The submanifold compatibility equations in magnetic geometry

TL;DR

This paper extends classical submanifold compatibility equations to magnetic geometry by introducing magnetic curvature and second fundamental form, establishing analogous Gauss, Ricci, and Codazzi-Mainardi equations.

Contribution

It introduces magnetic curvature and second fundamental form, and derives their compatibility equations, bridging classical submanifold theory with magnetic geometric settings.

Findings

Magnetic Gauss, Ricci, and Codazzi-Mainardi equations established.

Analogues of classical compatibility equations in magnetic geometry.

Framework for further study of submanifolds in magnetic spaces.

Abstract

With the notions of magnetic curvature and magnetic second fundamental form recently introduced by Assenza and Albers-Benedetti-Maier, respectively, we establish analogues of the Gauss, Ricci, and Codazzi-Mainardi compatibility equations from submanifold theory in the magnetic setting.