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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.
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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]]].
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A [[Riemannian space|Riemannian space]]  $  M $
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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  $  M _ {0} \times M _ {1} \times \dots \times M _ {k} $
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of a Euclidean space  $  M _ {0} $
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and complete simply-connected irreducible Riemannian spaces  $  M _ {i} $,
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$  i > 0 $;
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this decomposition is unique up to a rearrangement of factors.
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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 $
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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====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 08:10, 6 June 2020


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
How to Cite This Entry:
Reducible Riemannian space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reducible_Riemannian_space&oldid=12217
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article