Two-numbers and their applications - A survey

TL;DR

This survey reviews the concept of two-numbers in Riemannian geometry, their mathematical significance, and various applications, highlighting open problems and future research directions.

Contribution

It provides a comprehensive overview of two-numbers, summarizing their development, applications, and unresolved questions in the field.

Findings

Two-numbers are closely related to important mathematical areas.

The survey identifies key open problems and conjectures.

Two-numbers have diverse applications in geometry and related fields.

Abstract

The notion of two-numbers of connected Riemannian manifolds was introduced about 35 years ago in [Un invariant geometrique riemannien, C. R. Acad. Sci. Paris Math. 295 (1982), 389--391] by B.-Y. Chen and T. Nagano. Later, two-numbers have been studied by a number of mathematicians and it was then proved that two-numbers related closely with several important areas in mathematics. The main purpose of this article is to survey on two-numbers and their applications. At the end of this survey, we present several open problems and conjectures on two-numbers.