Reducible Riemannian space

A Riemannian space $ M $ whose linear (or, in other words, homogeneous) holonomy group is reducible, i.e. has non-trivial invariant subspaces. A Riemannian space with an irreducible holonomy group is called irreducible. A complete simply-connected reducible Riemannian space is decomposable (de Rham's decomposition theorem), i.e. splits into a direct product of Riemannian spaces of positive dimension. Moreover, any complete simply-connected Riemannian space is isometric to the direct product $ M _ {0} \times M _ {1} \times \dots \times M _ {k} $ of a Euclidean space $ M _ {0} $ and complete simply-connected irreducible Riemannian spaces $ M _ {i} $, $ i > 0 $; this decomposition is unique up to a rearrangement of factors.

A weak version of this theorem holds for pseudo-Riemannian spaces: A pseudo-Riemannian space is called weakly irreducible if all non-trivial subspaces of the tangent space that are invariant with respect to the holonomy group $ \Gamma $ are isotropic, i.e. the scalar product induced on them is degenerate. Any complete simply-connected pseudo-Riemannian space splits into a direct product of weakly irreducible pseudo-Riemannian spaces. If the subspace of vectors that are fixed under the holonomy group is non-isotropic, this decomposition is unique up to a rearrangement of factors. A weakly irreducible pseudo-Riemannian space does not split necessarily into a direct product of pseudo-Riemannian spaces [3].

References

[1]	A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated from French)
[2]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)
[3]	H. Wu, "On the de Rham decomposition theorem" Illinois J. Math. , 8 : 2 (1964) pp. 291–311
[4]	Ya.L. Shapiro, "Reducible Riemannain spaces and two-sheeted structures on them" Soviet Math. Dokl. , 13 : 5 (1972) pp. 1345–1348 Dokl. Akad. Nauk SSSR , 206 : 4 (1972) pp. 831–833

Comments

For de Rham's paper see [a1].

References

[a1]	G. de Rham, "Sur la réductibilité d'un espace de Riemann" Comm. Math. Helvetica , 26 (1952) pp. 328–344