The Bochner curvature tensor is the Kähler analog of the Weyl curvature tensor in the curvature decomposition of a Kähler, discovered by Bochner in 1949. The article on Bochner-Kähler metrics by Robert Bryant is a pleasant read (see below). A Kähler metric with vanishing Bochner tensor is said to be Bochner-Kähler, and compact Bochner-Kähler manifolds are compact quotients of symmetric Bochner-Kähler manifolds; symmetric Bochner-Kähler manifolds are products of space forms $M_c^p \times M_{-c}^{n-p}$ (where $M_c^p$ denotes the complex space form of constant holomorphic sectional curvature $c$ and dimension $p$).
A natural question to ask is the following:
Are the Kähler manifolds whose Bochner tensor has constant norm, classified? Are many examples known?
Robert Bryant, Bochner-Kähler metrics, Journal of the American Mathematical Society, Jul. 2001, vol. 14, no. 3, pp. 623-715.
