The zamkovoy canonical paracontact connection on LP-Kenmotsu manifolds

Introduction

The differential geometry of pseudo-Riemannian manifolds equipped with paracontact metric structures plays an important role in mathematical physics, in the study of general relativity, cosmological models, and non-symmetric gravitational theories. S. Zamkovoy [1] introduced a canonical linear connection on Paracontact metric manifolds, referred to as the Zamkovoy canonical Paracontact connection. This connection serves as the Paracontact of the Tanaka-Webster connection on contact metric manifolds. When dealing with a Lorentzian Para-Kenmotsu (LP-Kenmotsu) manifold, this connection provides profound structural insights that differ significantly from the standard Levi-Civita connection. This paper presents a rigorous exposition of the Zamkovoy Canonical Paracontact connection on LP-Kenmotsu manifolds, the propositions, theorems, and their proofs. It also address a fundamental algebraic property of the structure tensor 𝜙 specifically, its symmetry with respect to the Lorentzian metric which is crucial for establishing correct geometric identities and resolving mathematical discrepancies present in earlier works. Furthermore, it connect these results with recent developments on the curvature properties of Lorentzian Para-Kenmotsu manifolds. Particularly, it highlights the extensive research carried out by F. N. Mburu and collaborators [6][7][8][9] on several generalized curvature tensors, such as the W₃, W-, W₇, and W₉ curvature tensors, on LP-Kenmotsu manifolds. Their findings provide valuable geometric context and inspire deeper inquiries into how the choice of linear connections, like the Zamkovoy connection, affects the behavior of such curvature tensors. Alongside these purely geometric aims, this paper situates its results within the broader, emerging conversation on Responsible AI. Responsible AI research increasingly calls for formal, mathematically grounded vocabularies for describing when a transformation applied to a system, a dataset, or a risk classification is well-behaved: consistent with an underlying reference structure, free of unaccounted-for distortion, and traceable back to first principles. The Zamkovoy canonical Paracontact connection, precisely because its metric compatibility, torsion, and parallelism properties can be characterized exactly, offers a candidate formal analogue for these desiderata. Section 7 returns to this theme once the requisite geometric machinery has been established, using the connection's structural behavior as a lens on auditable, structure-preserving transformations relevant to AI governance frameworks.

Preliminaries

Let M be an n-dimensional smooth manifold with an almost paracontact metric structure (𝜙, ξ, 𝜂, 𝑔), where 𝜙 is a (1,1) -tensor field, ξ is the Reeb vector field, 𝜂 is a 1-form, and g is a Lorentzian metric. Definition 2.1. An n-dimensional almost Paracontact metric manifold (𝑀, 𝜙, ξ, 𝜂, 𝑔) is known as a Lorentzian Para-Kenmotsu (LP-Kenmotsu) manifold if the following conditions hold for all vector fields X, Y ∈ χ(M) [2,5] :

where ∇ denotes the Levi-Civita connection of 𝑔.

On an LP-Kenmotsu manifold, the covariant differentiation of the Reeb vector field ξ and the 1-form η satisfies the following standard relations:

Some authors adopt the opposite sign convention for the Reeb vector field derivative: For almost contact geometry where 𝜂(ξ) = 0, the structure tensor 𝜙 is skew-symmetric that is 𝑔(𝜙𝑋, 𝑌) = -𝑔(𝑋, 𝜙𝑌). Yet, on Lorentzian Paracontact manifolds where𝜂(ξ) = -1, the compatibility relation (2.2) mathematically forces 𝜙 to be symmetric. This distinction is pivotal for avoiding sign errors in subsequent derivations.

The Zamkovoy Connection

The Zamkovoy connection ▽ 𝑧 on an almost Paracontact metric manifold is defined by [1,2] ▽ 𝑋 𝑧 𝑌 =▽ 𝑋 𝑌 + 𝜂(𝑋)𝜙𝑌 -𝜂(𝑌) ▽ 𝑋 𝜉 + (▽ 𝑋 𝜂)(𝑌)𝜉 (3.1)

∀ vector fields 𝑋, 𝑌 ∈ 𝜒(𝑀). Consider this connection to LP-Kenmotsu manifolds under both conventions.

Connection under Convention (i)

▽ 𝑿 𝝃 = -𝑿 -𝜼(𝑿)𝝃 Using relations in (2.4) and (2.5), substitute these into (3.1)

Hence, connection reduced to

Then, the connection simplifies to

It is clear the two simplified expressions (3.3) and (3.5) differ only in the signs of the second and the third terms. For paracontact structure tr(ϕ) = 0 Hence 𝑟 𝑍 = 𝑟 + (𝑛 -1)(3𝑛 -4) (5.8)

Examining the metric compatibility of the

Submanifolds and Solitons

If 𝑀 is an anti-invariant submanifold of an Lorentzian Para Kenmotsu manifold then, 𝑔(𝑌, 𝜙𝑍) = 0 condition holds. Thus, the Ricci tensor 𝑆 𝑍 is completely symmetric: [3] 𝑆 𝑍 (𝑌, 𝑍) = 𝑆(𝑌, 𝑍) + 3𝑛 -5)𝑔(𝑌, 𝑍) + (2𝑛 -4)𝜂(𝑌)𝜂(𝑍) (6.1)

Allowing the study of Einstein solitons and η-Einstein solitons w.r.t Zamkovoy connection on such submanifolds. An Einstein soliton (𝑔, 𝑉, 𝜆) is defined by [4] ℒ𝑣𝑔 + 2𝑆 + (2𝜆 -𝑟) = 0 (6.2)

A manifold admits an Einstein soliton w.r.t the Zamkovoy connection, gives:

From these findings, depending on the scalar, the soliton is: Expanding if 𝑟 > -(3𝑛 -8)(𝑛 -1), Steady if 𝑟 = -(3𝑛 -8)(𝑛 -1) and Shrinking if 𝑟 < -(3𝑛 -8)(𝑛 -1) These results are enriched by the curvature investigations of Mburu et al. [6-9], that describes specialized W-curvature tensors on these manifolds.

Responsible AI Perspective: Structural Geometry as a Vocabulary for Trustworthy Transformations

The results developed above are geometric in their own right, but they also carry a suggestive analogy for Responsible AI, understood here as the interdisciplinary effort to make AI systems' internal transformations traceable, consistent, and auditable. aThree structural features of the Zamkovoy canonical Paracontact connection on an LP-Kenmotsu manifold are of particular interest from this vantage point. First, metric compatibility (Section 3) means the connection preserves the geometry's own notion of length and angle as it transports structure across the manifold. In a Responsible AI setting, this is analogous to requiring that a model update, a retraining step, or a risk-reclassification procedure preserve the relationships a governance framework has already certified as meaningful, rather than silently distorting them. Second, the explicit torsion tensor computed in Section 4 gives a quantitative handle on exactly how, and by how much, a transformation departs from being symmetric; a vanishing or bounded torsion term is the geometric counterpart of an audit trail showing that a system update introduces no unaccounted-for asymmetry. Third, the classification of Einstein solitons in Section 6 into expanding, steady, and shrinking regimes offers a formal vocabulary for describing whether a system governed by such a connection is drifting away from, holding at, or converging toward a reference equilibrium over time, a question directly analogous to monitoring whether an AI system's risk profile is stabilizing under continued operation. These are offered as a structural analogy rather than a claim that LP-Kenmotsu geometry directly models any deployed AI system; the connection between differential geometry and Responsible AI governance frameworks, such as those surveyed by Jobin et al. [10] , remains at the level of shared formal concerns: compatibility with a reference structure, quantifiable and bounded distortion, and well-characterized long-run behavior. Making this analogy precise, for instance by formalizing a risk-classification manifold on which an explicit connection of this kind acts, is left as a direction for future interdisciplinary work between differential geometry and Responsible AI research.

References

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