Berger's list is as follows:  Manifolds with holonomy Sp("n")·Sp(1) were simultaneously studied in 1965 by Edmond Bonan and Vivian Yoh Kraines and they constructed the parallel 4-form. Manifolds with holonomy G2 or Spin(7) were firstly introduced by Edmond Bonan in 1966, who constructed all the parallel forms and showed that those manifolds were Ricci-flat. (Berger's original list also included the possibility of Spin(9) as a subgroup of SO(16). Riemannian manifolds with such holonomy were later shown independently by D. Alekseevski and Brown-Gray to be necessarily locally symmetric, i.e., locally isometric to the Cayley plane F4/Spin(9) or locally flat. See below.) It is now known that all of these possibilities occur as holonomy groups of Riemannian manifolds. The last two exceptional cases were the most difficult to find. See G2 manifold and Spin(7) manifold. Note that Sp("n") ⊂ SU(2"n") ⊂ U(2"n") ⊂ SO(4"n"), so every hyperkähler manifold is a Calabi–Yau manifold, every Calabi–Yau manifold is a Kähler manifold, and every Kähler manifold is orientable. The strange list above was explained by Simons's proof of Berger's theorem. A simple and geometric proof of Berger's theorem was given by Carlos E. Olmos in 2005. One first shows that if a Riemannian manifold is "not" a locally symmetric space and the reduced holonomy acts irreducibly on the tangent space, then it acts transitively on the unit sphere. The Lie groups acting transitively on spheres are known: they consist of the list above, together with 2 extra cases: the group Spin(9) acting on R16, and the group "T" · Sp("m") acting on R4"m". Finally one checks that the first of these two extra cases only occurs as a holonomy group for locally symmetric spaces (that are locally isomorphic to the Cayley projective plane), and the second does not occur at all as a holonomy group. Berger's original classification also included non-positive-definite pseudo-Riemannian metric non-locally symmetric holonomy.