Integrable G2 manifolds

In the first part of this thesis consisting of three parts, basic definitions andtheorems necessary for the rest of the work are presented.In the second part, the Lie group G2 is defined as the automorphism groupof the octonion algebra together with two equivalent characterizations. Then theclassification of Riemannian manifolds with structure group G2 by means of somelow dimensional representations of G2 is stated.In the third and last part, the conditions for the existence of metric covariantderivatives on the tangent bundle of a Riemannian manifold are dealt with. Study-ing the existence of covariant derivatives is equivalent to studying the existence ofconnection 1-forms on a special subbundle of the frame bundle of that manifold.In particular, the work done for an arbitrary Riemannian manifold is repeated fora 7-dimensional Riemannian manifold with the fundamental 3-form on its tangentbundle. It is shown that a G2-manifold has a covariant derivative compatible withthe fundamental 3-form if and only if the manifold is an integrable G2-manifold. Inthis case, the covariant derivative is unique.