A holonomy invariant anisotropic surface energy in a Riemannian manifold

In this paper, we investigate a holonomy invariant elliptic anisotropic surface energy for hypersurfaces in a complete Riemannian manifold, where “holonomy invariant” means that the elliptic parametric Lagrangian (i.e., a Finsler metric) of the Riemannian manifold used to define the anisotropic surface energy is constant along each holonomy subbundle of the tangent bundle of the Riemannian manifold. First we obtain the first variational formula for this anisotropic surface energy. Next we shall introduce the notions of an anisotropic convex hypersurface, an anisotropic equifocal hypersurface and an anisotropic isoparametric hypersurface for this anisotropic surface energy. Also, we shall introduce the notion of an anisotropic tube for this anisotropic surface energy. We prove that anisotropic tubes over a one-point set in a symmetric space are anisotropic convex hypersurfaces and that anisotropic tubes over a certain kind of reflective submanifold in a symmetric space are anisotropic isoparametric and anisotropic equifocal hypersurfaces.