On the way to his classification of Riemannian holonomy groups, Berger developed two criteria that must be satisfied by the Lie algebra of the holonomy group of a torsion-free affine connection which is not locally symmetric : one of them, known as " Berger's first criterion ", is a consequence of the Ambrose Singer theorem, that the curvature generates the holonomy algebra; the other, known as " Berger's second criterion ", comes from the requirement that the connection should not be locally symmetric.
