An example of the geometry of a 5th-order ODE: the metric on the space of conics in ${\mathbb{CP}}^2$

TL;DR

This paper explores the geometric structure of the space of conics in complex projective plane via a 5th-order ODE, deriving its metric and connection to support twistor and Radon transform applications.

Contribution

It introduces a novel application of a method to explicitly compute the metric and connection on the space of conics in ${ m CP}^2$, linking differential equations with geometric structures.

Findings

Derived the metric on the space of conics in ${ m CP}^2$

Connected the geometric structure to twistor construction of Radon transform

Provided additional examples demonstrating the method

Abstract

As an application of the method of [4], we find the metric and connection on the space of conics in $\mathbb{CP}^2$ determined as the solution space of the ODE eqn(1). These calculations underpin the twistor construction of the Radon transform on conics in $\mathbb{CP}^2$ described in [5]. Two further examples of the method are provided.