Moving frame and spin field representations of submanifolds in flat space

TL;DR

This paper introduces a spin field method compatible with the Cartan moving frame to describe submanifolds in flat space, linking connection and curvature, and providing explicit solutions and a generative technique for immersions.

Contribution

The paper develops a novel spin field approach using Killing spin equations to represent submanifolds, enabling explicit solutions and a new method to generate immersions.

Findings

Established a linear relation between connection and extrinsic curvature.

Provided explicit solutions to the Killing spin field equation.

Demonstrated a technique to generate new submanifold immersions.

Abstract

We introduce a spin field approach, that is compatible with the Cartan moving frame method, to describe the submanifold in a flat space. In fact, we consider a kind of spin field $\psi$, that satisfies a Killing spin field equation (analogous to a Killing spinor equation) written in terms of the Clifford algebra, and we use the spin field to locally rotate the orthonormal basis $\{\hat{e}_\mathtt{I}\}$. Then, the deformed orthonormal frame $\{\tilde{\psi}\hat{e}_\mathtt{I}\psi\}$ can be seen as the moving frame of a submanifold. We find some solutions to the Killing spin field equation and demonstrate an explicit example. Using the product of the spin fields, one can easily generate a new immersion submanifold, and this technique should be useful for studies in geometry and physics. Through the spin field, we find a linear relation between the connection and the extrinsic curvature of the submanifold. We propose a conjecture that any solution of the Killing spin field equation can be locally written as the product of the solutions we find.