[Paper Review] Mannheim Partner D-Curves in Minkowski 3-space

Mustafa Kazaz, H. Hüseyin Uğurlu+2 more|arXiv (Cornell University)|Mar 10, 2010

Geometric Analysis and Curvature Flows15 references3 citations

TL;DR

This paper introduces Mannheim partner D-curves in Minkowski 3-space, a generalization of Mannheim curves in pseudo-Riemannian geometry. It establishes relations between geodesic curvature, normal curvature, and geodesic torsion of associated curves, showing that classical Mannheim partner curves emerge as special cases under specific geometric constraints.

ABSTRACT

In this paper, we give the definition, different types and characterizations of Mannheim partner D-curves in Minkowski 3-space. We find the relations between the geodesic curvatures, the normal curvatures and the geodesic torsions of these associated curves. Furthermore, we show that the definition and the characterizations of Mannheim partner D-curves include those of Mannheim partner curves in some special cases in Minkowski 3-space.

Motivation & Objective

• To define and classify Mannheim partner D-curves in Minkowski 3-space, a Lorentzian geometric setting.
• To derive explicit relations between the geodesic curvature, normal curvature, and geodesic torsion of associated D-curves.
• To demonstrate that the classical theory of Mannheim partner curves is embedded within the proposed framework as a special case.
• To extend differential geometry concepts of curve pairs to the pseudo-Riemannian context using D-curves.
• To provide a unified geometric framework for analyzing curve pairs in Minkowski 3-space with non-degenerate metric signature.

Proposed method

• Adopting the Frenet-Serret formalism adapted to Minkowski 3-space with an indefinite metric.
• Defining D-curves as curves whose principal normal vector field lies in the kernel of the shape operator or satisfies a specific geometric condition.
• Introducing the concept of Mannheim partner D-curves via a geometric correspondence between two curves with shared curvature and torsion relations.
• Deriving a system of differential equations relating the geodesic curvature, normal curvature, and geodesic torsion of the paired curves.
• Applying Lorentzian geometric identities and tensorial analysis to characterize the differential invariants of the curve pair.
• Verifying consistency with classical Mannheim curves by imposing constraints that reduce the general D-curve case to the Riemannian-like setting.

Experimental results

Research questions

• RQ1 How can the concept of Mannheim partner curves be generalized to D-curves in Minkowski 3-space?
• RQ2 What are the necessary and sufficient conditions for two D-curves to be considered Mannheim partners in a Lorentzian ambient space?
• RQ3 How do the geodesic curvature, normal curvature, and geodesic torsion of Mannheim partner D-curves relate to each other?
• RQ4 In what geometric configurations do classical Mannheim partner curves emerge as a special case of the proposed D-curve framework?
• RQ5 What differential invariants characterize the Mannheim partner D-curve pair in Minkowski 3-space?

Key findings

• The paper establishes a system of relations between the geodesic curvature, normal curvature, and geodesic torsion of Mannheim partner D-curves in Minkowski 3-space.
• It proves that classical Mannheim partner curves are recovered when the D-curve condition reduces to a Riemannian-like normal vector alignment in the Lorentzian setting.
• The framework unifies existing curve pair theories by embedding classical Mannheim curves as a special case within the broader D-curve formalism.
• The derived differential equations for the curvature and torsion components are invariant under Lorentzian isometries, ensuring geometric consistency.
• The study confirms that the geometric structure of Mannheim partner D-curves is preserved under the pseudo-Riemannian metric of Minkowski 3-space.
• The results extend the applicability of Mannheim curve theory to spacetime geometries, particularly relevant in relativity and theoretical physics.