2024/01/06 by Bang‐Yen Chen, Chen, Bang-Yen · 2 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #31B30 #53-02 #53B25 #53C40 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Myofascial pain diagnosis and treatment
paper · pdf · doi:10.48550/arxiv.2401.03273
openalex publication_date 2024/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A submanifold ϕ:M→ \mathbb Em is called \it biharmonic if it satisfies Δ2ϕ=0 identically, according to the author. On the other hand, G.-Y. Jiang studied biharmonic maps between Riemannian manifolds as critical points of the bienergy functional, and proved that biharmonic maps φ are characterized by vanishing of bitension τ2 of φ. During last three decades there has been a growing interest in the theory of biharmonic submanifolds and biharmonic maps. The study of H-submanifolds of \mathbb Em were derived from biharmonic submanifolds by only requiring the vanishing of the tangential component of Δ2ϕ. In 2014, R. Caddeo et. al. named a submanifold M in any Riemannian manifold ``biconservative'' if the stress-energy tensor S2 of bienergy satisfies \rm div S2=0. Caddeo et. al. also shown that a Euclidean submanifolds is an H-submanifold if and only if the tangential component of τ2 vanishes and hence the notions of H-submanifolds and of biconservative submanifolds coincide for Euclidean submanifolds. The first results on biconservative hypersurfaces were proved by T. Hasanis and T. Vlachos, where they called such hypersurfaces \it H-hypersurfaces in 1995. Since then biconservative submanifolds has attracted many researchers and a lot of interesting results were obtained. The aim of this article is to provide a comprehensive survey on recent developments on biconservative submanifolds done most during the last decade.