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man manifold admitting G2 as weak holonomy group (There are no known examples of Riemannian manifolds with holonomy group G2 )

In the introduction of the mentioned article, among other things, it is moreover conjectured that a compact Riemannian manifold M^ with weak holonomy group G2 and positive sectional curvature must be a manifold of constant sectional curvature, therefore locally isometric to a sphere

The aim of this note is to show that not only this conjecture is true but indeed it is possible to characterize the Riemannian manifolds with weak lonomy group G2 and non zero Ricci curvature (that is with holonomy group not contained in G2) as manifolds isometric to manifolds with constant sectional curvature

The idea of the demonstration consists m considering the Ricci identity, which gives relations between the components of the curvature tensor and of Its second covariant derivative If one supposes that the holonomy group of the manifold is a certain group G, then the Ricci identity yields no new mation. However new information is obtained from the Ricci identity when one assumes that the manifold has weak holonomy group G

Finally we note that almost the same proof works in the case of a pseudo- Riemanman manifold with weak holonomy group G| where Gf is the pact form of G 2

2 . Preliminaries

We recall briefly^ some essential facts about the compact Lie group G2

Let V be an euclidean vector space of real dimension seven, with positive

defined scalar product <, >

A (2-fold) vector cross product P on F is a 2-fold vectorial form

P VxV->V

verifying moreover the following properties :

( 2 1) P{x, y) IS orthogonal to both vectors x, y

( 2 2) the norm of P{x,y) equals the norm of the bivector хлу, that is

\Р { х , уГ = \х\'\у\'-(х,уУ

In [2] It IS shown that, isometryless, such a vectorial product is unique, and one can build it in a convenient way starting from the product between the purely imaginary Cayley octomons Then, the group G2 can be characterized as the subgroup of SO (7) of the automorphisms of V which preserve the product P

A special subspace of V with respect to the action of G2 is any dimensional subspace closed under the product P We will indicate V/ the generic special subspace.

Each pair of linearly independent vectors x,yeV belongs to a well mined special subspace F/, the one generated by the triple x, y, P(x, y)

* Referring to the article of Gray [3] and others ш the bibliography, which we quote when needed, for the demonstrations of the results of this number and of the next