[Paper Review] Geometry of world sheets in Lorentz-Minkowski space

Shyūichi Izumiya|arXiv (Cornell University)|Nov 26, 2014

Advanced Differential Geometry Research9 references3 citations

TL;DR

This paper develops a differential geometric framework for world sheets—timelike submanifolds formed as one-parameter families of spacelike submanifolds—in Lorentz-Minkowski space using the theory of big wave fronts and Legendrian singularities. It introduces lightcone Gauss maps, height functions, and pedal maps, and establishes that the unfolded lightcone pedal map forms a big wave front, enabling classification of singularities via $s$-$P$-Legendrian equivalence and $P$-$\mathcal{K}$-equivalence of associated functions.

ABSTRACT

A world sheet in Lorentz-Minkowski space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds in Lorentz-Minkowski space. In this paper we investigate differential geometry of world sheets in Lorentz-Minkowski space as an application of the theory of big wave fronts.

Motivation & Objective

• To develop a differential geometric theory of world sheets in Lorentz-Minkowski space as one-parameter families of spacelike submanifolds.
• To analyze the contact of world sheets with lightlike hyperplanes and lightcones using singularity theory.
• To introduce and study the lightcone Gauss map, lightcone height functions, and unfolded lightcone pedal maps as geometric invariants.
• To establish a correspondence between geometric singularities of world sheets and $s$-$P$-Legendrian equivalence of associated big wave fronts.
• To provide a classification of singularities on world sheets via $P$-$\mathcal{K}$-equivalence of extended height functions.

Proposed method

• Uses Lorentz-Minkowski space $\mathbb{R}^{n+1}_1$ with pseudo-Riemannian metric to model spacetime geometry.
• Defines world sheets as timelike submanifolds parameterized by time, with each slice a spacelike submanifold.
• Introduces the lightcone Gauss map and its curvature via contact with lightlike hyperplanes.
• Constructs the lightcone height functions family and its extension to define the unfolded lightcone pedal map.
• Applies the theory of big wave fronts and Legendrian singularities to analyze the singularities of the pedal map.
• Establishes equivalence of singularities via $s$-$P$-Legendrian stability and $P$-$\mathcal{K}$-equivalence of generating functions.

Experimental results

Research questions

• RQ1 How can the geometry of world sheets in Lorentz-Minkowski space be described using contact with lightlike hyperplanes and lightcones?
• RQ2 What is the geometric meaning of singular points in the lightcone pedal map of a world sheet?
• RQ3 How do the singularities of the extended lightcone height function relate to the intrinsic geometry of the world sheet?
• RQ4 Under what conditions are two world sheet germs geometrically equivalent in terms of their singularity types?
• RQ5 Can the unfolded lightcone pedal map be interpreted as a big wave front, and what does this imply for singularity classification?

Key findings

• The image of the unfolded lightcone pedal map is a big wave front of a certain big Legendrian submanifold in the projectivized cotangent bundle of $LC^* \times I$.
• The singularities of the unfolded lightcone pedal map are classified via $s$-$P$-Legendrian equivalence of the associated big Legendrian submanifolds.
• The $P$-$\mathcal{K}$-equivalence of the extended lightcone height functions corresponds to the $s$-$P$-Legendrian equivalence of the big wave fronts.
• The $s$-$P$-Legendrian stability of the big wave front is generic for $n \leq 5$, ensuring robustness of the singularity classification.
• The geometric invariants derived from the lightcone Gauss map and pedal map provide a complete classification of singularities under $P$-$\mathcal{K}$-equivalence.
• The key equivalence theorem (Theorem 6.8) establishes that five conditions—$s$-$P$-Legendrian equivalence, $P$-$\mathcal{K}$-equivalence of height functions, $s$-$P$-$\mathcal{K}$-equivalence of families, $s$-$P$-diffeomorphism of images, and equality of curvature invariants—are all equivalent under $s$-$P$-Legendrian stability.