Generalised Surfaces in ${\Bbb{R}}^3$

TL;DR

This paper explores the relationship between line families in three-dimensional space and complex surfaces, extending classical results to more general surfaces through geometric and complex analysis techniques.

Contribution

It introduces a generalized framework linking line congruences in  to complex surfaces, broadening the understanding of surface geometry beyond convex cases.

Findings

Established correspondence between line families and complex surfaces.

Proved theorems generalizing convex surface results.

Analyzed properties of congruences from global sections of tangent bundles.

Abstract

The correspondence between 2-parameter families of oriented lines in ${\Bbb{R}}^3$ and surfaces in $T{\Bbb{P}}^1$ is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences generated by global sections of $T{\Bbb{P}}^1$ are investigated and a number of theorems are proven that generalise results for closed convex surfaces in ${\Bbb{R}}^3$.