Difference between revisions of "Reducible Riemannian space"
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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 | + | {{TEX|auto}} |
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+ | A [[Riemannian space|Riemannian space]] $ M $ | ||
+ | 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} $ | ||
+ | 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 [[#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> | ||
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====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 |
Reducible Riemannian space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reducible_Riemannian_space&oldid=12217