The normalized $L^2$-norm of the traceless part of the Ricci curvature defines a Riemannian functional on the space of metrics. In this paper, we will consider this functional on 3-manifolds.
Minyo Katagiri. "On critical Riemannian metrics for a curvature functional on 3-manifolds." Proc. Japan Acad. Ser. A Math. Sci. 78 (4) 43 - 45, April 2002. https://doi.org/10.3792/pjaa.78.43