An Algebraic Inequality with Applications to Certain Chen Inequalities

Ion Mihai,Radu-Ioan Mihai

doi:10.3390/axioms10010007

Abstract

We give a simple proof of the Chen inequality for the Chen invariant δ(2,…,2)︸k terms of submanifolds in Riemannian space forms.

Highlights

In [1,2], B.-Y
Chen introduced a string of Riemannian invariants, known as Chen invariants, which are different in nature from the classical Riemannian invariants
As an application we shall give a simple proof of the Chen inequality for the invariant δk(2, . . . , 2)

Summary

Introduction

In [1,2], B.-Y. Chen introduced a string of Riemannian invariants, known as Chen invariants, which are different in nature from the classical Riemannian invariants. For any p ∈ M and π ⊂ Tp M a plane section, the sectional curvature K(π) of π is defined by K(π) = R(e1, e2, e1, e2), where we use the convention R(e1, e2, e1, e2) = g(R(e1, e2)e2, e1), with {e1, e2} an orthonormal basis of π. The Chen first invariant δM is defined by δM(p) = τ(p) − inf{K(π)|π ⊂ Tp M plane section}. The Chen invariant δ(2, 2), given by δ(2, 2)(p) = τ(p) − inf{K(π1) + K(π2)|π1, π2 ⊂ Tp M orthogonal plane sections}, was studied in [3]. We shall prove an algebraic inequality and study its equality case. As an application we shall give a simple proof of the Chen inequality for the invariant δk(2, .

The equality holds if and only if

The Gauss equation also implies