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

Sergey Stepanov; Irina Tsyganok

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arXiv · 2505.17566

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

Sergey Stepanov·Irina Tsyganok

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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.

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Sergey Stepanov, Irina Tsyganok. 2025-05-23. Geometric Interpretations and Applications of the Berger-Ebin and York $L^2$-Orthogonal Decompositions. https://arxiv.org/abs/2505.17566

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