Geometric Applications of the Berger–Ebin and York Orthogonal Decompositions of the Space of Symmetric Two-forms on a Compact Riemannian Manifold

Year 2025, Volume: 18 Issue: 2, 196 - 207, 19.10.2025

Sergey Stepanov , Irina Tsyganok

Abstract

The Berger–Ebin and York 𝑳𝟐-orthogonal decompositions of symmetric bilinear differential two-forms on a compact Riemannian manifold are fundamental tools in global Riemannian geometry. In the present paper, we consider geometric interpretations and applications of these decompositions. Namely, 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 𝑳𝟐-orthogonal decompositions. In addition, we explore applications of the York 𝑳𝟐-orthogonal decomposition to submanifold theory and use the Berger–Ebin 𝑳𝟐-orthogonal decomposition to study harmonic maps of compact Riemannian manifolds.

Keywords

Compact Riemannian manifold , vector space of differential two-forms , Berger–Ebin and York $L^{2}$-orthogonal decompositions

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