Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A survey on the DDVV-type inequalities

Jianquan Ge, Fagui Li, Zizhou Tang, Yi Zhou

Abstract

In this paper, we give a survey on the history and recent developments on the DDVV-type inequalities.

A survey on the DDVV-type inequalities

Abstract

In this paper, we give a survey on the history and recent developments on the DDVV-type inequalities.
Paper Structure (13 sections, 21 theorems, 51 equations, 3 tables)

This paper contains 13 sections, 21 theorems, 51 equations, 3 tables.

Table of Contents

  1. Introduction of DDVV-type inequality
  2. The origin and applications in geometry
  3. DDVV inequality in submanifold geometry
  4. Simons-type integral inequality in Riemannian submersion geometry
  5. New progress on DDVV-type inequalities
  6. DDVV-type inequality for complex matrices
  7. DDVV-type inequality for quaternionic matrices
  8. DDVV-type inequality for Clifford system and Clifford algebra
  9. Lu inequality
  10. Open questions
  11. Questions about DDVV-type inequalities
  12. Generalized Peng-Terng 2nd-Gap.
  13. Lu-Wenzel Conjectures

Key Result

Theorem 1.2

Let $B_1,\cdots,B_m$ be arbitrary $n\times n$ real symmetric matrices $(m,n\geq2)$. Then The equality holds if and only if there exists a $(P,R)\in K(n, m)$ such that where for some $\lambda\geq0$,

Theorems & Definitions (36)

  • Conjecture 1.1: DDVV Conjecture DDVV99
  • Theorem 1.2: DDVV inequality GT08Lu11
  • Conjecture 1.3: BW conjecture BW05
  • Theorem 1.4: BW inequality LW17
  • Theorem 2.1: Ge-TangGT08, LuLu11
  • Theorem 2.2: Chern-do Carmo-KobayashiCdK, SimonsSimons
  • Theorem 2.3: Chern-do Carmo-KobayashiCdK, LawsonLaw69
  • Theorem 2.4: Chen-XuCX93, Li-LiLL92
  • Theorem 2.5: LuLu11
  • Theorem 2.6: GeGe14
  • ...and 26 more