ ^{\shortmid }g^{d_{s}b_{s}}\ \ ^{\shortmid }e^{c_{s}}\ ^{\shortmid }N_{d_{s}k_{s}}), \nonumber \\ \ ^{\shortmid }{\widehat{C}}_{\ j_{s}}^{i_{s}\ c_{s}}= & {} \frac{1}{2}\ ^{\shortmid }g^{i_{s}k_{s}}\ ^{\shortmid }e^{c_{s}}\ ^{\shortmid }g_{j_{s}k_{s}},\nonumber \\ \ \ ^{\shortmid }{\widehat{C}}_{\ a_{s}}^{b_{s}\ c_{s}}= & {} \frac{1 }{2}\ ^{\shortmid }g_{a_{s}d_{s}}(\ ^{\shortmid }e^{c_{s}}\ ^{\shortmid }g^{b_{s}d_{s}}+\ ^{\shortmid }e^{b_{s}}\ ^{\shortmid }g^{c_{s}d_{s}}-\ ^{\shortmid }e^{d_{s}}\ ^{\shortmid }g^{b_{s}c_{s}}). \end{aligned}$$

(A.17)

In a similar form, we can prove that all N-adapted coefficient formulas are necessary formulating and finding solutions of physically important field and evolution equations in theories with MDRs and LIVs.

1.5 Proof of Theorem 4.2

The coefficients $g_{i_{1}}=e^{\psi (x^{k_{1}})}$ for the first dyadic shell $s=1$ are defined by solutions of the corresponding 2-d Poisson equation (41) for any given source $\, _{1}^{\shortmid } {\widehat{\Upsilon }}(x^{k_{1}})$.

The system (42)–(44) for the second dyadic shell $s=2$ can be solved following the same procedure following formulas (49)–(58) in section 2.3.6 of [6] (similar results were published in [36] and [2]). We have to re-define the coordinates and letters for the d-metric and s-connection coefficients following conventions of this paper. We obtain, respectively, from formula for $\, _{2}^{\shortmid }\gamma$ and Eqs. (42)–(44), this nonlinear system:

$$\begin{aligned} (\, _{2}^{\shortmid }\Psi )^{\diamond }g_{4}^{\diamond }= & {} 2g_{3}g_{4}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\, _{2}^{\shortmid }\Psi ), \end{aligned}$$

(A.18)

$$\begin{aligned} \sqrt{|g_{3}g_{4}|}\, _{2}^{\shortmid }\Psi= & {} g_{4}^{\diamond } \end{aligned}$$

(A.19)

$$\begin{aligned} \ (\, _{2}^{\shortmid }\Psi )^{\diamond }w_{i_{1}}-\partial _{i_{1}}(\, _{2}^{\shortmid }\Psi )= & {} \ 0 \end{aligned}$$

(A.20)

$$\begin{aligned} \ n_{i_{1}}^{\diamond \diamond }+\left( \ln \frac{|\ g_{4}|^{3/2}}{|g_{3}|} \right) ^{\diamond }n_{i_{1}}^{\diamond }= & {} 0,\ \end{aligned}$$

(A.21)

for $g_{4}^{\diamond }=\partial _{3}g_{4}=\partial g_{4}/\partial y^{3}=\partial g_{4}/\partial \varphi .$ Prescribing generating function and source, $\, _{2}^{\shortmid }\Psi$ and $\, _{2}^{\shortmid }\widehat{\Upsilon },$ we can integrate in general form this system with decoupling of equations. Let us prove this in new variables which are different from those used in section 2.3.6 of [6]. Let introduce:

$$\begin{aligned} \rho ^{2}:=-g_{3}g_{4} \end{aligned}$$

(A.22)

which allows us to write (A.18) and (A.19), respectively, in the form:

$$\begin{aligned} (\, _{2}^{\shortmid }\Psi )^{\diamond }g_{4}^{\diamond }=-2\rho ^{2}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\, _{2}^{\shortmid }\Psi )\quad \text{ and }\quad g_{4}^{\diamond }=\rho \, _{2}^{\shortmid }\Psi . \end{aligned}$$

(A.23)

Substituting in this line the value of $g_{4}^{\diamond }$ from the second equation into the first equation, we get:

$$\begin{aligned} \rho =-(\, _{2}^{\shortmid }\Psi )^{\diamond }/2(\, _{2}^{\shortmid }\widehat{ \Upsilon }). \end{aligned}$$

(A.24)

Introducing this $\rho$ into the second equation in (A.23) and integrating on $y^{3},$ we obtain:

$$\begin{aligned} \ g_{4}=g_{4}^{[0]}(x^{k_{1}})-\int \mathrm{d}y^{3}[(\, _{2}^{\shortmid }\Psi )^{2}]^{\diamond }/4(\, _{2}^{\shortmid }{\widehat{\Upsilon }}). \end{aligned}$$

(A.25)

This formula can be used in (A.22) and (A.24), which allows us to compute:

$$\begin{aligned} g_{3}=-\frac{1}{4g_{4}}\left( \frac{(\, _{2}^{\shortmid }\Psi )^{\diamond }}{ \, _{2}^{\shortmid }{\widehat{\Upsilon }}}\right) ^{2}=-\left( \frac{(\, _{2}^{\shortmid }\Psi )^{\diamond }}{2\, _{2}^{\shortmid }{\widehat{\Upsilon }}} \right) ^{2}\left( g_{4}^{[0]}(x^{k_{1}})-\int \mathrm{d}y^{3}\frac{[(\, _{2}^{\shortmid }\Psi )^{2}]^{\diamond }}{4(\, _{2}^{\shortmid }\widehat{ \Upsilon })}\right) ^{-1}. \end{aligned}$$

(A.26)

Having computed $g_{3}$ and $g_{4},$ we can integrate two times on $y^{3}$ Eq. (A.21), $\ n_{i_{1}}^{\diamond \diamond }+\left( \ln \frac{|g_{4}|^{3/2}}{|\ g_{3}|}\right) ^{\diamond }n_{i_{1}}^{\diamond }=0,$ when:

$$\begin{aligned} n_{k_{1}}(x^{k_{1}},y^{3})= & {} \, _{1}n_{k_{1}}+\, _{2}n_{k_{1}}\int \mathrm{d}y^{3}\ \frac{g_{3}}{|\ g_{4}|^{3/2}}=\, _{1}n_{k_{1}}+\, _{2}n_{k_{1}}\int \mathrm{d}y^{3}\left( \frac{(\, _{2}^{\shortmid }\Psi )^{\diamond }}{2\, _{2}^{\shortmid }{\widehat{\Upsilon }}}\right) ^{2}|\ g_{4}|^{-5/2} \nonumber \\= & {} \, _{1}n_{k_{1}}+\, _{2}n_{k_{1}}\int \mathrm{d}y^{3}\left( \frac{(\, _{2}^{\shortmid }\Psi )^{\diamond }}{2\, _{2}^{\shortmid }{\widehat{\Upsilon }}}\right) ^{2}\left| g_{4}^{[0]}(x^{k_{1}})-\int \mathrm{d}y^{3}[(\, _{2}^{\shortmid }\Psi )^{2}]^{\diamond }/4(\, _{2}^{\shortmid }{\widehat{\Upsilon }})\right| ^{-5/2}, \nonumber \\ \end{aligned}$$

(A.27)

with two integration functions $\, _{1}n_{k_{1}}(x^{i_{1}})$ and a re-defined $\, _{2}{\widetilde{n}}_{k_{1}}(x^{i_{1}})=\, _{2}{\widetilde{n}} _{k_{1}}(x^{i_{1}}).$ Finally, for the shell $s=2,$ we can solve the algebraic system (A.20) and find:

$$\begin{aligned} w_{i_{1}}=\partial _{i_{1}}\ (\, _{2}^{\shortmid }\Psi )/(\, _{2}^{\shortmid }\Psi )^{\diamond }. \end{aligned}$$

(A.28)

At the next step, we provide a proof for the shell $s=4$ (constructions for $s=3$ are similar but with less fiber momentum coordinates). Such constructions have not yet considered for cotangent bundles or Hamilton geometries. Introducing the values of coefficients $\alpha _{i_{3}},\, _{4}^{\shortmid }\beta ,\, _{4}^{\shortmid }\gamma$ into (48)–(50), we obtain a system of nonlinear PDEs with a decoupling property:

$$\begin{aligned} \begin{aligned} \ \ \, _{4}^{\shortmid }\Psi ^{*}\ (g^{7})^{*}&= 2\ g^{7}g^{8}\ (\, _{4}^{\shortmid }{\widehat{\Upsilon }})\, _{4}^{\shortmid }\Psi ,\\ \sqrt{|g^{7}g^{8}|}\, _{4}^{\shortmid }\Psi&=(g^{7})^{*}, \\ \ n_{i_{3}}^{**}+\left( \ln \frac{|g^{8}|}{|\ g^{7}|^{3/2}}\right) ^{*}n_{i_{3}}^{*}&=0,\ \\ \ \, _{4}^{\shortmid }\Psi ^{*}w_{i_{3}}-\partial _{i_{3}}(\, _{4}^{\shortmid }\Psi )&=\ 0,\ \end{aligned} \end{aligned}$$

(A.29)

for $(g^{7})^{*}=\partial ^{8}g^{7}=\partial g^{7}/\partial p_{8}=\partial g^{7}/\partial E.$ This system can be integrated in very general forms by prescribing $\, _{4}^{\shortmid }\Psi (x^{k_{1}},y^{3},t,p_{a_{3}},E)$ and $\ \ \, _{4}^{\shortmid }\widehat{ \Upsilon }(x^{k_{1}},y^{3},t,p_{a_{3}},E),$ where $k_{1}=1,2;a_{3}=5,6;i_{3}=1,2,\ldots 6;a_{4}=7,8;$ and $x^{4}=y^{4}=t$ and $p_{8}=E.$ Introducing the function:

$$\begin{aligned} \ (\, _{4}\rho )^{2}:=-g^{7}g^{8}, \end{aligned}$$

(A.30)

(the sign - is motivated by the pseudo-Euclidean signature; this sign is + for $s=3$), we express the first two equations in above system in the form:

$$\begin{aligned} \, _{4}^{\shortmid }\Psi ^{*}\ (g^{7})^{*}=-2\ (\, _{4}\rho )^{2}\ (\, _{4}^{\shortmid }{\widehat{\Upsilon }})\, _{4}^{\shortmid }\Psi \text{ and } \ \ (g^{7})^{*}=\, _{4}\rho (\, _{4}^{\shortmid }\Psi ). \end{aligned}$$

(A.31)

Then, we substitute in the first equation of (A.29) the value of $(g^{7})^{*}$ given by the second equation (A.31) and obtain:

$$\begin{aligned} \, _{4}\rho =-\, _{4}^{\shortmid }\Psi ^{*}/2\ (\, _{4}^{\shortmid } {\widehat{\Upsilon }}). \end{aligned}$$

(A.32)

Using this value and the second equation of (A.31) and integrating on E,  we obtain:

$$\begin{aligned} \ g^{7}=g_{[0]}^{7}(x^{k_{1}},y^{3},t,p_{a_{3}})-\int \mathrm{d}E[(\, _{4}^{\shortmid }\Psi )^{2}]^{*}/4(\, _{4}^{\shortmid }{\widehat{\Upsilon }}), \end{aligned}$$

(A.33)

where $g_{[0]}^{7}(x^{k_{3}})$ is an integration function. At the next step, considering formulas (A.30), (A.32), (A.33), we compute:

$$\begin{aligned} \ g^{8}=-\frac{1}{4\ g^{7}}\left( \frac{\ \, _{4}^{\shortmid }\Psi ^{*}}{\, _{4}^{\shortmid }{\widehat{\Upsilon }}}\right) ^{2}=-\frac{1}{4}\left( \frac{\ \, _{4}^{\shortmid }\Psi ^{*}}{\, _{4}^{\shortmid }{\widehat{\Upsilon }}} \right) ^{2}\Bigg /\left( g_{[0]}^{7}-\frac{1}{4}\int \mathrm{d}E\frac{[(\, _{4}^{\shortmid }\Psi )^{2}]^{*}}{\, _{4}^{\shortmid }{\widehat{\Upsilon }}}\right) . \end{aligned}$$

(A.34)

Now, we can perform an integration of the third equation in (A.29). The first subset of the N-connection coefficients, $n_{i_{3}}^{*},$ are found by integrating two times on E in that equations written in the form:

$$\begin{aligned} \ n_{i_{3}}^{**}=(n_{i_{3}}^{*})^{*}=-\ n_{i_{3}}^{*}(\ln |g^{7}|^{3/2}/|g^{8}|)^{*} \end{aligned}$$

for the coefficient $\, _{4}^{\shortmid }\gamma$ defined in for Eq. (49) of the theorem. Using explicit values (A.33) and (A.34), we find:

$$\begin{aligned}&n_{k_{3}}(x^{k_{1}},y^{3},t,p_{a_{3}},E) \\&\quad =\, _{1}n_{k_{3}}+\, _{2}n_{k_{3}}\int \mathrm{d}E\ \frac{|\ g^{8}|}{|g^{7}|^{3/2}}=\, _{1}n_{k_{3}}+\, _{2} {\widetilde{n}}_{k_{3}}\int \mathrm{d}E\ \left( \frac{\ \, _{4}^{\shortmid }\Psi ^{*} }{\, _{4}^{\shortmid }{\widehat{\Upsilon }}}\right) ^{2}|\ g_{7}|^{-5/2} \\&\quad =\, _{1}n_{k_{1}}+\, _{2}n_{k_{1}}\int \mathrm{d}E\left( \frac{\ \, _{4}^{\shortmid }\Psi ^{*}}{\, _{4}^{\shortmid }{\widehat{\Upsilon }}}\right) ^{2}\left| g_{[0]}^{7}(x^{k_{1}},y^{3},t,p_{a_{3}})-\int \mathrm{d}E[(\, _{4}^{\shortmid }\Psi )^{2}]^{*}/4(\, _{4}^{\shortmid }{\widehat{\Upsilon }} )\right| ^{-5/2}, \end{aligned}$$

where two integration functions are parameterized $\, _{1}n_{k_{3}}(x^{i_{3}})=$$\, _{1}n_{k_{3}}(x^{k_{1}},y^{3},t,p_{a_{3}})$ and a re-defined $\, _{2}{\widetilde{n}}_{k_{3}}(x^{i_{3}})=\, _{2}{\widetilde{n}} _{k_{3}}(x^{k_{1}},y^{3},t,p_{a_{3}}).$

The second subset of N-connection coefficients, $w_{i_{3}},$ can be easily found as a solution of the linear algebraic equations in (A.29), $w_{i_{3}}=\partial _{i_{3}}\ (\, _{4}^{\shortmid }\Psi )/(\, _{4}^{\shortmid }\Psi )^{*}$.

Putting together above formulas for the s-metric and N-connection coefficients on the shell $s=4,$ we construct a general solution of the system (48)–(50):

$$\begin{aligned} \ \ g^{7}= & {} g_{[0]}^{7}-\int \mathrm{d}E[(\, _{4}^{\shortmid }\Psi )^{2}]^{*}/4(\, _{4}^{\shortmid }{\widehat{\Upsilon }});\ \nonumber \\ \ g^{8}= & {} -\frac{1}{4\ g^{7}}\left( \frac{\ \, _{4}^{\shortmid }\Psi ^{*}}{\, _{4}^{\shortmid }{\widehat{\Upsilon }}}\right) ^{2}=-\frac{1}{4} \left( \frac{\ \, _{4}^{\shortmid }\Psi ^{*}}{\, _{4}^{\shortmid }{\widehat{\Upsilon }}} \right) ^{2}\Bigg /\left( g_{[0]}^{7}-\frac{1}{4}\int \mathrm{d}E\frac{[(\, _{4}^{\shortmid }\Psi )^{2}]^{*}}{\, _{4}^{\shortmid }{\widehat{\Upsilon }}}\right) ; \nonumber \\ n_{k_{3}}= & {} \, _{1}n_{k_{1}}+\, _{2}n_{k_{1}}\int \mathrm{d}E\left( \frac{\ \, _{4}^{\shortmid }\Psi ^{*}}{\, _{4}^{\shortmid }{\widehat{\Upsilon }}} \right) ^{2}\left| g_{[0]}^{7}(x^{k_{1}},y^{3},t,p_{a_{3}})-\int \mathrm{d}E[(\, _{4}^{\shortmid }\Psi )^{2}]^{*}/4(\, _{4}^{\shortmid }{\widehat{\Upsilon }} )\right| ^{5/2}; \nonumber \\ w_{i_{3}}= & {} \partial _{i_{3}}\ (\, _{4}^{\shortmid }\Psi )/(\, _{4}^{\shortmid }\Psi )^{*}. \end{aligned}$$

(A.35)

Finally, introducing all s-metric and N-connection elements for all shell, we can construct the quadratic line element (51).

1.6 Proof of Theorem 5.1

We provide some details of geometric computations to show how such nonholonomic parametric deformations can be constructed in explicit form. There used the formulas (68), (69), and Definition 5.2.

Let us consider the shell $s=1.$ Writing:

$$\begin{aligned} \ ^{\shortmid }g_{i_{1}}=\zeta _{i_{1}}(1+\varepsilon \chi _{i_{1}})\ ^{\shortmid }\mathring{g}_{i_{1}}=e^{\psi (x^{k_{1}})}\approx e^{\psi _{0}(x^{k_{1}})(1+\varepsilon \ ^{\psi }\chi (x^{k_{1}}))}\approx e^{\psi _{0}(x^{k_{1}})}(1+\varepsilon \ ^{\psi }\chi ), \end{aligned}$$

we obtain $\zeta _{i_{1}}\mathring{g}_{i_{1}}=e^{\psi _{0}(x^{k_{1}})}$ and $\chi _{i_{1}}\ ^{\shortmid }\mathring{g}_{i_{1}}=\ ^{\psi }\chi .$

For $s=2,$ when $\ ^{\shortmid }\eta _{\alpha _{2}}=\zeta _{\alpha _{2}}(1+\varepsilon \chi _{\alpha _{2}})$ and$\ ^{\shortmid }\eta _{i_{1}}^{a_{2}}=\zeta _{i_{1}}^{a_{2}}(1+\varepsilon \chi _{i_{1}}^{a_{2}}),$ i.e., for $\ ^{\shortmid }\eta _{3}=\zeta _{3}(1+\varepsilon \chi _{3})$ and $\ ^{\shortmid }\eta _{4}=\zeta _{4}(1+\varepsilon \chi _{4}),$ we can express all $\varepsilon$-polarizations as functionals on generating data $\zeta _{4}$ and $\chi _{4}.$ We perform such computations for respective coefficients of a d-metric and N-connections:

$$\begin{aligned}&\ ^{\shortmid }\eta _{3}\ ^{\shortmid }\mathring{g}_{3}\\&\quad =-\frac{[(\ ^{\shortmid }\eta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }]^{2}}{ |\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\ ^{\shortmid }\eta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }|\ (\ ^{\shortmid }\eta _{4}\ ^{\shortmid }\mathring{g}_{4})}=-4\frac{[(|\ ^{\shortmid }\eta _{4}\ ^{\shortmid }\mathring{g}_{4}|^{1/2})^{\diamond }]^{2}}{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\ ^{\shortmid }\eta _{4}\ ^{\shortmid } \mathring{g}_{4})^{\diamond }|\ }\\&\quad \simeq \zeta _{3}(1+\varepsilon \chi _{3})\ ^{\shortmid }\mathring{g}_{3}\\&\quad =-4 \frac{[(|\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4}(1+\varepsilon \chi _{4})|^{1/2})^{\diamond }]^{2}}{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }\widehat{ \Upsilon })[(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})(1+\varepsilon \chi _{4})]^{\diamond }|} \\&\quad =-4\frac{[(|\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4}|^{1/2}\ |1+\varepsilon \chi _{4}\ |^{1/2})^{\diamond }]^{2}}{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})[(\ \zeta _{4}\ ^{\shortmid }\mathring{g }_{4})+\varepsilon \ (\zeta _{4}\chi _{4}\ ^{\shortmid }\mathring{g} _{4})]^{\diamond }|}\\&\quad =-4\frac{[(|\ \zeta _{4}\ ^{\shortmid }\mathring{g} _{4}|^{1/2}|1+\frac{\varepsilon }{2}\chi _{4}|)^{\diamond }]^{2}}{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})\{(\ \zeta _{4}\ ^{\shortmid } \mathring{g}_{4})^{\diamond }+\varepsilon [(\zeta _{4}\chi _{4})\ ^{\shortmid }\mathring{g}_{4})]^{\diamond }\}|} \\&\quad =-4\frac{[(|\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4}|^{1/2}+\frac{ \varepsilon }{2}\chi _{4}|\ \zeta _{4}\ ^{\shortmid }\mathring{g} _{4}|^{1/2})^{\diamond }]^{2}}{|\int \mathrm{d}y^{3}[(\, _{2}^{\shortmid }\widehat{ \Upsilon })(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }]+\varepsilon \int \mathrm{d}y^{3}\{(\, _{2}^{\shortmid }{\widehat{\Upsilon }})[(\zeta _{4}\chi _{4})\ ^{\shortmid }\mathring{g}_{4})]^{\diamond }\}|} \\&\quad =-4\frac{[(|\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4}|^{1/2})^{\diamond }+\frac{\varepsilon }{2}(\chi _{4}|\ \zeta _{4}\ ^{\shortmid }\mathring{g} _{4}|^{1/2})^{\diamond }]^{2}}{|\int \mathrm{d}y^{3}[(\, _{2}^{\shortmid }\widehat{ \Upsilon })(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }\}|(1+\varepsilon \frac{\int \mathrm{d}y^{3}\{(\, _{2}^{\shortmid }{\widehat{\Upsilon }} )[\ (\zeta _{4}\chi _{4})\ ^{\shortmid }\mathring{g}_{4})]^{\diamond }\}}{ \int \mathrm{d}y^{3}\{(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }\}}\}|} \\&\quad =-4\frac{[(|\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4}|^{1/2})^{\diamond }+ \frac{\varepsilon }{2}(\chi _{4}|\ \zeta _{4}\ ^{\shortmid }\mathring{g} _{4}|^{1/2})^{\diamond }]^{2}}{|\int \mathrm{d}y^{3}\{(\, _{2}^{\shortmid }\widehat{ \Upsilon })(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }\}|\left( 1+\varepsilon \frac{\int \mathrm{d}y^{3}\{(\, _{2}^{\shortmid }{\widehat{\Upsilon }} )[(\zeta _{4}\chi _{4})\ ^{\shortmid }\mathring{g}_{4})]^{\diamond }\}}{\int \mathrm{d}y^{3}[(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\ \zeta _{4}\ ^{\shortmid } \mathring{g}_{4})^{\diamond }]}\right) } \\&\quad =-4\frac{[(|\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4}|^{1/2})^{\diamond }]^{2}\left[ 1+\frac{\varepsilon }{2}\frac{(\chi _{4}|\ \zeta _{4}\ ^{\shortmid } \mathring{g}_{4}|^{1/2})^{\diamond }}{(|\ \zeta _{4}\ ^{\shortmid }\mathring{ g}_{4}|^{1/2})^{\diamond }}\right] ^{2}\left( 1-\varepsilon \frac{\int \mathrm{d}y^{3}\{(\, _{2}^{\shortmid }{\widehat{\Upsilon }})[\ (\zeta _{4}\chi _{4})\ ^{\shortmid } \mathring{g}_{4})]^{\diamond }\}}{\int \mathrm{d}y^{3}[(\, _{2}^{\shortmid }\widehat{ \Upsilon })(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }]}\right) }{ |\int \mathrm{d}y^{3}[(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }]|} \\&\quad =-4\frac{[(|\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4}|^{1/2})^{\diamond }]^{2}}{|\int \mathrm{d}y^{3}[(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }]|}\\&\qquad \left[ 1+\varepsilon \left( \frac{(\chi _{4}|\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4}|^{1/2})^{ \diamond }}{(|\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4}|^{1/2})^{\diamond } }-\frac{\int \mathrm{d}y^{3}\{(\, _{2}^{\shortmid }{\widehat{\Upsilon }})[(\zeta _{4}\chi _{4})\ ^{\shortmid }\mathring{g}_{4})]^{\diamond }\}}{\int \mathrm{d}y^{3}[(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\ \zeta _{4}\ ^{\shortmid } \mathring{g}_{4})^{\diamond }]}\right) \right] . \end{aligned}$$

From these formulas, we express:

$$\begin{aligned} \zeta _{3}\ ^{\shortmid }\mathring{g}_{3}= & {} -4\frac{[(|\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4}|^{1/2})^{\diamond }]^{2}}{|\int \mathrm{d}y^{3}[(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\ \zeta _{4}\ ^{\shortmid }\mathring{g} _{4})^{\diamond }]|}\quad \text{ and } \\ \text{ proportional } \text{ to } \varepsilon :\chi _{3}= & {} \frac{(\chi _{4}|\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4}|^{1/2})^{\diamond }}{4(|\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4}|^{1/2})^{\diamond }}-\frac{\int \mathrm{d}y^{3}\{(\, _{2}^{\shortmid }{\widehat{\Upsilon }})[(\zeta _{4}\ ^{\shortmid }\mathring{g} _{4})\chi _{4}]^{\diamond }\}}{\int \mathrm{d}y^{3}[(\, _{2}^{\shortmid }\widehat{ \Upsilon })(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }]}. \end{aligned}$$

For the N-connection coefficients, we have:

$$\begin{aligned} \ ^{\shortmid }\eta _{i_{1}}^{3}\ ^{\shortmid }\mathring{N}_{i_{1}}^{3}= & {} \frac{\partial _{i_{1}}\ \int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }} )\ (\ ^{\shortmid }\eta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }}{(\, _{2}^{\shortmid }{\widehat{\Upsilon }})\ (\ ^{\shortmid }\eta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }} \\ \simeq \zeta _{i_{1}}^{3}(1+\varepsilon \chi _{i_{1}}^{3})\ ^{\shortmid }\mathring{N }_{i_{1}}^{3}= & {} \frac{\partial _{i_{1}}\ \int \mathrm{d}y^{3}(\, _{2}^{\shortmid } {\widehat{\Upsilon }})\ [\zeta _{4}(1+\varepsilon \chi _{4})]^{\diamond }}{(\, _{2}^{\shortmid }{\widehat{\Upsilon }})\ [\zeta _{4}(1+\varepsilon \chi _{4})]^{\diamond }}\\= & {} \frac{\partial _{i_{1}}\ \int \mathrm{d}y^{3}(\, _{2}^{\shortmid } {\widehat{\Upsilon }})\ \{(\zeta _{4})^{\diamond }+\varepsilon [(\zeta _{4}\chi _{4})]^{\diamond }\}}{(\, _{2}^{\shortmid }{\widehat{\Upsilon }})\ \{(\zeta _{4})^{\diamond }+\varepsilon [(\zeta _{4}\chi _{4})]^{\diamond }\}} \\= & {} \frac{\partial _{i_{1}}\ \left\{ \int \mathrm{d}y^{3}(\, _{2}^{\shortmid }\widehat{\Upsilon })(\zeta _{4})^{\diamond }\ \left[ 1+\varepsilon \frac{(\zeta _{4}\chi _{4})^{\diamond }}{(\zeta _{4})^{\diamond }}\right] \right\} }{(\, _{2}^{\shortmid } {\widehat{\Upsilon }})\ (\zeta _{4})^{\diamond }\left[ 1+\varepsilon \frac{(\zeta _{4}\chi _{4})^{\diamond }}{(\zeta _{4})^{\diamond }}\right] }\\= & {} \frac{\partial _{i_{1}}\ [\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4})^{\diamond }]\ +\varepsilon \partial _{i_{1}}[\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\chi _{4})^{\diamond }]}{(\, _{2}^{\shortmid }{\widehat{\Upsilon }})\ (\zeta _{4})^{\diamond }}\times \left[ 1-\varepsilon \frac{(\zeta _{4}\chi _{4})^{\diamond }}{(\zeta _{4})^{\diamond }}\right] \\= & {} \frac{\{\partial _{i_{1}}\ [\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }\widehat{ \Upsilon })(\zeta _{4})^{\diamond }]\}\left\{ 1+\varepsilon \frac{\partial _{i_{1}}[\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\chi _{4})^{\diamond }]}{\partial _{i_{1}}\ [\int \mathrm{d}y^{3}(\, _{2}^{\shortmid } {\widehat{\Upsilon }})(\zeta _{4})^{\diamond }]}\right\} }{(\, _{2}^{\shortmid } {\widehat{\Upsilon }})\ (\zeta _{4})^{\diamond }}\left\{ 1-\varepsilon \frac{(\zeta _{4}\chi _{4})^{\diamond }}{(\zeta _{4})^{\diamond }}\right\} \\= & {} \frac{\partial _{i_{1}}\ [\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }\widehat{ \Upsilon })(\zeta _{4})^{\diamond }]}{(\, _{2}^{\shortmid }{\widehat{\Upsilon }} )\ (\zeta _{4})^{\diamond }}\left\{ 1+\varepsilon \left( \frac{\partial _{i_{1}}[\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\chi _{4})^{\diamond }]}{\partial _{i_{1}}\ [\int \mathrm{d}y^{3}(\, _{2}^{\shortmid } {\widehat{\Upsilon }})(\zeta _{4})^{\diamond }]}-\frac{(\zeta _{4}\chi _{4})^{\diamond }}{(\zeta _{4})^{\diamond }}\right) \right\} . \end{aligned}$$

It should be noted that there is not summation on repeating indices, because they are not arranged on the rule “up-low”. From these formulas, we obtain:

$$\begin{aligned} \zeta _{i_{1}}^{3}\ ^{\shortmid }\mathring{N}_{i_{1}}^{3}=\frac{\partial _{i_{1}}[\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})\ (\zeta _{4})^{\diamond }]}{(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4})^{\diamond }}\quad \text{ and }\quad \varepsilon :\chi _{i_{1}}^{3}=\frac{\partial _{i_{1}}[\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\chi _{4})^{\diamond }]}{\partial _{i_{1}}\ [\int \mathrm{d}y^{3}(\, _{2}^{\shortmid } {\widehat{\Upsilon }})(\zeta _{4})^{\diamond }]}-\frac{(\zeta _{4}\chi _{4})^{\diamond }}{(\zeta _{4})^{\diamond }}. \end{aligned}$$

Finally (for this shell), we compute:

$$\begin{aligned}&\ ^{\shortmid }\eta _{k_{1}}^{4}\ ^{\shortmid }\mathring{N}_{k_{1}}^{4}\\&\quad =\, _{1}n_{k_{1}}+\, _{2}n_{k_{1}}\int \mathrm{d}y^{3}\frac{[(\ ^{\shortmid }\eta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }]^{2}}{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\ ^{\shortmid }\eta _{4}\ ^{\shortmid } \mathring{g}_{4})^{\diamond }|\ (\ ^{\shortmid }\eta _{4}\ ^{\shortmid } \mathring{g}_{4})^{5/2}} \\&\quad =\, _{1}n_{k_{1}}+16\ \, _{2}n_{k_{1}}\int \mathrm{d}y^{3}\frac{\left( [(\ ^{\shortmid }\eta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }\right) ^{2}}{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\ ^{\shortmid }\eta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }|\ }\\&\quad \simeq \zeta _{k_{1}}^{4}(1+\varepsilon \chi _{k_{1}}^{4})\ ^{\shortmid }\mathring{N }_{k_{1}}^{4}\\&\quad =\, _{1}n_{k_{1}}+16\, _{2}n_{k_{1}}\left[ \int \mathrm{d}y^{3}\frac{ \left( \{[(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})(1+\varepsilon \chi _{4})]^{-1/4}\}^{\diamond }\right) ^{2}}{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid } {\widehat{\Upsilon }})[(\zeta _{4}(1+\varepsilon \chi _{4})\ ^{\shortmid } \mathring{g}_{4})]^{\diamond }|\ }\right] \\&\quad =\, _{1}n_{k_{1}}+16\, _{2}n_{k_{1}}\left[ \int \mathrm{d}y^{3}\frac{\{\left[ (\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}\left( 1-\frac{\varepsilon }{4}\chi _{4}\right) \right] ^{\diamond }\}^{2}}{|\int \mathrm{d}y^{3}\{(\, _{2}^{\shortmid }\widehat{ \Upsilon })(\zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }+\varepsilon (\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\chi _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }\}|\ }\right] \\&\quad =\, _{1}n_{k_{1}}+16\, _{2}n_{k_{1}}\left[ \int \mathrm{d}y^{3}\frac{\{[(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}-\frac{\varepsilon }{4}(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}\chi _{4})]^{\diamond }\}^{2}}{ |\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }|\left\{ 1+\varepsilon \frac{\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\chi _{4}\ ^{\shortmid } \mathring{g}_{4})^{\diamond }}{\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }\widehat{ \Upsilon })(\zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }}\right\} }\right] \\&\quad = \, _{1}n_{k_{1}}+16\, _{2}n_{k_{1}}\left[ \int \mathrm{d}y^{3}\frac{\{[(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }-\frac{\varepsilon }{ 4}[(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}\chi _{4})]^{\diamond }\}^{2}}{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }|\left\{ 1+\varepsilon \frac{\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\chi _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }}{\int \mathrm{d}y^{3}(\, _{2}^{\shortmid } {\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }} \right\} }\right] \\&\quad =\, _{1}n_{k_{1}}+16\, _{2}n_{k_{1}}\left[ \int \mathrm{d}y^{3}\frac{\{[(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }\left( 1-\frac{ \varepsilon }{4}\frac{[(\ \zeta _{4}\ ^{\shortmid }\mathring{g} _{4})^{-1/4}\chi _{4})]^{\diamond }}{[(\ \zeta _{4}\ ^{\shortmid }\mathring{g }_{4})^{-1/4}]^{\diamond }}\right) \}^{2}}{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid } {\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }|\left\{ 1+\varepsilon \frac{\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }} )(\zeta _{4}\chi _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }}{\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid } \mathring{g}_{4})^{\diamond }}\right\} }\right] \\&\quad =\, _{1}n_{k_{1}}+16\, _{2}n_{k_{1}}\left[ \int \mathrm{d}y^{3}\frac{\left( [(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }\right) ^{2}\left( 1- \frac{\varepsilon }{2}\frac{[(\ \zeta _{4}\ ^{\shortmid }\mathring{g} _{4})^{-1/4}\chi _{4})]^{\diamond }}{[(\ \zeta _{4}\ ^{\shortmid }\mathring{g }_{4})^{-1/4}]^{\diamond }}\right) }{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid } {\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }|\left\{ 1+\varepsilon \frac{\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }} )(\zeta _{4}\chi _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }}{\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid } \mathring{g}_{4})^{\diamond }}\right\} }\right] \\&\quad =\, _{1}n_{k_{1}}+16\, _{2}n_{k_{1}}\left\{ \int \mathrm{d}y^{3}\frac{\left( [(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }\right) ^{2}}{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid } \mathring{g}_{4})^{\diamond }|}\right. \\&\qquad \left. \left[ 1-\varepsilon \left( \frac{[(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}\chi _{4})]^{\diamond }}{2[(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }}+\frac{\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\chi _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }}{\int \mathrm{d}y^{3}(\, _{2}^{\shortmid } {\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }} \right) \right] \right\} \\ \end{aligned}$$

$$\begin{aligned}&\quad =\, _{1}n_{k_{1}}+16\, _{2}n_{k_{1}}\left[ \int \mathrm{d}y^{3}\frac{\left( [(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }\right) ^{2}}{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid } \mathring{g}_{4})^{\diamond }|}\right] \\&\qquad -\varepsilon 16\, _{2}n_{k_{1}}\int \mathrm{d}y^{3}\frac{\left( [(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }\right) ^{2}}{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid } \mathring{g}_{4})^{\diamond }|}\left( \frac{[(\ \zeta _{4}\ ^{\shortmid } \mathring{g}_{4})^{-1/4}\chi _{4})]^{\diamond }}{2[(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }}\right. \\&\qquad \left. +\frac{\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\chi _{4}\ ^{\shortmid } \mathring{g}_{4})^{\diamond }}{\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }\widehat{ \Upsilon })(\zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }}\right) \\&\quad =\left\{ \, _{1}n_{k_{1}}+16\, _{2}n_{k_{1}}\left[ \int \mathrm{d}y^{3}\frac{\left( [(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }\right) ^{2}}{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid } \mathring{g}_{4})^{\diamond }|}\right] \right\} \\&\qquad \times \left[ 1-\varepsilon \frac{16\, _{2}n_{k_{1}}\int \mathrm{d}y^{3}\frac{\left( [(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }\right) ^{2}}{ |\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }|}\left( \frac{[(\ \zeta _{4}\ ^{\shortmid } \mathring{g}_{4})^{-1/4}\chi _{4})]^{\diamond }}{2[(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }}+\frac{\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\chi _{4}\ ^{\shortmid } \mathring{g}_{4})^{\diamond }}{\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }\widehat{ \Upsilon })(\zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }}\right) }{ \, _{1}n_{k_{1}}+16\, _{2}n_{k_{1}}[\int \mathrm{d}y^{3}\frac{\left( [(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }\right) ^{2}}{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid } \mathring{g}_{4})^{\diamond }|}]}\right] \end{aligned}$$

$$\begin{aligned} \text{ where } \zeta _{k_{1}}^{4}\ ^{\shortmid }\mathring{N}_{k_{1}}^{4}= & {} \, _{1}n_{k_{1}}+16\, _{2}n_{k_{1}}[\int \mathrm{d}y^{3}\{\frac{\left( [(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }\right) ^{2}}{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid } \mathring{g}_{4})^{\diamond }|} \text{ and } \\ \varepsilon :\chi _{k_{1}}^{4}= & {} \ -\frac{16\, _{2}n_{k_{1}}\int \mathrm{d}y^{3}\frac{ \left( [(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }\right) ^{2}}{|\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }|}\left( \frac{[(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}\chi _{4})]^{\diamond }}{2[(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }}+\frac{\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})[(\zeta _{4}\chi _{4}\ ^{\shortmid }\mathring{g}_{4})]^{\diamond }}{\int \mathrm{d}y^{3}(\, _{2}^{\shortmid } {\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }} \right) }{\, _{1}n_{k_{1}}+16\, _{2}n_{k_{1}}\left[ \int \mathrm{d}y^{3}\frac{\left( [(\ \zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{-1/4}]^{\diamond }\right) ^{2}}{ |\int \mathrm{d}y^{3}(\, _{2}^{\shortmid }{\widehat{\Upsilon }})(\zeta _{4}\ ^{\shortmid }\mathring{g}_{4})^{\diamond }|}\right] }. \end{aligned}$$

Computations for shells $s=3$ and $s=4$ are similar but with re-definition of corresponding symbols for shall coordinates and indices.

Relativistic models of Finsler–Lagrange–Hamilton phase spaces

In this Appendix, we summarize necessary results on Finsler–Lagrange–Hamilton geometries modeled on (co) tangent bundles TV and $T^{*}V$ on 4-d Lorentz manifolds, see detail with proofs in [1] and references therein.

1.1 Generating functions determined by MDR indicators

For any MDR of type (1), we can construct an effective Hamiltonian:

$$\begin{aligned} H(p):=E=\pm (c^{2}\overrightarrow{{\mathbf {p}}}^{2}+c^{4}m^{2}-\varpi (E, \overrightarrow{{\mathbf {p}}},m;\ell _{P}))^{1/2}. \end{aligned}$$

(B.1)

This Hamiltonian describes relativistic point particles propagating in a typical cofiber of a $T^{*}V$ over a point $x=\{x^{i}\}\in V.$ Globalizing the constructions for a Lorentz manifold basis, with an effective phase space endowed with local coordinates $(x^{i},p_{a}),$ we obtain indicators $\varpi (x^{i},E,\overrightarrow{{\mathbf {p}}},m;\ell _{P})$ depending both on spacetime and phase space coordinates. Considering general systems of frames/coordinates and their transforms on total dual bundle $T^{*}V,$ the Hamiltonian (B.1) can be written in the form H(x, p).

Similarly to relativistic mechanics, we can define (inverse) Legendre transforms and the concept of L-duality for certain Lagrange and Hamilton densities. The Legendre transforms, $L\rightarrow H,$ are constructed:

$$\begin{aligned} H(x,p):=p_{a}v^{a}-L(x,v) \end{aligned}$$

(B.2)

and $v^{a}$ determining solutions of the equations $p_{a}=\partial L(x,v)/\partial v^{a}.$ In a similar manner, the inverse Legendre transforms, $H\rightarrow L,$ are constructed:

$$\begin{aligned} L(x,v):=p_{a}v^{a}-H(x,p) \end{aligned}$$

(B.3)

and $p_{a}$ determining solutions of the equations $y^{a}=\partial H(x,p)/\partial p_{a}.$

Our main assumption is that MGTs with MDRs are described by basic Lorentzian and non-Riemannian total phase space geometries determined by nonlinear quadratic line elements:

$$\begin{aligned} \mathrm{d}s_{L}^{2}= & {} L(x,y), \text{ for } \text{ models } \text{ on } TV; \end{aligned}$$

(B.4)

$$\begin{aligned} d\ ^{\shortmid }s_{H}^{2}= & {} H(x,p), \text{ for } \text{ models } \text{ on } T^{*}V. \end{aligned}$$

(B.5)

The functions (B.4) and (B.5) are corresponding called Lagrange and Hamilton fundamental (equivalent, generating) functions. For localized zero indicators in (1), $\varpi =0,$ (B.4) and (B.5) transform correspondingly into linear quadratic elements (2) and (3) and in (pseudo) Riemannian geometry extended on tangent bundles.

A relativistic 4-d model of Lagrange space $L^{3,1}=(TV,L(x,y))$ is defined by a fundamental/generating Lagrange function, $TV\ni (x,y)\rightarrow L(x,y)\in {\mathbb {R}},$ which is a real-valued and differentiable function on ${\widetilde{TV}}:=TV/\{0\},$ for $\{0\}$ being the null section of TV,  and continuous on the null section of $\pi :TV\rightarrow V.$ We say that such a model is regular if the vertical metric (v-metric, Hessian):

$$\begin{aligned} {\widetilde{g}}_{ab}(x,y):=\frac{1}{2}\frac{\partial ^{2}L}{\partial y^{a}\partial y^{b}} \end{aligned}$$

(B.6)

is non-degenerate, i.e., $\det |{\widetilde{g}}_{ab}|\ne 0,$ and of constant signature.

In a similar form, we can introduce a 4-d relativistic model of Hamilton space $H^{3,1}=(T^{*}V,H(x,p))$ which is determined by a fundamental/generating Hamilton function on a Lorentz manifold V. Such a real-valued function is constructed as $T^{*}V\ni (x,p)\rightarrow H(x,p)\in {\mathbb {R}}$ subjected to the conditions that it is differentiable on $\widetilde{T^{*}V}:=T^{*}V/\{0^{*}\},$ for $\{0^{*}\}$ being the null section of $T^{*}V,$ and continuous on the null section of $\pi ^{*}:\ T^{*}V\rightarrow V.$ Such a model is regular if the covertical metric (cv-metric, Hessian):

$$\begin{aligned} \ \ ^{\shortmid }{\widetilde{g}}^{ab}(x,p):=\frac{1}{2}\frac{\partial ^{2}H}{ \partial p_{a}\partial p_{b}} \end{aligned}$$

(B.7)

is non-degenerate, i.e., $\det |\ ^{\shortmid }{\widetilde{g}}^{ab}|\ne 0,$ and of constant signature.

We shall use tilde “$^{\sim }$”, for instance, for values ${\widetilde{g}} _{ab}$ and $\ ^{\shortmid }{\widetilde{g}}^{ab},$ to emphasize that certain geometric objects are defined canonically by respective Lagrange and Hamilton generating functions. In their turn, such fundamental functions may encode various types of MDRs and LIVs terms, etc. For general frame/coordinate transforms on TV and/or $T^{*}V,$ we can express any “tilde” value in a “non-tilde” form. In such cases, we shall write $g_{ab}(x,v),$ for a v-metric, and $\ ^{\shortmid }g^{ab}(x,p),$ for a cv-metric. Inversely, prescribing any v-metric or cv-metric structure, we can consider such (co) frame/coordinate systems when the geometric values can be transformed into certain canonical ones with “tilde”. Here, we note that, in general, a v-metric $g_{ab}$ is different from the inverse of $\ ^{\shortmid }g^{ab}$ and from $\ ^{\shortmid }g_{ab}.$

It is important to emphasize that a relativistic 4-d model of Finsler space is an example of relativistic Lagrange space when a regular $L=F^{2}$ is defined by a fundamental (generating) Finsler function subjected to certain additional conditions:

• the fundamental/generating Finsler function F is a real positive valued one which is differential on ${\widetilde{TV}}$ and continuous on the null section of the projection $\pi :TV\rightarrow V;$

• F satisfies the homogeneity condition $F(x,\lambda v)=|\lambda |$F(x, v),  for a nonzero real value $\lambda ;$ and

• Hessian (B.6) is defined by $F^{2}$ in such a form that in any point $(x_{(0)},v_{(0)})$ the v-metric is of signature $(+++-).$

In a similar form, there are defined relativistic 4-d Cartan spaces $C^{3,1}=(V,C(x,p)),$ when $H=C^{2}(x,p)$ is 1-homogeneous on cofiber coordinates $p_{a}.$ In principle, we can always introduce on a TV and/or $T^{*}V$ a subclass of Finsler and/or Cartan variables using respective classes for frame/coordinate transforms. For simplicity, we shall prefer to use in this work terms like Lagrange and/or Hamilton geometry, or Lagrange–Hamilton geometry, considering that constructions with Finsler and/or Cartan structures as certain particular examples.

1.2 Canonical N-connections and adapted metrics

There are possible coordinate-free and N-adapted, or local, coefficient forms for defining geometric objects in Lagrange–Hamilton geometry. Certain formulas and results will be written in coefficient forms, with respect to N-adapted frames, which is important for constructing in explicit form exact and parametric solutions following the AFDM.

1.2.1 N-connections for Lagrange–Hamilton spaces

Any MDR (1) defines L-dual, i.e., related via Legendre transforms, canonical relativistic models of Hamilton space ${\widetilde{H}} ^{3,1}=(T^{*}V,{\widetilde{H}}(x,p))$ and Lagrange space ${\widetilde{L}} ^{3,1}=(TV,{\widetilde{L}}(x,y)).$

Following standard geometric and variational calculus, we can prove such results: the dynamics of a probing point particle in L-dual effective phase spaces ${\widetilde{H}}^{3,1}$ and ${\widetilde{L}}^{3,1}$ is described by fundamental generating functions ${\widetilde{H}}$ and ${\widetilde{L}}$ subjected to solve respective via Hamilton–Jacobi equations:

$$\begin{aligned}&\frac{\mathrm{d}x^{i}}{\mathrm{d}\tau }=\frac{\partial {\widetilde{H}}}{\partial p_{i}} \quad \text{ and }\quad \frac{\mathrm{d}p_{i}}{\mathrm{d}\tau }=-\frac{\partial {\widetilde{H}}}{\partial x^{i}}, \\&\text{ or } \text{ Euler--Lagrange } \text{ equations, } \frac{d}{\mathrm{d}\tau }\frac{\partial {\widetilde{L}}}{\partial y^{i}}-\frac{\partial {\widetilde{L}}}{\partial x^{i}} =0. \end{aligned}$$

In their turn, such relativistic effective mechanics equations are equivalent to the nonlinear geodesic (semi-spray) equations:

$$\begin{aligned} \frac{d^{2}x^{i}}{\mathrm{d}\tau ^{2}}+2{\widetilde{G}}^{i}(x,y)=0,\quad \text{ for } {\widetilde{G}}^{i}=\frac{1}{2}{\widetilde{g}}^{ij}\left( \frac{\partial ^{2} {\widetilde{L}}}{\partial y^{i}}y^{k}-\frac{\partial {\widetilde{L}}}{\partial x^{i}}\right) , \end{aligned}$$

(B.8)

with ${\widetilde{g}}^{ij}$ being inverse to ${\widetilde{g}}_{ij}$ (B.6).

Equation (B.8) prove that point like probing particles in a relativistic effective phase space do not move along usual geodesics as on Lorentz manifolds, but follow some nonlinear geodesic equations determined canonically by MDRs.

By explicit constructions on open sets covering V, TV and $T^{*}V,$ it was proven that there are canonical N-connections determined by MDRs in L-dual form when:

$$\begin{aligned} \ \widetilde{{\mathbf {N}}}=\left\{ {\widetilde{N}}_{i}^{a}:=\frac{\partial {\widetilde{G}}}{\partial y^{i}}\right\} \quad \text{ and }\quad ^{\shortmid }\widetilde{ {\mathbf {N}}}=\left\{ \ ^{\shortmid }{\widetilde{N}}_{ij}:=\frac{1}{2}\left[ \{\ \ ^{\shortmid }{\widetilde{g}}_{ij},{\widetilde{H}}\}-\frac{\partial ^{2} {\widetilde{H}}}{\partial p_{k}\partial x^{i}}\ ^{\shortmid }{\widetilde{g}} _{jk}-\frac{\partial ^{2}{\widetilde{H}}}{\partial p_{k}\partial x^{j}}\ ^{\shortmid }{\widetilde{g}}_{ik}\right] , \right\} \end{aligned}$$

where $\ \ ^{\shortmid }{\widetilde{g}}_{ij}$ is inverse to $\ \ ^{\shortmid } {\widetilde{g}}^{ab}$ (B.7).

The canonical N-connections s $\widetilde{{\mathbf {N}}}$ and $\ ^{\shortmid } \widetilde{{\mathbf {N}}}$ define respective canonical systems of N-adapted (co) frames:

$$\begin{aligned} \widetilde{{\mathbf {e}}}_{\alpha }= & {} \left( \widetilde{{\mathbf {e}}}_{i}=\frac{ \partial }{\partial x^{i}}-{\widetilde{N}}_{i}^{a}(x,y)\frac{\partial }{ \partial y^{a}},e_{b}=\frac{\partial }{\partial y^{b}}\right) , \text{ on } TV; \nonumber \\ \widetilde{{\mathbf {e}}}^{\alpha }= & {} ({\widetilde{e}}^{i}=\mathrm{d}x^{i},\widetilde{ {\mathbf {e}}}^{a}=\mathrm{d}y^{a}+{\widetilde{N}}_{i}^{a}(x,y)\mathrm{d}x^{i}), \text{ on } (TV)^{*};\text{ and } \end{aligned}$$

(B.9)

$$\begin{aligned} \ ^{\shortmid }\widetilde{{\mathbf {e}}}_{\alpha }= & {} \left( \ ^{\shortmid } \widetilde{{\mathbf {e}}}_{i}=\frac{\partial }{\partial x^{i}}-\ ^{\shortmid } {\widetilde{N}}_{ia}(x,p)\frac{\partial }{\partial p_{a}},\ ^{\shortmid }e^{b}= \frac{\partial }{\partial p_{b}}\right) , \text{ on } T^{*}V; \nonumber \\ \ \ ^{\shortmid }\widetilde{{\mathbf {e}}}^{\alpha }= & {} (\ ^{\shortmid }e^{i}=\mathrm{d}x^{i},\ ^{\shortmid }{\mathbf {e}}_{a}=\mathrm{d}p_{a}+\ ^{\shortmid }\widetilde{ N}_{ia}(x,p)\mathrm{d}x^{i}) \text{ on } (T^{*}V)^{*}. \end{aligned}$$

(B.10)

We conclude that the nonholonomic structure of a Lorentz manifold and respective (co)tangent bundles nonholonomically deformed by MDR (1 ) can be described in equivalent forms using canonical data $({\widetilde{L}} ,\ \widetilde{{\mathbf {N}}};\widetilde{{\mathbf {e}}}_{\alpha },\widetilde{ {\mathbf {e}}}^{\alpha })$ and/or $({\widetilde{H}},\ ^{\shortmid }\widetilde{ {\mathbf {N}}};\ ^{\shortmid }\widetilde{{\mathbf {e}}}_{\alpha },\ ^{\shortmid } \widetilde{{\mathbf {e}}}^{\alpha }).$ For general frame and coordinate transforms, considering a general N-splitting without effective Lagrangians/Hamiltonians, the nonholonomic geometry is described in terms of geometric data $({\mathbf {N}};{\mathbf {e}}_{\alpha },{\mathbf {e}}^{\alpha })$ and/or $(\ ^{\shortmid }{\mathbf {N}};\ ^{\shortmid }{\mathbf {e}}_{\alpha },\ ^{\shortmid }{\mathbf {e}}^{\alpha }).$

Vector fields on $T{\mathbf {V}}$ and $T^{*}{\mathbf {V}}$ are called d-vectors if they are written in a form adapted to a prescribed N-connection structure. For instance, we consider tilde and non-tilde decompositions:

$$\begin{aligned} {\mathbf {X}}= & {} \widetilde{{\mathbf {X}}}^{\alpha }\widetilde{{\mathbf {e}}}_{\alpha }=\widetilde{{\mathbf {X}}}^{i}\widetilde{{\mathbf {e}}}_{i}+X^{b}e_{b}={\mathbf {X}} ^{\alpha }{\mathbf {e}}_{\alpha }={\mathbf {X}}^{i}{\mathbf {e}}_{i}+X^{b}e_{b}\in T \mathbf {TV}, \nonumber \\ \ ^{\shortmid }{\mathbf {X}}= & {} \ ^{\shortmid }\widetilde{{\mathbf {X}}}^{\alpha } \widetilde{{\mathbf {e}}}_{\alpha }=\ ^{\shortmid }\widetilde{{\mathbf {X}}}^{i}\ ^{\shortmid }\widetilde{{\mathbf {e}}}_{i}+\ ^{\shortmid }X_{b}\ ^{\shortmid }e^{b}=\ ^{\shortmid }{\mathbf {X}}^{\alpha }\ ^{\shortmid }{\mathbf {e}}_{\alpha }=\ ^{\shortmid }{\mathbf {X}}^{i}\ ^{\shortmid }{\mathbf {e}}_{i}+\ ^{\shortmid }X_{b}\ ^{\shortmid }e^{b}\in T{\mathbf {T}}^{*}\mathbf {V.}\qquad \end{aligned}$$

(B.11)

In brief, we can write conventional h–v and/or h–cv decompositions, ${\mathbf {X}}^{\alpha }=\widetilde{{\mathbf {X}}}^{\alpha }=(\widetilde{{\mathbf {X}}} ^{i},X^{b})=({\mathbf {X}}^{i},X^{b}),\ ^{\shortmid }{\mathbf {X}}^{\alpha }=\ ^{\shortmid }\widetilde{{\mathbf {X}}}^{\alpha }=(\ ^{\shortmid }\widetilde{ {\mathbf {X}}}^{i},\ ^{\shortmid }X_{b})=(\ ^{\shortmid }{\mathbf {X}}^{i},\ ^{\shortmid }X_{b}).$ It is possible to consider ${\mathbf {X}}$ and $\ ^{\shortmid }{\mathbf {X}}$ as 1-forms:

$$\begin{aligned} {\mathbf {X}}= & {} \widetilde{{\mathbf {X}}}_{\alpha }\ {\mathbf {e}}^{\alpha }=X_{i}\ e^{i}+\widetilde{{\mathbf {X}}}^{a}\widetilde{{\mathbf {e}}}_{a}=\widetilde{ {\mathbf {X}}}_{\alpha }{\mathbf {e}}^{\alpha }=X_{i}e^{i}+{\mathbf {X}}^{a}{\mathbf {e}} _{a}\ \in T^{*}\mathbf {TV} \\ \ ^{\shortmid }{\mathbf {X}}= & {} \ ^{\shortmid }\widetilde{{\mathbf {X}}}_{\alpha }\ ^{\shortmid }{\mathbf {e}}^{\alpha }=\ ^{\shortmid }X_{i}\ ^{\shortmid }e^{i}+\ ^{\shortmid }\widetilde{{\mathbf {X}}}^{a}\ ^{\shortmid }\widetilde{ {\mathbf {e}}}_{a}=\ ^{\shortmid }\widetilde{{\mathbf {X}}}_{\alpha }\ ^{\shortmid }{\mathbf {e}}^{\alpha }=\ ^{\shortmid }X_{i}\ ^{\shortmid }e^{i}+\ ^{\shortmid }{\mathbf {X}}^{a}\ ^{\shortmid }{\mathbf {e}}_{a}\ \in T^{*}{\mathbf {T}}^{*} \mathbf {V,} \end{aligned}$$

or, in brief, ${\mathbf {X}}_{\alpha }=\widetilde{{\mathbf {X}}}_{\alpha }=(X_{i}, \widetilde{{\mathbf {X}}}^{a})=(X_{i},{\mathbf {X}}^{a}),\ ^{\shortmid }\mathbf { X_{\alpha }=}\ ^{\shortmid }\widetilde{{\mathbf {X}}}_{\alpha }=(\ ^{\shortmid }X_{i},\ ^{\shortmid }\widetilde{{\mathbf {X}}}^{a})=(\ ^{\shortmid }X_{i},\ ^{\shortmid }{\mathbf {X}}^{a}).$

Using tensor products of N-adapted (co) frames, we can parameterize in various N-adapted forms arbitrary tensors fields, called as d-tensors (similarly, for d-connections, d-tensors, etc.).

1.2.2 Canonical d-metrics and d-connections for Lagrange–Hamilton spaces

There are canonical d-metric structures $\widetilde{{\mathbf {g}}}$ and $\ ^{\shortmid }\widetilde{{\mathbf {g}}}$ completely determined by a MDR (1) and respective data $({\widetilde{L}},\ \ \widetilde{{\mathbf {N}}}; \widetilde{{\mathbf {e}}}_{\alpha },\widetilde{{\mathbf {e}}}^{\alpha };\widetilde{ g}_{jk},{\widetilde{g}}^{jk})$ and $({\widetilde{H}},\ ^{\shortmid }\widetilde{ {\mathbf {N}}};\ ^{\shortmid }\widetilde{{\mathbf {e}}}_{\alpha },\ ^{\shortmid } \widetilde{{\mathbf {e}}}^{\alpha };\ \ ^{\shortmid }{\widetilde{g}}^{ab},\ \ ^{\shortmid }{\widetilde{g}}_{ab}):$

$$\begin{aligned} \widetilde{{\mathbf {g}}}= & {} \widetilde{{\mathbf {g}}}_{\alpha \beta }(x,y) \widetilde{{\mathbf {e}}}^{\alpha }\mathbf {\otimes }\widetilde{{\mathbf {e}}} ^{\beta }={\widetilde{g}}_{ij}(x,y)e^{i}\otimes e^{j}+{\widetilde{g}}_{ab}(x,y) \widetilde{{\mathbf {e}}}^{a}\otimes \widetilde{{\mathbf {e}}}^{a}\quad \text{ and/or } \end{aligned}$$

(B.12)

$$\begin{aligned} \ ^{\shortmid }\widetilde{{\mathbf {g}}}= & {} \ ^{\shortmid }\widetilde{{\mathbf {g}} }_{\alpha \beta }(x,p)\ ^{\shortmid }\widetilde{{\mathbf {e}}}^{\alpha }\mathbf { \otimes \ ^{\shortmid }}\widetilde{{\mathbf {e}}}^{\beta }=\ \ ^{\shortmid } {\widetilde{g}}_{ij}(x,p)e^{i}\otimes e^{j}+\ ^{\shortmid }{\widetilde{g}} ^{ab}(x,p)\ ^{\shortmid }\widetilde{{\mathbf {e}}}_{a}\otimes \ ^{\shortmid } \widetilde{{\mathbf {e}}}_{b}. \end{aligned}$$

(B.13)

By straightforward N-adapted calculus with $\widetilde{{\mathbf {e}}}_{\alpha }=(\widetilde{{\mathbf {e}}}_{i},e_{b})$ (B.9) and$\ ^{\shortmid } \widetilde{{\mathbf {e}}}_{\alpha }=(\ ^{\shortmid }\widetilde{{\mathbf {e}}} _{i},\ ^{\shortmid }e^{b})$ (B.10) and (A.7), we prove non-trivial indicators for MDRs and respective canonical N-connection structures induce canonical nonholonomic frame structures on $\mathbf {TV}$ and/or ${\mathbf {T}}^{*}\mathbf {V.}$ Such nonholonomic frame bases are characterized by corresponding anholonomy relations:

$$\begin{aligned}{}[\widetilde{{\mathbf {e}}}_{\alpha },\widetilde{{\mathbf {e}}}_{\beta }]= \widetilde{{\mathbf {e}}}_{\alpha }\widetilde{{\mathbf {e}}}_{\beta }-\widetilde{ {\mathbf {e}}}_{\beta }\widetilde{{\mathbf {e}}}_{\alpha }={\widetilde{W}}_{\alpha \beta }^{\gamma }\widetilde{{\mathbf {e}}}_{\gamma }, \end{aligned}$$

with (antisymmetric) anholonomy coefficients ${\widetilde{W}} _{ia}^{b}=\partial _{a}{\widetilde{N}}_{i}^{b}$ and ${\widetilde{W}}_{ji}^{a}= {\widetilde{\Omega }}_{ij}^{a},$ and:

$$\begin{aligned}{}[\ ^{\shortmid }\widetilde{{\mathbf {e}}}_{\alpha },\ ^{\shortmid } \widetilde{{\mathbf {e}}}_{\beta }]=\ ^{\shortmid }\widetilde{{\mathbf {e}}} _{\alpha }\ ^{\shortmid }\widetilde{{\mathbf {e}}}_{\beta }-\ ^{\shortmid } \widetilde{{\mathbf {e}}}_{\beta }\ ^{\shortmid }\widetilde{{\mathbf {e}}}_{\alpha }=\ ^{\shortmid }{\widetilde{W}}_{\alpha \beta }^{\gamma }\ ^{\shortmid } \widetilde{{\mathbf {e}}}_{\gamma }, \end{aligned}$$

with anholonomy coefficients $\ ^{\shortmid }{\widetilde{W}}_{ib}^{a}=\partial \ ^{\shortmid }{\widetilde{N}}_{ib}/\partial p_{a}$ and $\ ^{\shortmid } {\widetilde{W}}_{jia}=\ \mathbf {\ ^{\shortmid }}{\widetilde{\Omega }}_{ija}.$ Explicit definitions and formulas for the Neijenhuis tensors $\widetilde{ \Omega }_{ij}^{a}$ and $\ \mathbf {\ ^{\shortmid }}{\widetilde{\Omega }}_{ija}$ can be found, for instance, in [1], see also (A.7).

There are the so-called Cartan–Lagrange and Cartan–Hamilton d-connections induced directly by an indicator of MDR and determined, respectively, by coefficients of Lagrange and Hamilton d-metrics (B.12) and (B.13) when all coefficients are generated by “tilde” objects with identifications of d-metric coefficients with corresponding base and (co)fiber indices:

$$\begin{aligned} \text{ on } T\mathbf {TV},\ \widetilde{{\mathbf {D}}}= & {} \left\{ \widetilde{\varvec{ \Gamma }}_{\ \alpha \beta }^{\gamma }=({\widetilde{L}}_{jk}^{i},{\widetilde{L}} _{bk}^{a},{\widetilde{C}}_{jc}^{i},{\widetilde{C}}_{bc}^{a})\right\} ,\nonumber \\&\quad \text{ for } \mathbf {[}\widetilde{{\mathbf {g}}}_{\alpha \beta }=({\widetilde{g}}_{jr}, {\widetilde{g}}_{ab}),\widetilde{{\mathbf {N}}}_{i}^{a}={\widetilde{N}}_{i}^{a}], \nonumber \\ {\widetilde{L}}_{jk}^{i}= & {} \frac{1}{2}{\widetilde{g}}^{ir}\left( \widetilde{ {\mathbf {e}}}_{k}{\widetilde{g}}_{jr}+\widetilde{{\mathbf {e}}}_{j}{\widetilde{g}} _{kr}-\widetilde{{\mathbf {e}}}_{r}{\widetilde{g}}_{jk}\right) ,\ {\widetilde{L}} _{bk}^{a} \text{ as } {\widetilde{L}}_{jk}^{i}, \nonumber \\ \ {\widetilde{C}}_{bc}^{a}= & {} \frac{1}{2}{\widetilde{g}}^{ad}\left( e_{c} {\widetilde{g}}_{bd}+e_{b}{\widetilde{g}}_{cd}-e_{d}{\widetilde{g}}_{bc}\right) \text{ being } \text{ similar } \text{ to } {\widetilde{C}}_{jc}^{i}; \end{aligned}$$

(B.14)

$$\begin{aligned} \text{ and, } \text{ on } T{\mathbf {T}}^{*}{\mathbf {V}},\ \ ^{\shortmid }\widetilde{ {\mathbf {D}}}= & {} \left\{ \ ^{\shortmid }\widetilde{\varvec{\Gamma }}_{\ \alpha \beta }^{\gamma }=(\ ^{\shortmid }{\widetilde{L}}_{jk}^{i},\ ^{\shortmid }\widetilde{ L}_{a\ k}^{\ b},\ ^{\shortmid }{\widetilde{C}}_{\ j}^{i\ c},\ ^{\shortmid } {\widetilde{C}}_{\ j}^{i\ c})\right\} ,\nonumber \\&\text{ for } \mathbf {[}\ ^{\shortmid } \widetilde{{\mathbf {g}}}_{\alpha \beta }=(\ ^{\shortmid }{\widetilde{g}}_{jr},\ ^{\shortmid }{\widetilde{g}}^{ab}),\ ^{\shortmid }\widetilde{{\mathbf {N}}} _{ai}=\ ^{\shortmid }{\widetilde{N}}_{ai}], \nonumber \\ \ ^{\shortmid }{\widetilde{L}}_{jk}^{i}= & {} \frac{1}{2}\ ^{\shortmid } {\widetilde{g}}^{ir}(\ ^{\shortmid }\widetilde{{\mathbf {e}}}_{k}\ ^{\shortmid } {\widetilde{g}}_{jr}+\ ^{\shortmid }\widetilde{{\mathbf {e}}}_{j}\ ^{\shortmid } {\widetilde{g}}_{kr}-\ ^{\shortmid }\widetilde{{\mathbf {e}}}_{r}\ ^{\shortmid } {\widetilde{g}}_{jk}),\ \text{ with } \text{ similar } \ ^{\shortmid }{\widetilde{L}}_{a\ k}^{\ b}, \nonumber \\ \ \ \ ^{\shortmid }{\widetilde{C}}_{\ a}^{b\ c}= & {} \frac{1}{2}\ ^{\shortmid } {\widetilde{g}}_{ad}(\ ^{\shortmid }e^{c}\ ^{\shortmid }{\widetilde{g}}^{bd}+\ ^{\shortmid }e^{b}\ ^{\shortmid }{\widetilde{g}}^{cd}-\ ^{\shortmid }e^{d}\ ^{\shortmid }{\widetilde{g}}^{bc}) \text{ being } \text{ similar } \text{ to } ^{\shortmid } {\widetilde{C}}_{\ j}^{i\ c}. \nonumber \\ \end{aligned}$$

(B.15)

These d-connections are similar, respectively, to the canonical d-connections (A.14) and (A.15) from Corollary A.3, but possess an important property that they are also canonical almost symplectic connections. It is difficult to find rich classes of exact physically important solutions working directly with $\widetilde{{\mathbf {D}}}$ (B.14) or $\ ^{\shortmid }\widetilde{{\mathbf {D}}}$ (B.15).