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The Weitzenböck formula and curvature decompositions play the key role in the classification of Einstein four-manifolds with positive curvature. In this talk we will provide a new proof of the Weitzenböck formula for Einstein metrics using Berger curvature decomposition, and establish a unified framework for the Weitzenböck formula for a large class of canonical metrics on four-manifolds, which are called generalized m-quasi-Einstein metrics (or "Einstein metrics" on smooth metric measure spaces, including gradient Ricci soliton, quasi-Einstein metrics, and conformally Einstein metrics). As applications we will discuss several rigidity results for four-dimensional Einstein manifolds, conformally Einstein manifolds, gradient shrinking Ricci solitons, and quasi-Eisntein manifolds.
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