2013/09/30 by Naoyuki Koike
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Anisotropy #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holonomy #Hypersurface #Invariant (physics) #Mathematical analysis #Mathematical physics #Mathematics #Physics #Quantum mechanics #Riemannian manifold #math.DG
paper · pdf · doi:10.1016/j.difgeo.2015.11.003
published as Differential Geometry and its Applications vol. 44 (2016) pp. 98-121 · 27pages
arxiv created 2014/07/26 · openalex publication_date 2015/11/21 · arxiv updated 2015/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.