Minimal Norm Tensors Principle and its Applications

TL;DR

This paper introduces the minimal norm tensors principle for third and fourth covariant tensors, providing new insights into Weyl and Cotten tensors and deriving useful inequalities based on tensor norms.

Contribution

It presents a novel approach to understanding Weyl and Cotten tensors as minimal norm tensors, offering new explanations and inequalities in Riemannian geometry.

Findings

Weyl tensor is the minimal norm tensor of the Riemannian curvature tensor.

Cotten tensor is the minimal norm tensor of the divergence of the Riemannian curvature tensor.

Derived inequalities related to the norms of these minimal tensors.

Abstract

In this paper we study the minimal norm tensors for general third covariant tensors and fourth covariant tensors, using this we can give a new explanation of Weyl tensor and Cotten tensor: Weyl tensor is the minimal norm tensor of Riemannian curvature tensor and Cotten tensor is the minimal norm tensor of divergence of Riemannian curvature tensor, and we also get some useful inequalities by computation the norm of minimal norm tensors.