  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.
