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It is proved that there exist metrics with G2 and Spin(7), thus settling the remaining cases in Berger's list of possible groups. We first reformulate the holonomy H condition as a set of differential equations for an associated H-structure on a given manifold. We collect the needed algebraic facts about G2 and Spin(7). We then apply the machinery of over-determined partial differential equations (in the form of the Cartan-Kahler theorem) to prove the existence of solutions whose is G2 or Spin(7). We also provide explicit examples and some information about the generality of the space of
Robert L. Bryant (Sun,) studied this question.
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