# Characterizations of Normal and Binormal Surfaces in G3

## TL;DR

This paper characterizes certain surfaces in Galilean 3-space where specific geodesic-based constructions yield minimal surfaces and surfaces with constant negative Gaussian curvature, identifying them as planes or hyperboloids.

## Contribution

It introduces a novel characterization of normal and binormal surfaces in G3 linked to geodesic properties and minimal surface conditions.

## Key findings

- Surfaces with four geodesics through each point are either planes or circular hyperboloids.
- Constructed surfaces from normal and binormal lines exhibit minimal surface and negative Gaussian curvature properties.
- The key surface $ta$ must be isoparametric in G3.

## Abstract

In this paper, our aim is to give surfaces in the Galilean 3-space G3 with the property that there exist four geodesics through each point such that every surface built with the normal lines and the binormal lines along these geodesics is a surface with a minimal surface and a constant negative Gaussian curvature. We show that $\psi $ should be an isoparametric surface in G3: A plane or a circular hyperboloid.