View source for Reducible Riemannian space

A [[Riemannian space|Riemannian space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080420/r0804201.png" /> whose linear (or, in other words, homogeneous) [[Holonomy group|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080420/r0804202.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080420/r0804203.png" /> and complete simply-connected irreducible Riemannian spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080420/r0804204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080420/r0804205.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080420/r0804206.png" /> 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 [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Lichnerowicz,   "Global theory of connections and holonomy groups" , Noordhoff  (1976)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''1''' , Interscience  (1963)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Wu,   "On the de Rham decomposition theorem"  ''Illinois J. Math.'' , '''8''' :  2  (1964)  pp. 291–311</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR></table>



====Comments====
For de Rham's paper see [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. de Rham,   "Sur la réductibilité d'un espace de Riemann"  ''Comm. Math. Helvetica'' , '''26'''  (1952)  pp. 328–344</TD></TR></table>