2005/07/14 by Jemal Guven
Engineering · Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Curvature #Elasticity and Material Modeling #Geometric Analysis and Curvature Flows #Geometry #Invariant (physics) #Invariant polynomial #Mathematical analysis #Mathematical physics #Mathematics #Matrix polynomial #Physics #Polynomial #Pure mathematics #Quadratic equation #Quadratic function #Quartic function #Riemann curvature tensor #cond-mat.soft #math-ph #math.MP
paper · pdf · doi:10.1088/0305-4470/38/37/002
16 pages
arxiv created 2005/07/14 · openalex publication_date 2005/08/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The most general conformally invariant bending energy of a closed four-dimensional surface, polynomial in the extrinsic curvature and its derivatives, is constructed. This invariance manifests itself as a set of constraints on the corresponding stress tensor. If the topology is fixed, there are three independent polynomial invariants: two of these are the straightforward quartic analogues of the quadratic Willmore energy for a two-dimensional surface; one is intrinsic (the Weyl invariant), the other extrinsic; the third invariant involves a sum of a quadratic in gradients of the extrinsic curvature—which is not itself invariant—and a quartic in the curvature. The four-dimensional energy quadratic in extrinsic curvature plays a central role in this construction.