The Fenchel-type inequality in the 3-dimensional Lorentz space and a Crofton formula

TL;DR

This paper extends classical geometric inequalities to spacelike curves in 3D Lorentz space, establishing bounds on total curvature and deriving a Crofton formula on de Sitter space, thus broadening the scope of Lorentzian geometry.

Contribution

It generalizes the Fenchel and Fary-Milnor theorems to Lorentzian settings and introduces a Crofton formula on de Sitter space, connecting curvature bounds with integral geometry.

Findings

Total curvature of certain Lorentzian curves is bounded by 2π.

Established a Crofton formula on de Sitter 2-sphere.

Generalized classical theorems to Lorentz space context.

Abstract

We generalize the Fenchel theorem to strong spacelike (which means that the tangent vector and the curvature vector span a spacelike 2-plane at each point) closed curves with index 1 in the 3-dimensional Lorentz space, showing that the total curvatures must be less than or equal to $2\pi$. A similar generalization of the Fary-Milnor theorem is also obtained. We establish the Crofton formula on the de Sitter 2-sphere which implies the above results.