The submanifold compatibility equations in magnetic geometry

TL;DR

The paper introduces a magnetic generalization of submanifold theory by equipping a Riemannian manifold with a closed 2-form $\sigma$ and a Lorentz force $Y$ defined by $g(Y(v),w)=\sigma(v,w)$. It then defines the $s$-magnetic curvature operator $\mathbf{M}^{g,\sigma}_s = R^{g,\sigma}_s + A^{g,\sigma}$ on the orthogonal bundle over the unit sphere bundle, and uses this to formulate magnetic analogues of the Gauss, Codazzi–Mainardi, and Ricci equations for submanifolds with induced magnetic data. The main result, Theorem mag-comp, provides explicit relations between the ambient magnetic curvature on $M$ and the induced magnetic curvature on a submanifold $N$, via the magnetic second fundamental form $\mathrm{II}^{\sigma}_s$ and magnetic shape operator $S^{\sigma}_s$, yielding magnetic Gauss and Ricci-type curvature formulas for hypersurfaces as corollaries. The work also situates these results within the context of Killing magnetic systems on spheres, including a computation of the Mañé critical value on $\mathbb{S}^3$, highlighting the dynamical implications of the magnetic submanifold framework and its potential for further geometric and dynamical applications.

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.

Figures (2)

• Figure 1: The relation $E^N_{(x,v)}= E^M_{(x,v)}\cap T_xN$, for $(x,v)\in SN$.
• Figure 2: The graph of the function $f\colon (0,\infty)\to {\mathbb{R}}$ given by $f(s) = ({\rm Ric}^{\mathtt{g}^\circ,\sigma^{\rm i}}_s)(x,\boldsymbol{E}_{\rm i}(x)) = 2s^2-2s+1/2 = 2(s-1/2)^2$, satisfying the optimal bound ${\rm Ric}^{\mathtt{g}^\circ,\sigma^{\rm i}}_s \geq f(s)$ for every $s>0$.

Theorems & Definitions (15)

• Theorem A
• Corollary B
• Corollary C
• Definition 4.1
• Remark 4.2
• Example 4.3
• Proposition 5.1
• proof
• Lemma 5.2
• proof