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Recent development in biconservative submanifolds

Bang-Yen Chen

Abstract

A submanifold $φ:M\to \mathbb E^{m}$ is called {\it biharmonic} if it satisfies $Δ^{2}φ=0$ 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 $τ_{2}$ 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 $Δ^{2}φ$. 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 $τ_{2}$ 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.

Recent development in biconservative submanifolds

Abstract

A submanifold is called {\it biharmonic} if it satisfies 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 are characterized by vanishing of bitension of . During last three decades there has been a growing interest in the theory of biharmonic submanifolds and biharmonic maps. The study of -submanifolds of were derived from biharmonic submanifolds by only requiring the vanishing of the tangential component of . In 2014, R. Caddeo et. al. named a submanifold in any Riemannian manifold ``biconservative'' if the stress-energy tensor of bienergy satisfies . Caddeo et. al. also shown that a Euclidean submanifolds is an -submanifold if and only if the tangential component of vanishes and hence the notions of -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.
Paper Structure (52 sections, 125 theorems, 98 equations)

This paper contains 52 sections, 125 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Stress-Energy tensor and biconservative submanifolds
  3. Basics of submanifolds
  4. Two basic formulas and biconservative equations
  5. Geometry of biconservative surfaces in $R^{3}(c)$
  6. Characterizations of biconservative surface
  7. Biconservative surfaces in $\mathbb E^{3}$
  8. Biconservative surfaces in $S^{3}$
  9. Biconservative surfaces in $H^{3}$
  10. Biconservative surfaces in $R^{3}(c)$
  11. Uniqueness of complete biconservative surfaces in $R^{3}(c)$
  12. Biconservative hypersurfaces in $\mathbb E^{n+1}$
  13. First result on biconservative hypersurfaces
  14. Biconservative hypersurfaces in $\mathbb E^{n+1}$
  15. $\delta(r)$-ideal biconservative hypersurfaces in $\mathbb E^{n+1}$
  16. ...and 37 more sections

Key Result

Theorem 4.1

beltramiIf $\phi :M \to \mathbb E^m_s$ is an isometric immersion of a pseudo-Riemannian $n$-manifold $M$ into a pseudo-Euclidean $m$-space $\mathbb E^m_s$, then we have

Theorems & Definitions (137)

  • Definition 2.1
  • Remark 3.1
  • Theorem 4.1
  • Corollary 4.2
  • Corollary 4.3
  • Theorem 4.4
  • Corollary 4.5
  • Theorem 4.6
  • Corollary 4.7
  • Proposition 4.8
  • ...and 127 more