1. Introduction
As the most successful gravity theory so far, general relativity (GR) is based on the Riemannian geometry [1], in which the affine connection is constrained to be torsion-free and metric-compatible, i.e. ${{T}^{\lambda }}_{\mu \nu }=0$ and ∇αgμν = 0. So that the gravity is completely attributed to the spacetime curvature, which is described by the Riemann tensor ${{R}^{\lambda }}_{\rho \mu \nu }$, as well as its traces Ricci tensor ${R}_{\mu \nu }={{R}^{\lambda }}_{\mu \lambda \nu }$ and curvature scalar R = gμνRμν. The dynamical evolution of the gravitational field is sourced by the matter content through the Einstein equation: ${R}_{\mu \nu }-\tfrac{1}{2}{{Rg}}_{\mu \nu }={T}_{\mu \nu }$, where Tμν is the energy-momentum tensor of matter [2–4] and the unit 8πG = c = 1 is taken. The Einstein equation can be derived from the Einstein–Hilbert action $S=\int \sqrt{-g}(R/2+{{ \mathcal L }}_{m})$, where ${{ \mathcal L }}_{m}$ is the Lagrangian density of matter fields. Up to now, many important predictions of GR have been confirmed by experiments and observations, especially the events of gravitational waves recently detected by the LIGO/Virgo collaborations.
However, there are other frameworks that can be used to hatch gravity theories but are not based on the Riemannian geometry. One example is the ‘Teleparallel’ framework, in which the connection is constrained to be curvature-free ${{R}^{\lambda }}_{\rho \mu \nu }=0$ and metric-compatible ∇αgμν = 0, so the gravity is attributed to the torsion ${{T}^{\lambda }}_{\mu \nu }$ [5–8]. Another similar example is the so-called ‘Symmetric Teleparallel’ framework, in which the connection is constrained to be curvature-free ${{R}^{\lambda }}_{\rho \mu \nu }=0$ and torsion-free ${{T}^{\lambda }}_{\mu \nu }=0$, and the gravity is attributed entirely to the non-metricity tensor Qαμν = ∇αgμν [9, 10]. Within both frameworks, gravity models can be constructed to be equivalent to GR, i.e. the Teleparallel equivalent General Relativity (TEGR) model [5–8] in the Teleparallel framework and Symmetric Teleparallel equivalent General Relativity (STGR) model [9, 10] in Symmetric Teleparallel framework, respectively. Interestingly, these two frameworks can be used to hatch some modified gravity models which are very difficult (if not impossible) to be constructed within the Riemannian geometry, for instance, the Nieh–Yan modified Teleparallel Gravity [11, 12] in the Teleparallel framework, and the parity-violating gravity models discussed in [13, 14] within the Symmetric Teleparallel framework. All these frameworks including the Riemannian geometry can be considered as special cases of the metric-affine geometry under different constraints.
There are frameworks more general than metric-affine geometry. The conception of ‘metric’ may also be generalized. In Riemannian geometry, the length of the world line of a particle is ${\rm{d}}{s}=\sqrt{{g}_{\mu \nu }(x){{\rm{d}}{x}}^{\mu }{{\rm{d}}{x}}^{\nu }}$. In a general case, the length of the world line of a particle may be ds = F(x, dx) as long as the function F is smooth enough and homogeneous for the displacement dx [15, 16]. Such generalization leads us to the so-called Finsler geometry. Some interesting phenomena may come out from such a generalization, e.g., the geodesic equations are direction-dependent now, or a curve is a geodesic curve from A to B, but not from B to A. Also, more abundant connection structures are contained in Finsler geometry and it may induce the normal metric-affine theory with a specially selected section of the tangent bundle of the base manifold. Hence, it would exist at least three different kinds of connection structure: the ‘GR type’ that may lead to Riemannian geometry, ‘Teleparallel type’ and ‘Symmetric Teleparallel type’ lead to corresponding frameworks, respectively. The ‘GR type’ at least includes two connections: the so-called ‘Cartan connection’ [17] and ‘Chern connection’ [18], as discussed a lot in the literature. On the other hand, the other two types of connection structures are rarely discussed. In this paper, we come up with an example of ‘Teleparallel type’ connection and show how to obtain a generalized teleparallel gravity model from this connection structure.
In this paper, we will use the metric signature $\left\{+,-,-,-\right\}$. As usual, the Greek letters μ, ν, ρ, ⋯ = 0, 1, 2, 3 are tensor indicators and Capital Latins A, B, C, ⋯ = 0, 1, 2, 3 are internal space indicators. In sections 2 and 3 we review the basic structure of Finsler Geometry. In section 4 , we present our connection as an example of ‘Teleparallel type’ connection. In section 5 , we use a toy model based on our connection to compare with TEGR or GR, and show the similarities and differences between them.
2. Basics of Finsler geometry
2.1. Metric structure
A Finsler space is an n-dimensional manifold Mn with a Finsler metric function $F:{TM}\mapsto {{\mathbb{R}}}^{+}$. The function F satisfies the following properties [15, 16]:
| 1. | 1. Regularity: F is C∞ on the entire tangent bundle except the origin TM\{0}. |
| 2. | 2. Positive homogeneity: F(x, λy) = λF(x, y) for all λ > 0. |
| 3. | 3. Signature: The Hessian Matrix $\begin{eqnarray}{g}_{\mu \nu }:= \displaystyle \frac{1}{2}\displaystyle \frac{{\partial }^{2}({F}^{2})}{\partial {y}^{\mu }\partial {y}^{\nu }}\end{eqnarray}$ has a Lorentzian signature.3(3 The original requirement of the Hessian matrix in Finsler Geometry is to be positive definite. However, considering the connection between the Hessian matrix and metric tensor, we change the requirement to having a Lorentzian signature.) |
Since for any point x ∈ M with local coordinate (xμ), any tangent vector can be expanded as $y={y}^{\lambda }\tfrac{\partial }{\partial {x}^{\lambda }}\in {T}_{x}M$, (xμ, yλ) can be chosen as a local coordinate on TM. Then a function F defined on TM can be locally expressed as F = F(xμ, yλ).
Besides the fundamental tensor (1 ), one can define the so-called Cartan tensor [15, 16]:
$\begin{eqnarray}{C}_{\mu \nu \lambda }:= \displaystyle \frac{1}{2}\displaystyle \frac{\partial {g}_{\mu \nu }}{\partial {y}^{\lambda }}.\end{eqnarray}$
It is clear that the Cartan tensor is totally symmetric. Moreover, the homogeneity leads to the following identities: $\begin{eqnarray}{g}_{\mu \nu }{y}^{\mu }{y}^{\nu }={F}^{2},\,{C}_{\mu \nu \lambda }{y}^{\lambda }=0.\end{eqnarray}$
The Finsler Function F defines the length of a curve C(t) in the base manifold M:1 ) is just aμν(x), and the length of a curve is the same as that in (pseudo-)Riemann geometry. So the fundamental tensor (1 ) plays a similar role in Finsler geometry to the metric tensor in (pseudo-)Riemann geometry, while the Cartan tensor (2 ) implies the difference between Finsler geometry and (pseudo-)Riemann Geometry.
$\begin{eqnarray}l={\int }_{b}^{a}F\left(x,\displaystyle \frac{{\rm{d}}x}{{\rm{d}}t}\right){\rm{d}}t.\end{eqnarray}$
It is easy to see that when taking $F=\sqrt{{a}_{\mu \nu }(x){y}^{\mu }{y}^{\nu }}$, the fundamental tensor (2.2. (α, β)-metric
A family of special metric functions is the so-called (α, β)-metric. These metric functions satisfy F = F(α, β), where $\alpha =\sqrt{{a}_{\mu \nu }(x){y}^{\mu }{y}^{\nu }}$ and β = Aμ(x)yμ. The homogeneity then requires that F(λα, λβ) = λF(α, β). Denoting that
$\begin{eqnarray}L:= \displaystyle \frac{1}{2}{F}^{2},\,{h}_{\mu \nu }:= {a}_{\mu \nu }-\displaystyle \frac{{y}_{\mu }{y}_{\nu }}{{\alpha }^{2}},\,{p}_{\mu }:= {A}_{\mu }-\displaystyle \frac{\beta }{{\alpha }^{2}}{y}_{\mu },\end{eqnarray}$
where yμ $:= $ aμνyν, we have the fundamental tensor $\begin{eqnarray}\begin{array}{l}{g}_{\mu \nu }(x,y)=\displaystyle \frac{{L}_{\alpha }}{\alpha }{h}_{\mu \nu }+\displaystyle \frac{{L}_{\alpha \alpha }}{{\alpha }^{2}}{y}_{\mu }{y}_{\nu }\\ \,+\,\displaystyle \frac{{L}_{\alpha \beta }}{\alpha }\left({y}_{\mu }{A}_{\nu }+{y}_{\nu }{A}_{\mu }\right)+{L}_{\beta \beta }{A}_{\mu }{A}_{\nu }\end{array}\end{eqnarray}$
and Cartan tensor $\begin{eqnarray}\begin{array}{l}2{C}_{\mu \nu \lambda }(x,y)=\displaystyle \frac{{L}_{\alpha \beta }}{\alpha }\left({p}_{\mu }{h}_{\nu \lambda }+{p}_{\nu }{h}_{\lambda \mu }+{p}_{\lambda }{h}_{\mu \nu }\right)\\ \,+\,{L}_{\beta \beta \beta }{p}_{\mu }{p}_{\nu }{p}_{\lambda },\end{array}\end{eqnarray}$
where the lower indices α, β of L indicate the partial differentiation with respect to α and β.The homogeneity also leads to an equivalent form: $F=\alpha f\left(\tfrac{\beta }{\alpha }\right)$, where f(x) is a smooth function on ${\mathbb{R}}\setminus \{0\}$. Such marks give the fundamental tensor
$\begin{eqnarray}\begin{array}{l}{g}_{\mu \nu }(x,y)={f}^{2}{a}_{\mu \nu }+\displaystyle \frac{{ff}^{\prime} }{\alpha }\left({y}_{\mu }{p}_{\nu }+{y}_{\nu }{p}_{\mu }-\beta {h}_{\mu \nu }\right)\\ \,+\,\left({ff}^{\prime\prime} +f^{\prime 2}\right){p}_{\mu }{p}_{\nu }\end{array}\end{eqnarray}$
and Cartan tensor $\begin{eqnarray}\begin{array}{l}2{C}_{\mu \nu \lambda }(x,y)=\displaystyle \frac{\alpha {ff}^{\prime} -\beta \left({ff}^{\prime\prime} +f^{\prime2}\right)}{{\alpha }^{2}}\\ \quad \times \left({p}_{\mu }{h}_{\nu \lambda }+{p}_{\nu }{h}_{\lambda \mu }+{p}_{\lambda }{h}_{\mu \nu }\right)+\displaystyle \frac{{ff}^{\prime\prime\prime} +3f^{\prime} f^{\prime\prime} }{\alpha }{p}_{\mu }{p}_{\nu }{p}_{\lambda },\end{array}\end{eqnarray}$
where the arguments of functions f are all β/α. When f ≡ 1, (pseudo-)Riemann geometry is recovered.As examples, we have two simpler cases: Randers metric [19] F = α + β or f(x) = 1 + x, in which
$\begin{eqnarray}\begin{array}{l}{g}_{\mu \nu }(x,y)=\left(1+\displaystyle \frac{\beta }{\alpha }\right){h}_{\mu \nu }+\left(\displaystyle \frac{{y}_{\mu }}{\alpha }+{A}_{\mu }\right)\left(\displaystyle \frac{{y}_{\nu }}{\alpha }+{A}_{\nu }\right)\\ \quad =\ \left(1+\displaystyle \frac{\beta }{\alpha }\right){h}_{\mu \nu }+{\left(1+\displaystyle \frac{\beta }{\alpha }\right)}^{2}\displaystyle \frac{{y}_{\mu }{y}_{\nu }}{{\alpha }^{2}}\\ \quad +\ \left(1+\displaystyle \frac{\beta }{\alpha }\right)\left(\displaystyle \frac{{y}_{\mu }}{\alpha }{p}_{\nu }+\displaystyle \frac{{y}_{\nu }}{\alpha }{p}_{\mu }\right)+{p}_{\mu }{p}_{\nu },\\ {C}_{\mu \nu \lambda }(x,y)=\displaystyle \frac{1}{2\alpha }\left({p}_{\mu }{h}_{\nu \lambda }+{p}_{\nu }{h}_{\lambda \mu }+{p}_{\lambda }{h}_{\mu \nu }\right);\end{array}\end{eqnarray}$
and Kropina metric [20] $F=\tfrac{{\alpha }^{2}}{\beta }$ or f(x) = x−1, in which $\begin{eqnarray}\begin{array}{l}{g}_{\mu \nu }(x,y)=2\displaystyle \frac{{\alpha }^{2}}{{\beta }^{2}}{h}_{\mu \nu }+6\displaystyle \frac{{y}_{\mu }{y}_{\nu }}{{\beta }^{2}}\\ \quad -4\displaystyle \frac{{\alpha }^{2}}{{\beta }^{3}}\left({y}_{\mu }{A}_{\nu }+{y}_{\nu }{A}_{\mu }\right)+3\displaystyle \frac{{\alpha }^{4}}{{\beta }^{4}}{A}_{\mu }{A}_{\nu }\\ \quad =2\displaystyle \frac{{\alpha }^{2}}{{\beta }^{2}}{h}_{\mu \nu }+\displaystyle \frac{{y}_{\mu }{y}_{\nu }}{{\beta }^{2}}-\displaystyle \frac{{\alpha }^{2}}{{\beta }^{3}}\left({y}_{\mu }{p}_{\nu }+{y}_{\nu }{p}_{\mu }\right)+3\displaystyle \frac{{\alpha }^{4}}{{\beta }^{4}}{p}_{\mu }{p}_{\nu },\\ \quad {C}_{\mu \nu \lambda }(x,y)=-2\displaystyle \frac{{\alpha }^{2}}{{\beta }^{3}}\\ \quad \left({p}_{\mu }{h}_{\nu \lambda }+{p}_{\nu }{h}_{\lambda \mu }+{p}_{\lambda }{h}_{\mu \nu }+3\displaystyle \frac{{\alpha }^{2}}{{\beta }^{2}}{p}_{\mu }{p}_{\nu }{p}_{\lambda }\right).\end{array}\end{eqnarray}$
2.3. Connection structure: basis and non-linear connection
Since the local coordinate on TM has been chosen to be (xμ, yλ), the natural basis on TM is also determined [16]:12 ) are [16]13 ):
$\begin{eqnarray}{\partial }_{\mu }:= \displaystyle \frac{\partial }{\partial {x}^{\mu }},\,{\dot{\partial }}_{\lambda }:= \displaystyle \frac{\partial }{\partial {y}^{\lambda }}.\end{eqnarray}$
However, under a coordinate transformation ${x}^{\mu }\to {\tilde{x}}^{\mu }={\tilde{x}}^{\mu }(x)$ on M, the coordinate system on TM transforms as $\begin{eqnarray*}{x}^{\mu }\to {\tilde{x}}^{\mu }={\tilde{x}}^{\mu }(x),\,{y}^{\lambda }\to {\tilde{y}}^{\lambda }={y}^{\alpha }\displaystyle \frac{\partial {\tilde{x}}^{\lambda }}{\partial {x}^{\alpha }},\end{eqnarray*}$
so the transformations of basis ( $\begin{eqnarray*}\begin{array}{l}{\tilde{\partial }}_{\mu }=\displaystyle \frac{\partial {x}^{\alpha }}{\partial {\tilde{x}}^{\mu }}{\partial }_{\alpha }+\displaystyle \frac{\partial {y}^{\alpha }}{\partial {\tilde{x}}^{\mu }}{\dot{\partial }}_{\alpha }\\ \quad =\,\displaystyle \frac{\partial {x}^{\alpha }}{\partial {\tilde{x}}^{\mu }}{\partial }_{\alpha }+\displaystyle \frac{{\partial }^{2}{x}^{\alpha }}{\partial {\tilde{x}}^{\mu }\partial {\tilde{x}}^{\nu }}{\tilde{y}}^{\nu }{\dot{\partial }}_{\alpha },\\ \quad {\tilde{\dot{\partial }}}_{\lambda }=\displaystyle \frac{\partial {y}^{\alpha }}{\partial {\tilde{y}}^{\lambda }}{\dot{\partial }}_{\alpha }=\displaystyle \frac{\partial {x}^{\alpha }}{\partial {\tilde{x}}^{\lambda }}{\dot{\partial }}_{\alpha }.\end{array}\end{eqnarray*}$
Hence they are not covariant under a coordinate transformation on the base manifold. In order to receive a group of covariant bases, a non-linear connection ${N}_{\mu }^{\lambda }$ should be introduced, and a group of covariant bases are $\begin{eqnarray}{\delta }_{\mu }=\displaystyle \frac{\delta }{\delta {x}^{\mu }}:= \displaystyle \frac{\partial }{\partial {x}^{\mu }}-{N}_{\mu }^{\lambda }\displaystyle \frac{\partial }{\partial {y}^{\lambda }},\,{\dot{\partial }}_{\lambda }:= \displaystyle \frac{\partial }{\partial {y}^{\lambda }}\,,\end{eqnarray}$
with transformations $\begin{eqnarray*}{\tilde{\delta }}_{\mu }=\displaystyle \frac{\partial {x}^{\alpha }}{\partial {\tilde{x}}^{\mu }}{\delta }_{\alpha },\,{\tilde{\dot{\partial }}}_{\lambda }=\displaystyle \frac{\partial {x}^{\alpha }}{\partial {\tilde{x}}^{\lambda }}{\dot{\partial }}_{\alpha }\,.\end{eqnarray*}$
These bases have the following commutative relations: $\begin{eqnarray}\begin{array}{rcl}\left[{\delta }_{\mu },{\delta }_{\nu }\right] & = & \left({\delta }_{\nu }{N}_{\mu }^{\lambda }-{\delta }_{\mu }{N}_{\nu }^{\lambda }\right){\dot{\partial }}_{\lambda },\\ \left[{\delta }_{\mu },{\dot{\partial }}_{\nu }\right] & = & {\dot{\partial }}_{\nu }{N}_{\mu }^{\lambda }\,{\dot{\partial }}_{\lambda },\\ \left[{\dot{\partial }}_{\mu },{\dot{\partial }}_{\nu }\right] & = & 0.\end{array}\end{eqnarray}$
In addition, we have the dual basis of ( $\begin{eqnarray}{\rm{d}}{x}^{\mu },\,\delta {y}^{\nu }:= {\rm{d}}{y}^{\nu }+{N}_{\mu }^{\nu }{\rm{d}}{x}^{\mu }.\end{eqnarray}$
2.4. Connection structure: linear connection on the bundle
The Vertical Bundle ${ \mathcal V }$ is the linear space spanned by the basis ${\dot{\partial }}_{\lambda }$ in (13 ), and its complement space is called the Horizontal Bundle ${ \mathcal H }$, i.e.17 , 18 ) defined on the entire TM.
$\begin{eqnarray*}T({TM})={ \mathcal H }\oplus { \mathcal V }.\end{eqnarray*}$
Then a linear connection can be defined on ${ \mathcal V }$ [15, 16]: $\begin{eqnarray}{{\omega }^{\lambda }}_{\mu }:= {{F}^{\lambda }}_{\mu \nu }{\rm{d}}{x}^{\nu }+{{V}^{\lambda }}_{\mu \nu }\delta {y}^{\nu }.\end{eqnarray}$
Such a connection induces two covariant derivatives of a vector X = Xμ(x, y)∂μ: the horizontal covariant derivative ∇HX $\begin{eqnarray}{{X}^{\lambda }}_{| \nu }=\displaystyle \frac{\delta {X}^{\lambda }}{\delta {x}^{\nu }}+{{F}^{\lambda }}_{\mu \nu }{X}^{\mu },\end{eqnarray}$
and the vertical covariant derivative ∇VX $\begin{eqnarray}{X}^{\lambda }{| }_{\nu }=\displaystyle \frac{\partial {X}^{\lambda }}{\partial {y}^{\nu }}+{{V}^{\lambda }}_{\mu \nu }{X}^{\mu }.\end{eqnarray}$
One can continue this connection to the entire tangent bundle if and only if the Deflection condition $\begin{eqnarray}{{F}^{\lambda }}_{\mu \nu }{y}^{\mu }={N}_{\nu }^{\lambda }\end{eqnarray}$
is satisfied [16]. At this time, we have the connection $\begin{eqnarray*}\omega ={{\omega }^{\lambda }}_{\mu }\left({\delta }_{\lambda }\otimes {\rm{d}}{x}^{\mu }+{\dot{\partial }}_{\lambda }\otimes \delta {y}^{\mu }\right)\end{eqnarray*}$
and two covariant derivatives (2.5. Curvature and torsion
On TM, we have an outer differential of a function f(x, y):16 ) one can immediately figure out the curvature and torsion with Cartan Structure equations.
$\begin{eqnarray*}{\rm{d}}f={\delta }_{\mu }f\,{\rm{d}}{x}^{\mu }+{\dot{\partial }}_{\mu }f\,\delta {y}^{\mu }.\end{eqnarray*}$
Now from connection (Curvature 2-form is defined as19 ). Then from the Cartan Curvature Structure equations we have [15, 16]
$\begin{eqnarray*}\begin{array}{l}{{{\rm{\Omega }}}^{\lambda }}_{\rho }=\displaystyle \frac{1}{2}{{R}^{\lambda }}_{\rho \mu \nu }{\rm{d}}{x}^{\mu }\wedge {\rm{d}}{x}^{\nu }+{{P}^{\lambda }}_{\rho \mu \nu }{\rm{d}}{x}^{\mu }\wedge \delta {y}^{\nu }\\ \,+\,\displaystyle \frac{1}{2}{{S}^{\lambda }}_{\rho \mu \nu }\delta {y}^{\mu }\wedge \delta {y}^{\nu }\end{array}\end{eqnarray*}$
with the tensor structure $\begin{eqnarray*}{\rm{\Omega }}={{{\rm{\Omega }}}^{\lambda }}_{\rho }\left({\delta }_{\lambda }\otimes {{\rm{d}}{x}}^{\rho }+{\dot{\partial }}_{\lambda }\otimes \delta {y}^{\rho }\right).\end{eqnarray*}$
Again, such a neat structure requires the Deflection condition ( $\begin{eqnarray}\begin{array}{l}{{{\rm{\Omega }}}^{\lambda }}_{\rho }={\rm{d}}{{\omega }^{\lambda }}_{\rho }+{{\omega }^{\lambda }}_{\sigma }\wedge {\omega }_{\rho }^{\sigma }\\ \quad =\,\left({\delta }_{{\rm{[}}\mu }{{F}^{\lambda }}_{| \rho | \nu {\rm{]}}}+{{F}^{\lambda }}_{\sigma {\rm{[}}\mu }{{F}^{\sigma }}_{| \rho | \nu {\rm{]}}}\right.\\ \quad \left.+\ {{V}^{\lambda }}_{\rho \sigma }{\delta }_{{\rm{[}}\mu }{N}_{\nu {\rm{]}}}^{\sigma }\right){\rm{d}}{x}^{\mu }\wedge {\rm{d}}{x}^{\nu }\\ \quad +\ \left({\delta }_{\mu }{{V}^{\lambda }}_{\rho \nu }-{\dot{\partial }}_{\nu }{{F}^{\lambda }}_{\rho \mu }+{{F}^{\lambda }}_{\sigma \mu }{{V}^{\sigma }}_{\rho \nu }\right.\\ \quad \left.-\ {{V}^{\lambda }}_{\sigma \nu }{{F}^{\sigma }}_{\rho \mu }-{{V}^{\lambda }}_{\rho \sigma }{\dot{\partial }}_{\nu }{N}_{\mu }^{\sigma }\right){\rm{d}}{x}^{\mu }\wedge \delta {y}^{\nu }\\ \quad +\ \left({\dot{\partial }}_{{\rm{[}}\mu }{{V}^{\lambda }}_{| \rho | \nu {\rm{]}}}+{{V}^{\lambda }}_{\sigma {\rm{[}}\mu }{{V}^{\sigma }}_{| \rho | \nu {\rm{]}}}\right)\delta {y}^{\mu }\wedge \delta {y}^{\nu },\end{array}\end{eqnarray}$
or $\begin{eqnarray}\begin{array}{l}{{R}^{\lambda }}_{\rho \mu \nu }={\delta }_{\mu }{{F}^{\lambda }}_{\rho \nu }-{\delta }_{\nu }{{F}^{\lambda }}_{\rho \mu }+{{F}^{\lambda }}_{\sigma \mu }{{F}^{\sigma }}_{\rho \nu }\\ \quad -{{F}^{\lambda }}_{\sigma \nu }{{F}^{\sigma }}_{\rho \mu }+{{V}^{\lambda }}_{\rho \sigma }\left({\delta }_{\mu }{N}_{\nu }^{\sigma }-{\delta }_{\nu }{N}_{\mu }^{\sigma }\right),\\ {{P}^{\lambda }}_{\rho \mu \nu }={\delta }_{\mu }{{V}^{\lambda }}_{\rho \nu }-{\dot{\partial }}_{\nu }{{F}^{\lambda }}_{\rho \mu }+{{F}^{\lambda }}_{\sigma \mu }{{V}^{\sigma }}_{\rho \nu }\\ \quad -{{V}^{\lambda }}_{\sigma \nu }{{F}^{\sigma }}_{\rho \mu }-{{V}^{\lambda }}_{\rho \sigma }{\dot{\partial }}_{\nu }{N}_{\mu }^{\sigma },\\ {{S}^{\lambda }}_{\rho \mu \nu }={\dot{\partial }}_{\mu }{{V}^{\lambda }}_{\rho \nu }-{\dot{\partial }}_{\nu }{{V}^{\lambda }}_{\rho \mu }\\ \quad +{{V}^{\lambda }}_{\sigma \mu }{{V}^{\sigma }}_{\rho \nu }-{{V}^{\lambda }}_{\sigma \nu }{{V}^{\sigma }}_{\rho \mu }.\end{array}\end{eqnarray}$
There is something different about the torsion. The tensor structure of the torsion is $\begin{eqnarray*}\tau ={\tau }^{\lambda }{\delta }_{\lambda }+{\dot{\tau }}^{\lambda }{\dot{\partial }}_{\lambda }\end{eqnarray*}$
with 2 torsion 2-forms $\begin{eqnarray*}\begin{array}{l}{\tau }^{\lambda }=\displaystyle \frac{1}{2}{{T}^{\lambda }}_{\mu \nu }{\rm{d}}{x}^{\mu }\wedge {\rm{d}}{x}^{\nu }+{{U}^{\lambda }}_{\mu \nu }{\rm{d}}{x}^{\mu }\wedge \delta {y}^{\nu }\\ \quad +\displaystyle \frac{1}{2}{{W}^{\lambda }}_{\mu \nu }\delta {y}^{\mu }\wedge {{\rm{d}}{x}}^{\nu },\\ {\dot{\tau }}^{\lambda }=\displaystyle \frac{1}{2}{{\dot{T}}^{\lambda }}_{\mu \nu }{\rm{d}}{x}^{\mu }\wedge {\rm{d}}{x}^{\nu }+{{\dot{U}}^{\lambda }}_{\mu \nu }{\rm{d}}{x}^{\mu }\wedge \delta {y}^{\nu }\\ \quad +\displaystyle \frac{1}{2}{{\dot{W}}^{\lambda }}_{\mu \nu }\delta {y}^{\mu }\wedge {{\rm{d}}{x}}^{\nu }.\end{array}\end{eqnarray*}$
Then from the Cartan Curvature Structure equations we have $\begin{eqnarray}\begin{array}{rcl}{\tau }^{\lambda } & = & {\rm{d}}\left({\rm{d}}{x}^{\lambda }\right)-{{\omega }^{\lambda }}_{\sigma }\wedge {\rm{d}}{x}^{\sigma }\\ & = & {{F}^{\lambda }}_{[\mu \nu ]}{\rm{d}}{x}^{\mu }\wedge {{\rm{d}}{x}}^{\nu }-{{V}^{\lambda }}_{\mu \nu }{\rm{d}}{x}^{\mu }\wedge \delta {y}^{\nu },\\ {\dot{\tau }}^{\lambda } & = & {\rm{d}}\left(\delta {y}^{\lambda }\right)-{{\omega }^{\lambda }}_{\sigma }\wedge \delta {y}^{\sigma }\\ & = & {\delta }_{{\rm{[}}\mu }{N}_{\nu {\rm{]}}}^{\lambda }{\rm{d}}{x}^{\mu }\wedge {\rm{d}}{x}^{\nu }-\left({\delta }_{\nu }{{V}^{\lambda }}_{\rho \mu }+{{F}^{\lambda }}_{\nu \mu }\right){\rm{d}}{x}^{\mu }\\ & & \wedge \delta {y}^{\nu }+{{V}^{\lambda }}_{[\mu \nu ]}\delta {y}^{\mu }\wedge {{\rm{d}}{x}}^{\nu };\end{array}\end{eqnarray}$
or $\begin{eqnarray}\begin{array}{rcl}{{T}^{\lambda }}_{\mu \nu } & = & {{F}^{\lambda }}_{\mu \nu }-{{F}^{\lambda }}_{\nu \mu },\\ {{U}^{\lambda }}_{\mu \nu } & = & {{V}^{\lambda }}_{\mu \nu },\\ {{W}^{\lambda }}_{\mu \nu } & = & 0;\\ {{\dot{T}}^{\lambda }}_{\mu \nu } & = & {\delta }_{\mu }{N}_{\nu }^{\lambda }-{\delta }_{\nu }{N}_{\mu }^{\lambda },\\ {{\dot{U}}^{\lambda }}_{\mu \nu } & = & -{{F}^{\lambda }}_{\nu \mu }-{\dot{\partial }}_{\nu }{N}_{\mu }^{\lambda },\\ {{\dot{W}}^{\lambda }}_{\mu \nu } & = & {{V}^{\lambda }}_{\mu \nu }-{{F}^{\lambda }}_{\nu \mu }.\end{array}\end{eqnarray}$
3. Finsler–Bathrel geometry
One can always induce a (pseudo-)Riemannian geometry from a Finsler geometry with a determined vector field. From this section on, we will use ‘̂’ to distinguish the parameters defined on the base manifold M from the ones on TM with similar names. For example, the curvature 2-form on M is denoted as
$\begin{eqnarray*}{{\hat{{\rm{\Omega }}}}^{\lambda }}_{\rho }=\displaystyle \frac{1}{2}{{\hat{R}}^{\lambda }}_{\rho \mu \nu }(x){{\rm{d}}{x}}^{\mu }\wedge {{\rm{d}}{x}}^{\nu }.\end{eqnarray*}$
The (pseudo-)Riemannian geometry induced from a given Finsler geometry is constructed as follows. We always have a natural map [16]24 ) to1 ):16 ) [15, 16]:28 - 30 ) are indeed constructed with the connection (27 ).
$\begin{eqnarray}\sigma :\,{TM}\mapsto M,\,(x,y)\mapsto x.\end{eqnarray}$
While a vector field Y(x) on M, or a section of TM, restricts the map ( $\begin{eqnarray}{\sigma }_{Y}:\,(x,Y(x))\mapsto x.\end{eqnarray}$
Then the parameters on M are defined as the pull-backs of those on TM with the same names, i.e. $\hat{A}:= {\sigma }_{Y}^{* }A$. For example, the metric on M is defined as the pull back of the fundamental tensor ( $\begin{eqnarray}{\hat{g}}_{\mu \nu }(x):= {g}_{\mu \nu }(x,Y)={g}_{\mu \nu }(x,y){| }_{y=Y(x)}.\end{eqnarray}$
And the connection is the pull back of linear connection ( $\begin{eqnarray}{{{\rm{\Gamma }}}^{\lambda }}_{\mu \nu }:= {{F}^{\lambda }}_{\mu \nu }(x,Y)+{{V}^{\lambda }}_{\mu \rho }(x,Y){Y}_{\nu }^{\rho },\end{eqnarray}$
where ${Y}_{\nu }^{\rho }:= {\partial }_{\nu }{Y}^{\rho }+{N}_{\nu }^{\rho }(x,Y)$. So the curvature on M is $\begin{eqnarray}\begin{array}{l}{{\hat{R}}^{\lambda }}_{\rho \mu \nu }(x)={{R}^{\lambda }}_{\rho \mu \nu }(x,Y)+{{P}^{\lambda }}_{\rho \mu \sigma }(x,Y){Y}_{\nu }^{\sigma }\\ \quad -{{P}^{\lambda }}_{\rho \nu \sigma }(x,Y){Y}_{\mu }^{\sigma }+{{S}^{\lambda }}_{\rho \alpha \beta }(x,Y){Y}_{\mu }^{\alpha }{Y}_{\nu }^{\beta }.\end{array}\end{eqnarray}$
The tensor is the pull back of τλ: $\begin{eqnarray}\begin{array}{l}{{\hat{T}}^{\lambda }}_{\mu \nu }:= 2{{{\rm{\Gamma }}}^{\lambda }}_{\left[\mu \nu \right]}={{F}^{\lambda }}_{\left[\mu \nu \right]}(x,Y)+{{V}^{\lambda }}_{{\rm{[}}\mu | \rho | }(x,Y){Y}_{\nu {\rm{]}}}^{\rho }\\ \quad ={{T}^{\lambda }}_{\mu \nu }(x,Y)+{{V}^{\lambda }}_{\mu \rho }(x,Y){Y}_{\nu }^{\rho }-{{V}^{\lambda }}_{\nu \rho }(x,Y){Y}_{\mu }^{\rho }.\end{array}\end{eqnarray}$
And the non-metricity tensor is $\begin{eqnarray}{Q}_{\alpha \mu \nu }:= {{\rm{\nabla }}}_{\alpha }{\hat{g}}_{\mu \nu }=\left({{\rm{\nabla }}}_{\alpha }^{H}{g}_{\mu \nu }\right)(x,Y)+\left({{\rm{\nabla }}}_{\rho }^{V}{g}_{\mu \nu }\right)(x,Y){Y}_{\alpha }^{\rho }.\end{eqnarray}$
One can examine that (4. The Finsler–Teleparallel connection
We have a group of connections that would lead to Riemannian geometry which hosts GR when the metric is restricted to be the Riemannian metric, and they have been discussed under different cases. On the other hand, as we know, there are other frameworks to host Teleparallel and Symmetric Teleparallel gravities. Then a question comes out that whether these frameworks can be popularized in Finsler Geometry. We will focus on the popularization of the framework hosting the Teleparallel gravity in this paper.
Such a popularization should keep the two conditions: curvature-free ${{{\rm{\Omega }}}^{\lambda }}_{\rho }=0$ and metric-compatible ${{\rm{\nabla }}}_{\alpha }^{H,V}{g}_{\mu \nu }=0$. Just like the case of Riemannian geometry, as has been raised in [21], the two conditions require that31 ), (32 ) imply the existence of local flat spacetime at each point and keep Einstein’s Equivalent Principle. And from (31 ), we have the connection coefficients
$\begin{eqnarray}{{{\rm{\Omega }}}^{\lambda }}_{\mu }={{E}_{A}}^{\lambda }{\rm{d}}{{E}^{A}}_{\mu },\end{eqnarray}$
$\begin{eqnarray}{g}_{\mu \nu }={\eta }_{{AB}}{{E}^{A}}_{\mu }{{E}^{B}}_{\nu },\end{eqnarray}$
where ${\eta }_{{AB}}={\rm{diag}}\left\{1,-1,-1,-1\right\}$ is the flat metric, ${{E}^{A}}_{\mu }$ is a matrix and its inverse is expressed as ${{E}_{A}}^{\lambda }$. As the natural generalization of ones in Teleparallel Geometry, equations ( $\begin{eqnarray}{{F}^{\lambda }}_{\mu \nu }={{E}_{A}}^{\lambda }{\delta }_{\nu }{{E}^{A}}_{\mu },\,{{V}^{\lambda }}_{\mu \nu }={{E}_{A}}^{\lambda }{\dot{\partial }}_{\nu }{{E}^{A}}_{\mu }\end{eqnarray}$
with ${{E}^{A}}_{\mu }$ to be determined. Here we have taken the Weitzenböck condition.We should point out that condition (32 ) is not easy to satisfy, because gμν is not a fundamental parameter, but is derived from the Finsler metric. Thus (32 ) would impose restrictions on the matrix ${{E}^{A}}_{\mu }$. In the following sections, we will try to determine ${{E}^{A}}_{\mu }$.
4.1. The non-linear connection
We will discuss the (α, β)-metric only, and denote that3 ) lead to equation
$\begin{eqnarray}{a}_{\mu \nu }={\eta }_{{AB}}{{{\rm{e}}}^{A}}_{\mu }{{{\rm{e}}}^{B}}_{\nu },\end{eqnarray}$
where ${{{\rm{e}}}^{A}}_{\mu }$ is the tetrad associated with aμν. Identities ( $\begin{eqnarray}{g}_{\mu \nu }{y}^{\mu }{y}^{\nu }={\eta }_{{AB}}{E}^{A}{E}^{B}={F}^{2}={\alpha }^{2}{f}^{2}\left(\displaystyle \frac{\beta }{\alpha }\right),\end{eqnarray}$
where ${E}^{A}:= {{E}^{A}}_{\mu }{y}^{\mu }$. From this equation we have $\begin{eqnarray}{E}^{A}={{{\rm{e}}}^{A}}_{\mu }{y}^{\mu }f\left(\displaystyle \frac{\beta }{\alpha }\right),\end{eqnarray}$
or $\begin{eqnarray}{{E}^{A}}_{\mu }={{{\rm{e}}}^{A}}_{\rho }{{m}^{\rho }}_{\mu },\end{eqnarray}$
where the matrix ${{m}^{\rho }}_{\mu }$ satisfies ${{m}^{\rho }}_{\mu }{y}^{\mu }={y}^{\rho }f\left(\tfrac{\beta }{\alpha }\right)$. From now on, we will omit the argument of functions when it is expressed as the ratio $\tfrac{\beta }{\alpha }$, e.g., when f appears alone, it refers to $f\left(\tfrac{\beta }{\alpha }\right)$.Now we can make up the non-linear connection from the Deflection condition (19 ). From (33 ) it is easy to derive42 ) is independent of it. So a conclusion comes that as long as the matrix ${{E}^{A}}_{\mu }$ takes the form (37 ) and the Deflection condition is satisfied, the non-linear connection always takes the form (42 ).
$\begin{eqnarray}\begin{array}{l}{{F}^{\lambda }}_{\mu \nu }={\left({m}^{-1}\right)}_{\alpha }^{\lambda }\left({{m}^{\beta }}_{\mu }{{{\rm{e}}}_{A}}^{\alpha }{{{\rm{e}}}^{A}}_{\beta ,\nu }\right.\\ \quad \left.+{\partial }_{\nu }{{m}^{\alpha }}_{\mu }-{N}_{\nu }^{\rho }{\dot{\partial }}_{\rho }{{m}^{\alpha }}_{\mu }\right).\end{array}\end{eqnarray}$
Then to contract it with yμ we have $\begin{eqnarray}\begin{array}{l}{N}_{\nu }^{\lambda }={y}^{\mu }{{F}^{\lambda }}_{\mu \nu }={\left({m}^{-1}\right)}_{\alpha }^{\lambda }\left\{{y}^{\beta }{{{\rm{e}}}_{A}}^{\alpha }{{{\rm{e}}}^{A}}_{\beta ,\nu }f+{y}^{\alpha }{\partial }_{\nu }\left(\displaystyle \frac{\beta }{\alpha }\right)f^{\prime} \right.\\ \quad \left.-{N}_{\nu }^{\rho }\left[{\delta }_{\rho }^{\alpha }f+{y}^{\alpha }{\dot{\partial }}_{\rho }\left(\displaystyle \frac{\beta }{\alpha }\right)f^{\prime} \right]\right\}+{N}_{\nu }^{\lambda }.\end{array}\end{eqnarray}$
One can see that the matrix ${\left({m}^{-1}\right)}_{\alpha }^{\lambda }$ is subtly eliminated, and the terms in the brace should vanish: $\begin{eqnarray}{N}_{\nu }^{\rho }\left[{\delta }_{\rho }^{\alpha }f+{y}^{\alpha }{\dot{\partial }}_{\rho }\left(\displaystyle \frac{\beta }{\alpha }\right)f^{\prime} \right]={y}^{\beta }{{{\rm{e}}}_{A}}^{\alpha }{{{\rm{e}}}^{A}}_{\beta ,\nu }f+{y}^{\alpha }{\partial }_{\nu }\left(\displaystyle \frac{\beta }{\alpha }\right)f^{\prime} .\end{eqnarray}$
It is a system of linear equations of ${N}_{\nu }^{\rho }$ with coefficient matrix ${\delta }_{\rho }^{\alpha }f+{y}^{\alpha }{\dot{\partial }}_{\rho }\left(\tfrac{\beta }{\alpha }\right)f^{\prime} $. Note that ${\dot{\partial }}_{\rho }\left(\tfrac{\beta }{\alpha }\right)=\tfrac{{p}_{\rho }}{\alpha }$ and pαyα = 0, we can come up with the inverse of the coefficient matrix: $\begin{eqnarray}\left[{\delta }_{\rho }^{\alpha }f+{y}^{\alpha }{\dot{\partial }}_{\rho }\left(\displaystyle \frac{\beta }{\alpha }\right)f^{\prime} \right]\left[\displaystyle \frac{1}{f}{\delta }_{\alpha }^{\lambda }-\displaystyle \frac{f^{\prime} }{{f}^{2}}\displaystyle \frac{{p}_{\alpha }{y}^{\lambda }}{\alpha }\right]={\delta }_{\rho }^{\lambda }.\end{eqnarray}$
Finally, we have the non-linear connection $\begin{eqnarray}{N}_{\nu }^{\lambda }={y}^{\rho }{{{\rm{e}}}_{A}}^{\lambda }{{{\rm{e}}}^{A}}_{\rho ,\nu }+{y}^{\lambda }\displaystyle \frac{f^{\prime} }{f}\displaystyle \frac{{y}^{\rho }}{\alpha }\left({A}_{\rho ,\nu }-{A}_{\sigma }{{{\rm{e}}}_{A}}^{\sigma }{{{\rm{e}}}^{A}}_{\rho ,\nu }\right).\end{eqnarray}$
Up to now we did not specify the form of the matrix ${{m}^{\rho }}_{\mu }$, and the result (From now on, we will rewrite the 1-form42 ) has a simpler form:
$\begin{eqnarray}\beta ={A}_{\mu }{{\rm{d}}{x}}^{\mu }={A}_{A}{{{\rm{e}}}^{A}}_{\mu }{{\rm{d}}{x}}^{\mu }.\end{eqnarray}$
Then we have $\begin{eqnarray}{\partial }_{\nu }{A}_{A}={\partial }_{\nu }{A}_{\rho }{{{\rm{e}}}_{A}}^{\rho }+{\partial }_{\nu }{{{\rm{e}}}_{A}}^{\rho }{A}_{\rho }={{{\rm{e}}}_{A}}^{\rho }\left({A}_{\rho ,\nu }-{A}_{\sigma }{{{\rm{e}}}_{B}}^{\sigma }{\partial }_{\nu }{{{\rm{e}}}^{B}}_{\rho }\right).\end{eqnarray}$
So the non-linear connection ( $\begin{eqnarray}{N}_{\nu }^{\lambda }={y}^{\rho }{{{\rm{e}}}_{A}}^{\lambda }{{{\rm{e}}}^{A}}_{\rho ,\nu }+{y}^{\lambda }\displaystyle \frac{f^{\prime} }{f}\displaystyle \frac{{y}^{\rho }}{\alpha }{{{\rm{e}}}^{A}}_{\rho }{\partial }_{\nu }{A}_{A}.\end{eqnarray}$
4.2. Matrix ${{m}^{\alpha }}_{\mu }$ and linear connection
We now need to determine the matrix ${{m}^{\alpha }}_{\mu }$ by the metric-compatible condition (32 ). The fundamental tensor (1 ) of a Finsler metric F = αf is
$\begin{eqnarray}\begin{array}{l}{g}_{\mu \nu }={g}^{2}{a}_{\mu \nu }+\displaystyle \frac{\beta }{\alpha }{ff}^{\prime} \displaystyle \frac{{y}_{\mu }{y}_{\nu }}{{\alpha }^{2}}\\ \quad +\displaystyle \frac{{ff}^{\prime} }{\alpha }\left({y}_{\mu }{p}_{\nu }+{y}_{\nu }{p}_{\mu }\right)+\left({ff}^{\prime\prime} +f^{\prime2}\right){p}_{\mu }{p}_{\nu }\end{array}\end{eqnarray}$
where we introduced another function $g(x)\,=\sqrt{{f}^{2}(x)-{xf}(x)f^{\prime} (x)}$ to shorten the equations, and also omitted its argument $\tfrac{\beta }{\alpha }$.It is not easy to find a matrix to satisfy (32 ). However, we may have a connection structure, which slightly breaks the metric-compatibility on TM, but keep this condition after being pulled back to the base manifold M. In this way, we have a satisfying matrix:46 ), there is a slight difference ff″pμpν, which would be seen that have no effect after being pulled back to the base manifold M. And the inverse of this matrix is
$\begin{eqnarray}{{m}^{\alpha }}_{\mu }=g{\delta }_{\mu }^{\alpha }+(f-g)\displaystyle \frac{{y}^{\alpha }{y}_{\mu }}{{\alpha }^{2}}+f^{\prime} \displaystyle \frac{{y}^{\alpha }{p}_{\mu }}{\alpha },\end{eqnarray}$
and it leads to $\begin{eqnarray}\begin{array}{l}{\eta }_{{AB}}{{E}^{A}}_{\mu }{{E}^{B}}_{\nu }={a}_{\alpha \beta }{{m}^{\alpha }}_{\mu }{{m}^{\beta }}_{\nu }={g}^{2}{a}_{\mu \nu }\\ \quad +\displaystyle \frac{\beta }{\alpha }{ff}^{\prime} \displaystyle \frac{{y}_{\mu }{y}_{\nu }}{{\alpha }^{2}}+\displaystyle \frac{{ff}^{\prime} }{\alpha }\left({y}_{\mu }{p}_{\nu }+{y}_{\nu }{p}_{\mu }\right)+f^{\prime2}{p}_{\mu }{p}_{\nu }.\end{array}\end{eqnarray}$
Compared with equation ( $\begin{eqnarray}{\left({m}^{-1}\right)}_{\alpha }^{\lambda }=\displaystyle \frac{1}{g}{\delta }_{\alpha }^{\lambda }+\left(\displaystyle \frac{1}{f}-\displaystyle \frac{1}{g}\right)\displaystyle \frac{{y}^{\lambda }{y}_{\alpha }}{{\alpha }^{2}}-\displaystyle \frac{f^{\prime} }{{fg}}\displaystyle \frac{{y}^{\lambda }{p}_{\alpha }}{\alpha }.\end{eqnarray}$
With the matrices (47 , 49 ), the linear connection is determined from (33 ):
$\begin{eqnarray}\begin{array}{l}{{F}^{\lambda }}_{\mu \nu }={{{\rm{e}}}_{A}}^{\lambda }{{{\rm{e}}}^{A}}_{\mu ,\nu }+\displaystyle \frac{f^{\prime} }{f}\displaystyle \frac{{y}^{\lambda }}{\alpha }{{{\rm{e}}}^{A}}_{\mu }{\partial }_{\nu }{A}_{A}\\ \quad +\left(\displaystyle \frac{g^{\prime} }{g}{{h}^{\lambda }}_{\mu }+\displaystyle \frac{f^{\prime\prime} g-f^{\prime} g^{\prime} }{{fg}}\displaystyle \frac{{y}^{\lambda }{p}_{\mu }}{\alpha }\right)\displaystyle \frac{{y}^{\rho }}{\alpha }{{{\rm{e}}}^{A}}_{\rho }{\partial }_{\nu }{A}_{A},\\ \quad {{V}^{\lambda }}_{\mu \nu }=\left[\displaystyle \frac{g^{\prime} }{g}{\delta }_{\mu }^{\lambda }+\left(\displaystyle \frac{f^{\prime} }{f}-\displaystyle \frac{g^{\prime} }{g}-\displaystyle \frac{f^{\prime} }{g}\right)\displaystyle \frac{{y}^{\lambda }{y}_{\mu }}{{\alpha }^{2}}\right]\displaystyle \frac{{p}_{\nu }}{\alpha }\\ \quad +\displaystyle \frac{f^{\prime\prime} g-f^{\prime} g^{\prime} -f^{\prime2}}{{fg}}\displaystyle \frac{{y}^{\lambda }{p}_{\mu }{p}_{\nu }}{{\alpha }^{2}}\\ \quad +\left[\left(\displaystyle \frac{f}{g}-1\right)\displaystyle \frac{{y}_{\mu }}{{\alpha }^{2}}+\displaystyle \frac{f^{\prime} }{g}\displaystyle \frac{{p}_{\mu }}{\alpha }\right]{{h}^{\lambda }}_{\nu }\\ \quad +\displaystyle \frac{{g}^{2}-{fg}}{{f}^{2}}\displaystyle \frac{{y}^{\lambda }{h}_{\mu \nu }}{{\alpha }^{2}},\end{array}\end{eqnarray}$
where ${{h}^{\lambda }}_{\mu }:= {h}_{\mu \nu }{a}^{\lambda \nu }$.Then the two covariant derivatives of the fundamental tensor (46 ) can be calculated. The calculation details can be found in the appendix , and here we just give the conclusions directly:
$\begin{eqnarray}{{\rm{\nabla }}}_{\alpha }^{V}{g}_{\mu \nu }={\dot{\partial }}_{\alpha }{g}_{\mu \nu }-{{V}^{\lambda }}_{\mu \alpha }{g}_{\nu \lambda }-{{V}^{\lambda }}_{\nu \alpha }{g}_{\mu \lambda }=\left\{{p}_{* }\right\},\end{eqnarray}$
$\begin{eqnarray}{{\rm{\nabla }}}_{\alpha }^{H}{g}_{\mu \nu }={\delta }_{\alpha }{g}_{\mu \nu }-{{F}^{\lambda }}_{\mu \alpha }{g}_{\nu \lambda }-{{F}^{\lambda }}_{\nu \alpha }{g}_{\mu \lambda }=\left\{{p}_{* }\right\},\end{eqnarray}$
where $\left\{{p}_{* }\right\}$ refers to the terms that are proportional to pμ, pμpν, or higher order of pμ. Thus we see that the two covariant derivatives are both proportional to pμ or its higher order. We will show that they vanish when pulled back to the base manifold in the following sections, which means that the metric-compatible condition is satisfied on the base manifold.4.3. The induced Finsler–Bathrel geometry
Now we pull back these parameters to the base manifold. First, we need a vector field Yλ. Since we already have a 1-form $\beta ={A}_{\mu }{{\rm{d}}{x}}^{\mu }={A}_{A}{{{\rm{e}}}^{A}}_{\mu }{{\rm{d}}{x}}^{\mu }$, it is a natural choice to take ${Y}^{\lambda }={A}^{\lambda }={a}^{\mu \lambda }{A}_{\mu }={A}^{A}{{e}_{A}}^{\lambda }$, where AA = ηABAB. Such a choice will greatly simplify the geometry. One can calculate that
$\begin{eqnarray*}\begin{array}{rcl}\hat{\beta } & := & \beta (x,A)={A}_{\mu }{A}^{\mu }={\eta }^{{AB}}{A}_{A}{A}_{B},\\ \hat{\alpha } & := & \alpha (x,A)=\sqrt{{a}^{\mu \nu }{A}_{\mu }{A}_{\nu }}=\sqrt{{\eta }^{{AB}}{A}_{A}{A}_{B}}.\end{array}\end{eqnarray*}$
Immediately, we have $\hat{\beta }={\hat{\alpha }}^{2}$. So the arguments of functions f and g become $\hat{\alpha }$, and it is clear that $\begin{eqnarray*}{\hat{p}}_{\mu }:= {p}_{\mu }(x,A)={A}_{\mu }-\displaystyle \frac{\hat{\beta }}{{\hat{\alpha }}^{2}}{A}_{\mu }=0.\end{eqnarray*}$
Then the non-linear connection is $\begin{eqnarray}{\hat{N}}_{\nu }^{\lambda }:= {N}_{\nu }^{\lambda }(x,A)={A}^{\rho }{{e}_{A}}^{\lambda }{{e}^{A}}_{\rho ,\nu }+{A}^{\lambda }\displaystyle \frac{f^{\prime} }{\hat{\alpha }f}{A}^{A}{\partial }_{\nu }{A}_{A}.\end{eqnarray}$
So we have $\begin{eqnarray*}{A}_{\nu }^{\lambda }:= {\partial }_{\nu }{A}^{\lambda }+{\hat{N}}_{\nu }^{\lambda }={\partial }_{\nu }{A}^{A}\left({{e}_{A}}^{\lambda }+\displaystyle \frac{f^{\prime} }{\hat{\alpha }f}{A}^{\lambda }{A}_{A}\right).\end{eqnarray*}$
And the induced tetrads are $\begin{eqnarray}{{\hat{E}}^{A}}_{\mu }:= {{E}^{A}}_{\mu }(x,A)=g(\hat{\alpha }){{e}^{A}}_{\mu }+(f-g)\displaystyle \frac{{A}_{\mu }{A}^{A}}{{\hat{\alpha }}^{2}},\end{eqnarray}$
$\begin{eqnarray}{{\hat{E}}_{A}}^{\lambda }:= {{E}_{A}}^{\lambda }(x,A)=\displaystyle \frac{1}{g}{{e}_{A}}^{\lambda }+\left(\displaystyle \frac{1}{f}-\displaystyle \frac{1}{g}\right)\displaystyle \frac{{A}_{A}{A}^{\lambda }}{{\hat{\alpha }}^{2}}.\end{eqnarray}$
The induced metric is $\begin{eqnarray}{\hat{g}}_{\mu \nu }={g}^{2}{a}_{\mu \nu }+\displaystyle \frac{{ff}^{\prime} }{\hat{\alpha }}{A}_{\mu }{A}_{\nu }.\end{eqnarray}$
One can verify that the two approaches to the induced metric, i.e. ${\hat{g}}_{\mu \nu }={g}_{\mu \nu }(x,A)$ and ${\hat{g}}_{\mu \nu }={\eta }_{{AB}}{{\hat{E}}^{A}}_{\mu }{{\hat{E}}^{B}}_{\nu }$, lead to the same result.The induced connection should be ${{{\rm{\Gamma }}}^{\lambda }}_{\mu \nu }\,={{F}^{\lambda }}_{\mu \nu }(x,A)+{{V}^{\lambda }}_{\mu \alpha }(x,A){A}_{\nu }^{\alpha }$, and ${{{\rm{\Gamma }}}^{\lambda }}_{\mu \nu }={{\hat{E}}_{A}}^{\lambda }{\partial }_{\nu }{{\hat{E}}^{A}}_{\mu }$ at the same time. Noticing that for a tensor T on TM, the derivative of its pull back $\hat{T}$ is57 ) is flat and compatible with the metric (56 ). On the other hand, the covariant derivative of metric (56 ) can be also written as
$\begin{eqnarray*}\begin{array}{l}{\partial }_{\mu }\hat{T}(x)={\partial }_{\mu }(T(x,A))=({\partial }_{\mu }T)(x,A)+({\dot{\partial }}_{\lambda }T)(x,A){\partial }_{\mu }{A}^{\lambda }\\ \,=\,({\delta }_{\mu }T)(x,A)+({\dot{\partial }}_{\lambda }T)(x,A){A}_{\mu }^{\lambda },\end{array}\end{eqnarray*}$
these two results are indeed equal to each other. Finally, the result is $\begin{eqnarray}\begin{array}{l}{{{\rm{\Gamma }}}^{\lambda }}_{\mu \nu }={{{\rm{e}}}_{A}}^{\lambda }{{{\rm{e}}}^{A}}_{\mu ,\nu }+\left(1-\displaystyle \frac{g}{f}\right)\displaystyle \frac{{A}^{\lambda }}{{\hat{\alpha }}^{2}}{{{\rm{e}}}^{A}}_{\nu }{\partial }_{\mu }{A}_{A}\\ \quad +\displaystyle \frac{g^{\prime} }{g}\left({\delta }_{\nu }^{\lambda }-\displaystyle \frac{{A}^{\lambda }{A}_{\nu }}{{\hat{\alpha }}^{2}}\right)\displaystyle \frac{{A}^{A}}{\hat{\alpha }}{\partial }_{\mu }{A}_{A}\\ \quad +\left(\displaystyle \frac{f}{g}-1\right)\displaystyle \frac{{A}_{\nu }}{{\hat{\alpha }}^{2}}{{{\rm{e}}}_{A}}^{\lambda }{\partial }_{\mu }{A}^{A}-(f-g)\displaystyle \frac{f^{\prime} }{{fg}}\displaystyle \frac{{A}^{\lambda }{A}_{\nu }}{{\hat{\alpha }}^{3}}{A}_{A}{\partial }_{\mu }{A}^{A}\end{array}\end{eqnarray}$
and induced torsion is $\begin{eqnarray}{{\hat{T}}^{\lambda }}_{\mu \nu }={{{\rm{\Gamma }}}^{\lambda }}_{[\mu \nu ]}={{\hat{E}}_{A}}^{\lambda }{\partial }_{[\nu }{{\hat{E}}^{A}}_{\mu ]}.\end{eqnarray}$
One can directly verify that this connection ( $\begin{eqnarray}({{\rm{\nabla }}}_{\alpha }{\hat{g}}_{\mu \nu })(x)=({{\rm{\nabla }}}_{\alpha }^{H}{g}_{\mu \nu })(x,A)+({{\rm{\nabla }}}_{\beta }^{V}{g}_{\mu \nu })(x,A){A}_{\alpha }^{\beta }(x).\end{eqnarray}$
While both ${{\rm{\nabla }}}_{\alpha }^{H}{g}_{\mu \nu }$ and ${{\rm{\nabla }}}_{\beta }^{V}{g}_{\mu \nu }$ are proportional to pμ or its higher order, ${\hat{p}}_{\mu }=0$, we also have ${{\rm{\nabla }}}_{\alpha }{\hat{g}}_{\mu \nu }=0$.So we have a connection of the Teleparallel type that is induced from Finsler geometry, and we may call it Finsler–Teleparallel–Bathrel geometry. Within this geometry, one may come up with a group of Teleparallel gravity models. These models should take the tetrad on the base manifold ${{e}^{A}}_{\mu }$ and the vector field Aλ as fundamental variables, for they are defined on the base manifold, i.e. the physical spacetime; while the Finsler metric F(α, β), or the function f, should rely on the model selection, and considered not to be a fundamental variable.
5. A toy model based on Finsler–Teleparallel–Bathrel connection
As a simple example, we lay out a toy model in which the induced metric (56 ) is dynamical, while the primordial metric aμν is ‘geometrical’, i.e. it is aμν instead of ${\hat{g}}_{\mu \nu }$ that couples to the matter fields. Such convention is to remain the dynamics of matter unchanged. Hence, this simple model has the action60 ) is reduced to GR when A(x) = 0 or $F(x,y)=\sqrt{{a}_{\mu \nu }(x){y}^{\mu }{y}^{\nu }}$.
$\begin{eqnarray}S=\displaystyle \frac{1}{2}\int \hat{E}\hat{{\mathbb{T}}}{{\rm{d}}}^{4}x+{S}_{{\rm{m}}},\end{eqnarray}$
where $\hat{E}$ is the determinant of the induce tetrad ${{\hat{E}}^{A}}_{\mu }$, $\begin{eqnarray}\begin{array}{l}\hat{{\mathbb{T}}}=-{\hat{g}}^{\mu \nu }{{\hat{T}}^{\alpha }}_{\mu \alpha }{{\hat{T}}^{\beta }}_{\nu \beta }+\displaystyle \frac{1}{2}{\hat{g}}^{\mu \nu }{{\hat{T}}^{\alpha }}_{\beta \mu }{{\hat{T}}^{\beta }}_{\alpha \nu }\\ \quad +\displaystyle \frac{1}{4}{\hat{g}}_{\alpha \beta }{\hat{g}}^{\mu \rho }{\hat{g}}^{\nu \sigma }{{\hat{T}}^{\alpha }}_{\mu \nu }{{\hat{T}}^{\beta }}_{\rho \sigma }\end{array}\end{eqnarray}$
is the torsion scalar determined by the induced tetrad ${{\hat{E}}^{A}}_{\mu }$. The action of matter Sm keeps the same as that in GR; as an example, considering a scalar field to mimic the matter field, with the following action $\begin{eqnarray}{S}_{{\rm{m}}}=\int {\rm{e}}\left[\displaystyle \frac{1}{2}{a}^{\mu \nu }{\partial }_{\mu }\phi {\partial }_{\nu }\phi -V(\phi )\right]{{\rm{d}}}^{4}x,\end{eqnarray}$
where $e=\sqrt{-{\rm{\det }}({a}_{\mu \nu })}$. In the previous section, we have proved that when A(x) = 0, the geometry is reduced to the Riemannian geometry; i.e. to take A(x) = 0 we have ${\hat{g}}_{\mu \nu }={a}_{\mu \nu }$, ${{\hat{E}}^{A}}_{\mu }={{{\rm{e}}}^{A}}_{\mu }$, ${{{\rm{\Gamma }}}^{\lambda }}_{\mu \nu }={{{\rm{e}}}_{A}}^{\lambda }{\partial }_{\nu }{{{\rm{e}}}^{A}}_{\mu }$, and the action is reduced to $\begin{eqnarray}S=\int {\rm{e}}\left(\displaystyle \frac{1}{2}{\mathbb{T}}+{{ \mathcal L }}_{{\rm{m}}}\right),\end{eqnarray}$
which is just the action of the TEGR model. Since the TEGR model is equivalent to GR, we can say that our action (For an arbitrary Finsler metric $F=\alpha f(\tfrac{\beta }{\alpha })$, the equations of motion (EoMs) appear after taking the variation of action (60 ) with respect to aμν:
$\begin{eqnarray}\begin{array}{l}{G}^{\alpha \beta }\left[{g}^{2}(\hat{\alpha }){\delta }_{(\alpha }^{\mu }{\delta }_{\beta )}^{\nu }+\left(\displaystyle \frac{{ff}^{\prime} (\hat{\alpha })}{\hat{\alpha }}-{ff}^{\prime\prime} (\hat{\alpha })\right.\right.\\ \quad \left.\left.-f^{\prime2}(\hat{\alpha },),\Space{0ex}{3.15ex}{0ex}\right){\hat{h}}_{\alpha \beta }{A}^{\mu }{A}^{\nu }\Space{0ex}{0.21ex}{0ex}\right]={T}^{\mu \nu }\end{array}\end{eqnarray}$
where Gαβ is the Einstein Tensor of gμν, and Tμν is the normal energy-momentum tensor. Obviously, the form of the function f, or the Finsler metric, greatly influences the evolution of spacetime, and the special case f ≡ 1 just gives the Einstein field equations in GR.6. Conclusion
In this paper, we came up with a connection based on the Finsler (α, β)-metric $F(\alpha ,\beta )=\alpha f(\tfrac{\beta }{\alpha })$:
$\begin{eqnarray*}{{\omega }^{\lambda }}_{\mu }={{F}^{\lambda }}_{\mu \nu }{\rm{d}}{x}^{\nu }+{{V}^{\lambda }}_{\mu \nu }\delta {y}^{\nu }\end{eqnarray*}$
with connection coefficients $\begin{eqnarray*}{{F}^{\lambda }}_{\mu \nu }={{E}_{A}}^{\lambda }{\delta }_{\nu }{{E}^{A}}_{\mu },\,{{V}^{\lambda }}_{\mu \nu }={{E}_{A}}^{\lambda }{\dot{\partial }}_{\nu }{{E}^{A}}_{\mu },\end{eqnarray*}$
where $\begin{eqnarray*}\begin{array}{l}{{E}^{A}}_{\mu }=\left[g\left(\displaystyle \frac{\beta }{\alpha }\right){\delta }_{\mu }^{\alpha }+(f-g)\left(\displaystyle \frac{\beta }{\alpha }\right)\displaystyle \frac{{y}^{\alpha }{y}_{\mu }}{{\alpha }^{2}}\right.\\ \,\left.+\,f^{\prime} \left(\displaystyle \frac{\beta }{\alpha }\right)\displaystyle \frac{{y}^{\alpha }{p}_{\mu }}{\alpha }\right]{{{\rm{e}}}^{A}}_{\alpha },\end{array}\end{eqnarray*}$
the function $g(x)=\sqrt{{f}^{2}(x)-{xf}(x)f^{\prime} (x)}$, and ${{{\rm{e}}}^{A}}_{\alpha }$ is the tetrad associated with the metric on the base manifold aμν. We showed that such a connection can indeed be used to induce the normal (pseud-)Riemann geometry when the functions are selected as F = α or f ≡ 1, and lead to TEGR or GR.We also used a toy model with action (60 ) to show this, with ${\hat{g}}_{\mu \nu }$ being dynamical and aμν coupled with matter. We showed that when f ≡ 1 or F = α, both the action and the EoMs are induced to the same as ones in TEGR or GR, so the model based on the connection (16 ) can indeed be considered as a generalization of Teleparallel gravity.
A group of gravity models can be built in terms of this scheme. These models can have new predictions beyond GR and how they are constrained by current observations are questions deserving of further studies. We will visit these problems in the future.
Acknowledgments
We thank the anonymous referees for useful suggestions, Prof. Mingzhe Li for instructions and Dr. Haomin Rao and Dr. Dehao Zhao for helpful discussions.
This work is supported in part by NSFC under Grant No.12075231 and 12047502.
Appendix. Proof of equations (51 ) and (52 )
For vertical covariant derivative, using (9 ), we have
$\begin{eqnarray}\begin{array}{l}{{\rm{\nabla }}}_{\alpha }^{V}{g}_{\mu \nu }={\dot{\partial }}_{\alpha }{g}_{\mu \nu }-{{V}^{\lambda }}_{\mu \alpha }{g}_{\nu \lambda }-{{V}^{\lambda }}_{\nu \alpha }{g}_{\mu \lambda }\\ \quad =\ \left(2{gg}^{\prime} -\displaystyle \frac{f^{\prime} {g}^{2}}{f}\right)\displaystyle \frac{{p}_{\mu }{h}_{\nu \alpha }+{p}_{\nu }{h}_{\mu \alpha }}{\alpha }-\displaystyle \frac{{f}^{2}f^{\prime\prime} }{g}\displaystyle \frac{{y}_{\mu }{p}_{\nu }+{y}_{\nu }{p}_{\mu }}{\alpha }\displaystyle \frac{{p}_{\alpha }}{\alpha }\\ \quad +\ \left[{ff}^{\prime\prime\prime} +f^{\prime} f^{\prime\prime} -\displaystyle \frac{2{ff}^{\prime\prime} }{g}\left(f^{\prime} +g^{\prime} \right)\right]\displaystyle \frac{{p}_{\mu }{p}_{\nu }{p}_{\alpha }}{\alpha },\end{array}\end{eqnarray}$
so it is proportional to pμ or its higher order.The horizontal covariant derivative is more complicated. The following results are needed:A2 –A5 ) we have57 ) is induced.
$\begin{eqnarray}{\delta }_{\nu }\left(\displaystyle \frac{\beta }{\alpha }\right)=\displaystyle \frac{{y}^{\rho }}{\alpha }\left({A}_{\rho ,\nu }-{A}_{\sigma }{{e}_{A}}^{\sigma }{{e}^{A}}_{\rho ,\nu }\right)=\displaystyle \frac{{y}^{\rho }}{\alpha }{{e}^{A}}_{\rho }{\partial }_{\nu }{A}_{A}\,,\end{eqnarray}$
$\begin{eqnarray}{\delta }_{\alpha }\left(\displaystyle \frac{{y}_{\mu }{y}_{\nu }}{{\alpha }^{2}}\right)=\displaystyle \frac{{y}_{\rho }{{e}_{A}}^{\rho }}{\alpha }\left({y}_{\mu }{{e}^{A}}_{\nu ,\alpha }+{y}_{\nu }{{e}^{A}}_{\mu ,\alpha }\right),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\delta }_{\alpha }\left(\displaystyle \frac{{y}_{\mu }{p}_{\nu }}{\alpha }\right)=\displaystyle \frac{{y}_{\mu }}{\alpha }{{e}^{A}}_{\nu }{\partial }_{\alpha }{A}_{A}-\displaystyle \frac{{y}_{\mu }{y}_{\nu }}{{\alpha }^{2}}\displaystyle \frac{{y}^{\rho }}{\alpha }{{e}^{A}}_{\rho }{\partial }_{\alpha }{A}_{A}\\ \quad -\displaystyle \frac{{y}_{\mu }{p}_{\rho }}{{\alpha }^{2}}{{e}_{A}}^{\rho }{{e}^{A}}_{\nu ,\alpha }+\displaystyle \frac{{y}_{\rho }{p}_{\nu }}{{\alpha }^{2}}{{e}_{A}}^{\rho }{{e}^{A}}_{\mu ,\alpha },\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\delta }_{\alpha }\left({p}_{\mu }{p}_{\nu }\right)=\left({p}_{\mu }{{e}^{A}}_{\nu ,\alpha }+{p}_{\nu }{{e}^{A}}_{\mu ,\alpha }\right){p}_{\rho }{{e}_{A}}^{\rho }\\ \quad +\left({p}_{\mu }{{e}^{A}}_{\nu }+{p}_{\nu }{{e}^{A}}_{\mu }\right){\partial }_{\alpha }{A}_{A}\\ \quad -\displaystyle \frac{{y}_{\mu }{p}_{\nu }+{y}_{\nu }{p}_{\mu }}{\alpha }\displaystyle \frac{{y}^{\rho }}{\alpha }{{e}^{A}}_{\rho }{\partial }_{\alpha }{A}_{A}.\end{array}\end{eqnarray}$
Noticing that the arguments of functions f, g, ⋯ are β/α, we have $\begin{eqnarray}\begin{array}{l}{\delta }_{\alpha }{g}_{\mu \nu }={\delta }_{\alpha }\left[{g}^{2}{a}_{\mu \nu }+\displaystyle \frac{\beta }{\alpha }{ff}^{\prime} \displaystyle \frac{{y}_{\mu }{y}_{\nu }}{{\alpha }^{2}}\right.\\ \quad \left.+\ {ff}^{\prime} \displaystyle \frac{{y}_{\mu }{p}_{\nu }+{y}_{\nu }{p}_{\mu }}{\alpha }+\left(f^{\prime2}+{ff}^{\prime\prime} \right){p}_{\mu }{p}_{\nu }\right]\\ \quad =\ 2{gg}^{\prime} {\delta }_{\alpha }\left(\displaystyle \frac{\beta }{\alpha }\right){a}_{\mu \nu }+{g}^{2}{\partial }_{\alpha }{a}_{\mu \nu }+\left({ff}^{\prime} +\displaystyle \frac{\beta }{\alpha }f^{\prime2}+\displaystyle \frac{\beta }{\alpha }{ff}^{\prime\prime} \right)\\ \quad \times \ \displaystyle \frac{{y}_{\mu }{y}_{\nu }}{{\alpha }^{2}}{\delta }_{\alpha }\left(\displaystyle \frac{\beta }{\alpha }\right)+\displaystyle \frac{\beta }{\alpha }{ff}^{\prime} {\delta }_{\alpha }\left(\displaystyle \frac{{y}_{\mu }{y}_{\nu }}{{\alpha }^{2}}\right)\\ \quad +\ {ff}^{\prime} {\delta }_{\alpha }\left(\displaystyle \frac{{y}_{\mu }{p}_{\nu }+{y}_{\nu }{p}_{\mu }}{\alpha }\right)+\left(f^{\prime2}+{ff}^{\prime\prime} \right)\displaystyle \frac{{y}_{\mu }{p}_{\nu }+{y}_{\nu }{p}_{\mu }}{\alpha }{\delta }_{\alpha }\\ \quad \times \ \left(\displaystyle \frac{\beta }{\alpha }\right)+\left({ff}^{\prime\prime\prime} +3f^{\prime} f^{\prime\prime} \right){p}_{\mu }{p}_{\nu }{\delta }_{\alpha }\left(\displaystyle \frac{\beta }{\alpha }\right)\\ \quad +\ \left(f^{\prime2}+{ff}^{\prime\prime} \right){\delta }_{\alpha }\left({p}_{\mu }{p}_{\nu }\right),\end{array}\end{eqnarray}$
Use results ( $\begin{eqnarray}\begin{array}{l}{\delta }_{\alpha }{g}_{\mu \nu }={g}^{2}{\eta }_{{AB}}\left({{{\rm{e}}}^{A}}_{\mu }{{{\rm{e}}}^{B}}_{\nu ,\alpha }+{{{\rm{e}}}^{A}}_{\mu ,\alpha }{{{\rm{e}}}^{B}}_{\nu }\right)+2{gg}^{\prime} {h}_{\mu \nu }\displaystyle \frac{{y}^{\rho }}{\alpha }{{{\rm{e}}}^{A}}_{\rho }{\partial }_{\alpha }{A}_{A}\\ \quad +\displaystyle \frac{\beta }{\alpha }{ff}^{\prime} \displaystyle \frac{{y}_{\rho }{{{\rm{e}}}_{A}}^{\rho }}{\alpha }\left({y}_{\mu }{{{\rm{e}}}^{A}}_{\nu ,\alpha }+{y}_{\nu }{{{\rm{e}}}^{A}}_{\mu ,\alpha }\right)\\ \quad +{ff}^{\prime} \left(\displaystyle \frac{{y}_{\mu }}{\alpha }{{{\rm{e}}}^{A}}_{\nu }+\displaystyle \frac{{y}_{\nu }}{\alpha }{{{\rm{e}}}^{A}}_{\mu }\right){\partial }_{\alpha }{A}_{A}\\ \quad +\left({ff}^{\prime\prime\prime }+3f^{\prime} f^{\prime\prime} \right){p}_{\mu }{p}_{\nu }\displaystyle \frac{{y}^{\rho }}{\alpha }{{{\rm{e}}}^{A}}_{\rho }{\partial }_{\alpha }{A}_{A}\\ \quad +\left(f^{\prime2}+{ff}^{\prime\prime} \right)\left[\left({p}_{\mu }{{{\rm{e}}}^{A}}_{\nu ,\alpha }+{p}_{\nu }{{{\rm{e}}}^{A}}_{\mu ,\alpha }\right){p}_{\rho }{{{\rm{e}}}_{A}}^{\rho }\right.\\ \quad \left.+\left({p}_{\mu }{{{\rm{e}}}^{A}}_{\nu }+{p}_{\nu }{{{\rm{e}}}^{A}}_{\mu }\right){\partial }_{\alpha }{A}_{A}\right].\end{array}\end{eqnarray}$
So the horizontal covariant derivative is $\begin{eqnarray}\begin{array}{l}{{\rm{\nabla }}}_{\alpha }^{H}{g}_{\mu \nu }={\delta }_{\alpha }{g}_{\mu \nu }-{{F}^{\lambda }}_{\mu \alpha }{g}_{\nu \lambda }-{{F}^{\lambda }}_{\nu \alpha }{g}_{\mu \lambda }\\ \quad ={ff}^{\prime\prime} \left({p}_{\mu }{{{\rm{e}}}^{A}}_{\nu }+{p}_{\nu }{{{\rm{e}}}^{A}}_{\mu }\right){\partial }_{\alpha }{A}_{A}-{ff}^{\prime} \displaystyle \frac{{y}_{\rho }}{\alpha }{{{\rm{e}}}_{A}}^{\rho }\\ \quad \times \left({p}_{\mu }{{{\rm{e}}}^{A}}_{\nu ,\alpha }+{p}_{\nu }{{{\rm{e}}}^{A}}_{\mu ,\alpha }\right)\\ \quad -{ff}^{\prime} {p}_{\rho }{{{\rm{e}}}_{A}}^{\rho }\displaystyle \frac{{y}_{\mu }{{{\rm{e}}}^{A}}_{\nu ,\alpha }+{y}_{\nu }{{{\rm{e}}}^{A}}_{\mu ,\alpha }}{\alpha }\\ \quad -\left(\displaystyle \frac{g^{\prime} {ff}^{\prime} }{g}+f^{\prime\prime} g-f^{\prime} g^{\prime} \right)\displaystyle \frac{{y}^{\rho }}{\alpha }{{{\rm{e}}}^{A}}_{\rho }{\partial }_{\alpha }{A}_{A}\left(\displaystyle \frac{{y}_{\mu }}{\alpha }{p}_{\nu }+\displaystyle \frac{{y}_{\nu }}{\alpha }{p}_{\mu }\right)\\ \quad +\left[{ff}^{\prime\prime\prime} +3f^{\prime} f^{\prime\prime} -\displaystyle \frac{g^{\prime} }{g}\left({ff}^{\prime\prime} +f^{\prime2}\right)-\displaystyle \frac{f^{\prime} }{f}\left(f^{\prime\prime} g-f^{\prime} g^{\prime} \right)\right]\\ \quad \times \displaystyle \frac{{y}^{\rho }}{\alpha }{{{\rm{e}}}^{A}}_{\rho }{\partial }_{\alpha }{A}_{A}{p}_{\mu }{p}_{\nu },\end{array}\end{eqnarray}$
so it is also proportional to pμ or its higher order. After being pulled back to the base manifold M, pμ = 0, thus both ${{\rm{\nabla }}}_{\alpha }^{H}{g}_{\mu \nu }$ and ${{\rm{\nabla }}}_{\alpha }^{V}{g}_{\mu \nu }$ vanish, and a metric-compatible connection (