Normal-Yang-Mills and Tangent-Yang-Mills submanifolds

Abstract

This paper investigates the variational problems associated with the $L^2$-norms of the normal and tangent curvature tensors for submanifolds immersed in a unit sphere. We define the critical points of these functionals under normal variations as Normal-Yang-Mills and Tangent-Yang-Mills submanifolds, for which we explicitly establish the Euler-Lagrange equations in terms of the second fundamental form. Furthermore, by investigating the focal submanifolds of OT-FKM isoparametric hypersurfaces, we construct infinitely many non-trivial examples of both Normal-Yang-Mills and Tangent-Yang-Mills submanifolds. Notably, the curvature tensors of these examples generally do not satisfy the classical Yang-Mills equations.