Parameterized Absolute Parallelism Geometry

• Parameterized Absolute Parallelism (PAP) geometry is a one-parameter family of affine connections that interpolates between torsion-free Riemannian and torsionful absolute parallelism models.
• It utilizes a tetrad formalism where a pseudo-Riemannian metric is induced from frame fields, and the affine connection is modified by a dimensionless parameter b multiplying the contortion.
• The framework underpins applications in cosmology, field equations, and particle motion by generating mixed curvature–torsion invariants and differential identities.

Parameterized Absolute Parallelism (PAP) geometry is a one-parameter extension of Absolute Parallelism (AP) geometry in which a tetrad-built pseudo-Riemannian metric is retained while the affine structure is deformed by a dimensionless parameter multiplying contortion. In the standard PAP presentation, the defining connection is

$$\nabla^{\alpha}{}_{\mu\nu}=\left\{^{\alpha}_{\mu\nu}\right\}+b\,\gamma^{\alpha}{}_{\mu\nu},$$

so that $b=0$ gives the Riemannian Levi-Civita limit and $b=1$ gives the Weitzenböck or canonical AP limit. PAP therefore interpolates, at the level of the affine connection itself, between torsion-free Riemannian geometry and torsionful absolute parallelism, while allowing intermediate geometries with both curvature and torsion (Dabash et al., 24 Aug 2025, Wanas et al., 2016).

1. Foundational construction

PAP geometry is built on the AP tetrad formalism. The basic fields are $n$ linearly independent frame vectors $\lambda_i^\mu$ and dual covariant components $\lambda_{i\mu}$, satisfying the standard completeness and orthonormality relations

$$\lambda_i^{\mu}\lambda^i_{\nu}=\delta^\mu_{\nu}, \qquad \lambda_i^{\nu}\lambda^j_{\nu}=\delta_i^{\,j}.$$

From these fields one constructs the metric

$$g_{\mu\nu}=\lambda^i_{\mu}\lambda^i_{\nu}, \qquad g^{\mu\nu}=\lambda_i^{\mu}\lambda_i^{\nu}.$$

In this sense PAP geometry is not introduced independently of the metric: it begins with a parallelization, induces a metric from it, and then modifies the affine structure relative to the Levi-Civita connection of that metric (Dabash et al., 24 Aug 2025).

The AP side of the construction uses the Weitzenböck connection

$$\Gamma^{\alpha}{}_{\mu\nu}=\lambda_i^{\alpha}\,\partial_\nu\lambda^i_{\mu},$$

which is generally non-symmetric and torsionful. The difference between this connection and the Levi-Civita connection $\{^{\alpha}_{\mu\nu}\}$ is the contortion tensor,

$b=0$0

The torsion tensor is then

$b=0$1

This tetrad $b=0$2 metric $b=0$3 Weitzenböck connection $b=0$4 contortion $b=0$5 torsion chain is the standard AP backbone on which PAP is erected. A global AP formulation expresses the same structure in terms of a parallelization basis, a canonical connection for which the frame fields are parallel, and a contortion defined as the difference between canonical and Levi-Civita connections; PAP inherits that architecture and parameterizes the non-Riemannian piece rather than replacing it (Dabash et al., 24 Aug 2025, Youssef et al., 2012).

2. Parameterization, limits, and characteristic identities

The defining step of PAP is the insertion of a dimensionless parameter $b=0$6 into the Levi-Civita–contortion split: $b=0$7 This is not merely a heuristic statement that torsion is “added.” It is a literal one-parameter family of affine connections. When $b=0$8,

$b=0$9

so PAP reduces to ordinary Riemannian geometry. When $b=1$0,

$b=1$1

so PAP reduces to the Weitzenböck or AP connection. Intermediate values yield a Riemann–Cartan-type regime with both curvature and torsion (Dabash et al., 24 Aug 2025, Wanas et al., 2016, Wanas et al., 2021).

The associated parameterized tensors scale accordingly. The parameterized contortion is

$b=1$2

the parameterized torsion is

$b=1$3

and the parameterized basic form is

$b=1$4

The parameterized connection is metric,

$b=1$5

and admits the usual dual and symmetric companions,

$b=1$6

Within the Wanas formulation, $b=1$7 has also been written phenomenologically as

$b=1$8

where $b=1$9, $n$0 is the fine-structure constant, and $n$1 is a dimensionless parameter to be fixed experimentally. This gives PAP a discrete-spectrum reading in some applications, although no universal empirical determination of $n$2 is supplied in the cited work (Wanas et al., 2016, Wanas et al., 2021).

A central identity of PAP geometry is the generalized differential identity

$n$3

It reduces to the Riemannian or AP identities at $n$4 or $n$5, and for symmetric tensors the contortion term drops out, yielding a conservation-type divergence law. This identity is one of the clearest formal expressions of PAP as an interpolation scheme rather than a disconnected collection of special cases (Wanas et al., 2016).

3. Curvature decomposition and invariant structure

The PAP curvature tensor is obtained from the full parameterized connection. In the formulation used for curvature invariants,

$n$6

where $n$7 is the Riemann-Christoffel curvature of the Levi-Civita connection and $n$8 is the contortion-built part,

$n$9

This decomposition is operationally decisive, because it separates the purely Riemannian sector from the torsional sector while preserving their coupling (Dabash et al., 24 Aug 2025).

The corresponding PAP-modified Kretschmann scalar is

$\lambda_i^\mu$0

and the substitution $\lambda_i^\mu$1 gives

$\lambda_i^\mu$2

The three pieces are explicitly interpreted as pure Riemannian curvature, curvature–torsion interaction, and pure torsion or non-Riemannian contribution. This formula shows that PAP does not substitute torsion for curvature; it produces mixed invariants in which torsion enters linearly and quadratically through the parameterized connection (Dabash et al., 24 Aug 2025).

A PAP-relevant neighboring development concerns conformal structure in AP geometry. Under the conformal change

$\lambda_i^\mu$3

the AP basic form transforms as

$\lambda_i^\mu$4

and the contortion transforms as

$\lambda_i^\mu$5

Trace-compensated tensors and conformal connections built from $\lambda_i^\mu$6, $\lambda_i^\mu$7, and the Levi-Civita curvature were then shown to be conformally invariant in AP geometry. Since PAP is defined by parameterizing the contortion contribution, this suggests a direct route to PAP conformal invariants, although that step is not carried out explicitly in the AP conformal paper itself (Youssef et al., 2016).

4. Field equations, variational constructions, and particle motion

One line of PAP work constructs field equations from a scalar built from the curvature of the dual PAP connection. In that formulation the parameterized canonical connection is written

$\lambda_i^\mu$8

and the scalar curvature of the dual connection becomes

$\lambda_i^\mu$9

Up to a divergence term, this yields

$\lambda_{i\mu}$0

so the PAP scalar is the Ricci scalar plus a quadratic contortion contribution. The associated Dolan–McCrea variation leads to a single non-symmetric field equation whose symmetric part can be written in Einstein-like form with a purely geometric source tensor. Its skew part is Maxwell-like, with

$\lambda_{i\mu}$1

but the same work states that in the linearized limit this $\lambda_{i\mu}$2 vanishes identically and therefore cannot represent the electromagnetic field tensor of Maxwell theory. PAP unification in this model is therefore formal rather than phenomenologically complete (Nwachioma et al., 2017).

A second line of work focuses on trajectories. Using the Bazanski method in PAP space, the path equation for a charged spinning test particle is

$\lambda_{i\mu}$3

Here $\lambda_{i\mu}$4 controls the spin-torsion coupling, $\lambda_{i\mu}$5 is interpreted as a charge parameter and may be taken as $\lambda_{i\mu}$6, $\lambda_{i\mu}$7 is identified as a geometric electromagnetic potential, and

$\lambda_{i\mu}$8

When $\lambda_{i\mu}$9, the equation reduces to the Riemannian geodesic. When $$\lambda_i^{\mu}\lambda^i_{\nu}=\delta^\mu_{\nu}, \qquad \lambda_i^{\nu}\lambda^j_{\nu}=\delta_i^{\,j}.$$0, it reduces to the previously known PAP modified geodesic for a neutral spinning particle (Wanas et al., 2021).

Earlier AP path theory had already shown that canonical, symmetric, and dual AP connections generate distinct path equations with torsion terms of different strengths, while AP spinning equations generalized the Papapetrou construction to those multiple absolute derivatives. This suggests why PAP was formulated as a one-parameter family: it converts a discrete collection of AP connection choices into a continuous interpolation with explicit control over torsion coupling (Nwachioma et al., 2017, Kahil, 2018).

5. Cosmology and the PAP-modified Kretschmann scalar

A recent cosmological application uses an FLRW-compatible tetrad adapted to Robertson’s cosmological principle and the isotropic-coordinate line element

$$\lambda_i^{\mu}\lambda^i_{\nu}=\delta^\mu_{\nu}, \qquad \lambda_i^{\nu}\lambda^j_{\nu}=\delta_i^{\,j}.$$1

where $$\lambda_i^{\mu}\lambda^i_{\nu}=\delta^\mu_{\nu}, \qquad \lambda_i^{\nu}\lambda^j_{\nu}=\delta_i^{\,j}.$$2 is the scale factor and $$\lambda_i^{\mu}\lambda^i_{\nu}=\delta^\mu_{\nu}, \qquad \lambda_i^{\nu}\lambda^j_{\nu}=\delta_i^{\,j}.$$3 is the spatial-curvature index. In this setting the PAP-modified Kretschmann scalar is reported as

$$\lambda_i^{\mu}\lambda^i_{\nu}=\delta^\mu_{\nu}, \qquad \lambda_i^{\nu}\lambda^j_{\nu}=\delta_i^{\,j}.$$4

while the classical FLRW result is

$$\lambda_i^{\mu}\lambda^i_{\nu}=\delta^\mu_{\nu}, \qquad \lambda_i^{\nu}\lambda^j_{\nu}=\delta_i^{\,j}.$$5

The PAP correction therefore enters through the multiplicative factor

$$\lambda_i^{\mu}\lambda^i_{\nu}=\delta^\mu_{\nu}, \qquad \lambda_i^{\nu}\lambda^j_{\nu}=\delta_i^{\,j}.$$6

Equality with the classical scalar requires

$$\lambda_i^{\mu}\lambda^i_{\nu}=\delta^\mu_{\nu}, \qquad \lambda_i^{\nu}\lambda^j_{\nu}=\delta_i^{\,j}.$$7

hence

$$\lambda_i^{\mu}\lambda^i_{\nu}=\delta^\mu_{\nu}, \qquad \lambda_i^{\nu}\lambda^j_{\nu}=\delta_i^{\,j}.$$8

At $$\lambda_i^{\mu}\lambda^i_{\nu}=\delta^\mu_{\nu}, \qquad \lambda_i^{\nu}\lambda^j_{\nu}=\delta_i^{\,j}.$$9 this is the Riemannian limit; at $$g_{\mu\nu}=\lambda^i_{\mu}\lambda^i_{\nu}, \qquad g^{\mu\nu}=\lambda_i^{\mu}\lambda_i^{\nu}.$$0 the paper interprets the agreement as a nontrivial cancellation of torsion contributions. Directly from the formula, one may also note that the modified scalar vanishes at $$g_{\mu\nu}=\lambda^i_{\mu}\lambda^i_{\nu}, \qquad g^{\mu\nu}=\lambda_i^{\mu}\lambda_i^{\nu}.$$1 and is not monotonic in $$g_{\mu\nu}=\lambda^i_{\mu}\lambda^i_{\nu}, \qquad g^{\mu\nu}=\lambda_i^{\mu}\lambda_i^{\nu}.$$2 over the whole real line (Dabash et al., 24 Aug 2025).

For the Big Bang–Big Rip model with

$$g_{\mu\nu}=\lambda^i_{\mu}\lambda^i_{\nu}, \qquad g^{\mu\nu}=\lambda_i^{\mu}\lambda_i^{\nu}.$$3

the paper gives

$$g_{\mu\nu}=\lambda^i_{\mu}\lambda^i_{\nu}, \qquad g^{\mu\nu}=\lambda_i^{\mu}\lambda_i^{\nu}.$$4

As $$g_{\mu\nu}=\lambda^i_{\mu}\lambda^i_{\nu}, \qquad g^{\mu\nu}=\lambda_i^{\mu}\lambda_i^{\nu}.$$5, the scalar diverges; as $$g_{\mu\nu}=\lambda^i_{\mu}\lambda^i_{\nu}, \qquad g^{\mu\nu}=\lambda_i^{\mu}\lambda_i^{\nu}.$$6, it tends to zero. The paper interprets the early-time divergence as the large-curvature Big Bang regime and the late-time decay as an approach to a flat or nearly flat state near the Big Rip. It also states that for $$g_{\mu\nu}=\lambda^i_{\mu}\lambda^i_{\nu}, \qquad g^{\mu\nu}=\lambda_i^{\mu}\lambda_i^{\nu}.$$7 the early-time peaks are suppressed relative to $$g_{\mu\nu}=\lambda^i_{\mu}\lambda^i_{\nu}, \qquad g^{\mu\nu}=\lambda_i^{\mu}\lambda_i^{\nu}.$$8, but the explicit prefactor shows that any claim of monotonic suppression must be restricted to specific parameter intervals rather than treated as global (Dabash et al., 24 Aug 2025).

A plausible implication, rather than a direct result, is that PAP cosmology should treat tetrad choice with the same care emphasized in teleparallel FRW models: in torsion-based theories the metric does not by itself fix the physically admissible frame, and naive diagonal tetrads can spoil homogeneity or isotropy at the level of torsion invariants (Ferraro et al., 2011).

6. Scope, misconceptions, and related extensions

A recurrent misconception is that PAP geometry is simply AP geometry with “some torsion added.” The defining literature does not support that reduction. PAP is a one-parameter family of affine connections, with explicitly parameterized contortion, parameterized torsion, and parameter-dependent curvature identities. Its characteristic move is not replacement of curvature by torsion, but controlled interpolation between Levi-Civita and Weitzenböck transport.

Several limitations are equally clear. In the FLRW Kretschmann analysis, the presentation is partly typographically corrupted, the reduction to the final closed form is not shown in full detail, and the parameter $$g_{\mu\nu}=\lambda^i_{\mu}\lambda^i_{\nu}, \qquad g^{\mu\nu}=\lambda_i^{\mu}\lambda_i^{\nu}.$$9 is left phenomenological: no field equations, matter couplings, or observational constraints are given to determine it. The Big Bang–Big Rip example does not remove the initial singularity, because the modified scalar still diverges at $$\Gamma^{\alpha}{}_{\mu\nu}=\lambda_i^{\alpha}\,\partial_\nu\lambda^i_{\mu},$$0; “smoothing” therefore means moderation of curvature in some regimes, not rigorous singularity resolution (Dabash et al., 24 Aug 2025).

The same caution applies to PAP unification programs. The scalar-curvature construction based on the dual PAP connection produces an Einstein-like symmetric sector and a Maxwell-like skew sector, yet its own weak-field analysis states that the electromagnetic tensor vanishes at linear order. PAP field theory, in that formulation, does not recover Maxwell electrodynamics (Nwachioma et al., 2017).

Related but distinct developments enlarge the AP/PAP landscape. A global treatment of AP geometry provides the canonical connection, torsion, contortion, dual and symmetric connections, and the Wanas tensor as the AP substrate from which PAP is naturally built (Youssef et al., 2012). Finslerized Parallelizable spaces replace $$\Gamma^{\alpha}{}_{\mu\nu}=\lambda_i^{\alpha}\,\partial_\nu\lambda^i_{\mu},$$1 by $$\Gamma^{\alpha}{}_{\mu\nu}=\lambda_i^{\alpha}\,\partial_\nu\lambda^i_{\mu},$$2, introduce horizontal and vertical contortion sectors, and recover classical AP in an FP-Riemannian limit; this suggests, but does not explicitly formulate, a Finsler-PAP analogue (Youssef et al., 2012). By contrast, another AP line of work argues for a unique no-free-parameter AP theory rather than a parameterized family, making PAP’s interpolation principle conceptually nontrivial rather than automatic (Zhogin, 2011).

Taken together, these results place PAP geometry in a specific niche: it is a tetrad-based, metric-compatible, parameterized Riemann–Cartan framework designed to keep both the Riemannian and absolute-parallelism limits visible within one formalism. Its present literature is strongest where the parameterized connection can be translated into explicit invariants, differential identities, or path equations, and weakest where a dynamical determination of the interpolation parameter is required.