Recent development in biconservative submanifolds

TL;DR

This paper surveys recent advances in the theory of biconservative submanifolds, focusing on their properties, classifications, and the connection to biharmonic maps, highlighting developments mainly from the last decade.

Contribution

It provides a comprehensive overview of recent research on biconservative submanifolds, summarizing key results and open problems in the field.

Findings

Biconservative submanifolds are characterized by the divergence-free stress-energy tensor of bienergy.

The notions of H-submanifolds and biconservative submanifolds coincide in Euclidean spaces.

Significant progress has been made in classifying and understanding biconservative hypersurfaces.

Abstract

A submanifold $\phi:M\to \mathbb E^{m}$ is called {\it biharmonic} if it satisfies $\Delta^{2}\phi=0$ 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 $\varphi$ are characterized by vanishing of bitension $\tau_{2}$ of $\varphi$. 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 $\Delta^{2}\phi$. 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 ${\rm div}\, \hat S_{2}=0$. 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.