for irreducible Einstein manifolds: only certain holonomies appear (SO(n), U(n), SU(n), Sp(n), Sp(n)Sp(1), G₂, Spin(7)).

Besse Einstein manifolds exhibit several interesting properties, including:

: The book provides an exhaustive survey of specialized structures that naturally satisfy the Einstein condition, including Kähler-Einstein manifolds , Quaternionic-Kähler manifolds , and homogeneous spaces.