Geometric Interpretations and Applications of the Berger-Ebin and York $L^2$-Orthogonal Decompositions

TL;DR

This paper explores the structure of Ricci tensors on compact Riemannian manifolds using Berger-Ebin and York $L^2$-orthogonal decompositions, with applications to Ricci almost solitons, submanifolds, and harmonic maps.

Contribution

It introduces new insights into Ricci tensor analysis on compact manifolds through orthogonal decompositions and applies these to various geometric structures.

Findings

Characterization of Ricci tensors on compact Ricci almost solitons

Applications of York $L^2$-decomposition to submanifold theory

Insights into harmonic map structures between Riemannian manifolds

Abstract

The Berger-Ebin and York $L^2$-orthogonal decompositions of the vector space of symmetric bilinear differential two-forms are fundamental tools in global Riemannian geometry. In this paper, we investigate the structure of Ricci tensors on compact Riemannian manifolds, with a particular focus on compact Ricci almost solitons, utilizing both the Berger-Ebin and York $L^2$-orthogonal decompositions. In addition, we explore applications of the York $L^2$-orthogonal decomposition to the theory of submanifolds and to the study of harmonic maps between Riemannian manifolds.