On affine connections whose holonomy is a tensor representation (Q1355115)

Among the problems related to prescribed holonomy is that of finding the reductive Lie groups \(G\subseteq Gl(V)\) which can be irreducibly acting holonomies of torsion-free connections. The present paper deals with certain holonomy representations of reductive Lie groups, the semi-simple part of which is not simple, and which are tensor products of irreducible representations, i.e., roughly speaking, there exist Lie algebras \({\mathfrak g}_i\) acting irreducibly on \(V_i\) \((i=1,2)\) such that the holonomy representation is equivalent to the induced representation of \({\mathfrak g}_1 \oplus {\mathfrak g}_2\) on \(V= V_1 \otimes V_2\). Here it is also assumed that \(\dim V_i \geq 3\). As a consequence of the main result (which is a bit too long and technical to be stated here), the case under consideration does not contain any exotic example. It should be noted that the class of such representations, with \(\dim V_1 =2\), contains conformal 4-manifolds, quaternionic Kähler manifolds and Grassmannians (in the metric case), as well as paraconformal structures and exotic holonomies (in the non-metric case). [We refer the interested reader also to the relevant article by \textit{R. L. Bryant} [Sémin. Congr. 1, 93-165 (1996; Zbl 0882.53014)], reviewed above].