Editing Holonomy (section)

===Reducible holonomy and the de Rham decomposition===
Let ''x'' ∈ ''M'' be an arbitrary point.  Then the holonomy group Hol(''M'') acts on the tangent space T<sub>x</sub>''M''.  This action may either be irreducible as a group representation, or reducible in the sense that there is a splitting of T<sub>x</sub>''M'' into orthogonal subspaces T<sub>x</sub>''M'' = T′<sub>x</sub>''M'' ⊕ T″<sub>x</sub>''M'', each of which is invariant under the action of Hol(''M'').  In the latter case, ''M'' is said to be '''reducible'''.

Suppose that ''M'' is a reducible manifold.  Allowing the point ''x'' to vary, the bundles T′''M'' and T″''M'' formed by the reduction of the tangent space at each point are smooth distributions which are [[Frobenius integration theorem|integrable in the sense of Frobenius]].  The [[integral manifold]]s of these distributions are totally geodesic submanifolds.  So ''M'' is locally a Cartesian product ''M′'' × ''M″''.  The (local) de Rham isomorphism follows by continuing this process until a complete reduction of the tangent space is achieved:<ref>{{harvnb|Kobayashi|Nomizu|1963|loc=§IV.5}}</ref>

: Let ''M'' be a [[simply connected]] Riemannian manifold,<ref>This theorem generalizes to non-simply connected manifolds, but the statement is more complicated.</ref> and T''M'' = T<sup>(0)</sup>''M'' ⊕ T<sup>(1)</sup>''M'' ⊕ ⋯ ⊕ T<sup>(''k'')</sup>''M'' be the complete reduction of the tangent bundle under the action of the holonomy group.  Suppose that T<sup>(0)</sup>''M'' consists of vectors invariant under the holonomy group (i.e., such that the holonomy representation is trivial).  Then locally ''M'' is isometric to a product
:: <math>V_0\times V_1\times \cdots\times V_k,</math>
: where ''V''<sub>0</sub> is an open set in a [[Euclidean space]], and each ''V<sub>i</sub>'' is an integral manifold for T<sup>(''i'')</sup>''M''.  Furthermore, Hol(''M'') splits as a direct product of the holonomy groups of each ''M<sub>i</sub>'', the maximal integral manifold of T<sup>(''i'')</sup> through a point.

If, moreover, ''M'' is assumed to be [[geodesically complete]], then the theorem holds globally, and each ''M<sub>i</sub>'' is a geodesically complete manifold.<ref>{{harvnb|Kobayashi|Nomizu|1963|loc=§IV.6}}</ref>