I am currently trying to read about Weyl tensors, Ricci curvature and Einstein manifolds, specifically on four-manifolds where there is a decomposition of the space of two-forms in a self-dual and anti-self-dual subspaces. I cannot seem to find any books at introductory level for these topics. I would appreciate any books, papers or lecture notes suggestions.
2026-03-25 06:34:59.1774420499
Books on the Riemannian geometry of four-manifolds
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I don't know if it would be viewed as an introductory level book, but the material you mentioned is discussed in Besse's Einstein Manifolds, chapter $1$, sections $G$ and $H$.
Also, you might be interested in chapter $6$, section $D$ which covers some necessary conditions for the existence of Einstein metrics on closed four-manifolds, namely the Hitchin-Thorpe inequality and Gromov's inequality.
Another excellent resource is LeBrun's paper Four-Dimensional Einstein Manifolds, and Beyond.