Recent development in biconservative submanifolds

Abstract

A submanifold $\varphi :M\to {\mathbb{E}}^m$ is called {\it biharmonic} if it satisfies ${\Delta }^2\varphi =0$ 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 $\phi$ are characterized by vanishing of bitension ${\tau }_2$ of $\phi$. 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 ${\mathbb{E}}^m$ were derived from biharmonic submanifolds by only requiring the vanishing of the tangential component of ${\Delta }^2\varphi$. In 2014, R. Caddeo et. al. named a submanifold $M$ in any Riemannian manifold ``biconservative'' if the stress-energy tensor ${\hat{S}}_2$ of bienergy satisfies $\mathrm{div}{\hat{S}}_2=0$. Caddeo et. al. also shown that a Euclidean submanifolds is an $H$-submanifold if and only if the tangential component of ${\tau }_2$ 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.

49 pages