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

TL;DR

This paper introduces Normal-Yang-Mills and Tangent-Yang-Mills submanifolds as critical points of specific curvature functionals in spheres, providing explicit equations and constructing numerous examples that often do not satisfy classical Yang-Mills conditions.

Contribution

It defines new classes of submanifolds based on variational problems of curvature tensors and constructs many non-trivial examples, expanding the understanding of Yang-Mills related geometries.

Findings

Explicit Euler-Lagrange equations for Normal-Yang-Mills and Tangent-Yang-Mills submanifolds.

Construction of infinitely many non-trivial examples from focal submanifolds.

Examples generally do not satisfy classical Yang-Mills equations.

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.