# Umbilical and zero curvature equations in a class of second order ODE's

## TL;DR

This paper explores geometric structures of a specific class of cubic second order ODEs, defining subclasses based on curvature properties and analyzing their invariants to understand their geometric behavior.

## Contribution

It introduces a geometric framework for classifying cubic second order ODEs into umbilical, zero mean curvature, and zero Gaussian curvature subclasses, focusing on invariants.

## Key findings

- Identification of subclasses based on curvature properties
- Analysis of invariants for zero mean curvature equations
- Geometric classification of a specific ODE class

## Abstract

The class of second order ODE's cubic with respect to the first order derivative is considered. Using geometric structures associated with these equations, the subclasses of umbilical equations, zero mean curvature equations, and zero Gaussian curvature equations are defined. Zero mean curvature equations are studied within the framework of the first case of intermediate degeneration with the stress on their pseudoscalar and scalar invariants.