Difference between revisions of "Semi-pseudo-Riemannian space"
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− | + | A manifold with a degenerate indefinite metric. The semi-pseudo-Riemannian manifold $ {} ^ {l _ {1} \cdots l _ {r} } V _ {n} ^ {m _ {1} \cdots m _ {r - 1 } } $ | |
+ | is defined as an $ n $-dimensional manifold with coordinates $ x ^ {i} $ | ||
+ | in which there are given $ r $ | ||
+ | line elements | ||
+ | |||
+ | $$ | ||
+ | ds _ {a} ^ {2} = \ | ||
+ | \sum _ { i,j= 1} ^ { {m _ a} - m _ {a- 1} } g _ {( a) ij } \ | ||
+ | dx ^ {i + m _ {a- 1} } dx ^ {j + m _ {a- 1} } , | ||
+ | $$ | ||
+ | |||
+ | where $ 0 = m _ {0} < m _ {1} < \dots < m _ {r} = n $; | ||
+ | $ a = 1, \dots, r $; | ||
+ | and where the index of the quadratic form $ g _ {( a) ij } $ | ||
+ | is $ l _ {a} $. | ||
+ | The line element $ ds _ {a} ^ {2} $ | ||
+ | is defined for those vectors for which all components with indices smaller than $ m _ {a - 1 } + 1 $ | ||
+ | or larger than $ m _ {a} $ | ||
+ | vanish. If $ l _ {1} = l _ {2} = \dots = 0 $, | ||
+ | a semi-pseudo-Riemannian space is a [[Semi-Riemannian space|semi-Riemannian space]]. The spaces $ V _ {n} ^ {m} $ | ||
+ | and $ {} ^ {kl} {V _ {n} ^ {m} } $ | ||
+ | are quasi-Riemannian spaces. The basic concepts of differential geometry (for example, curvature) are defined in semi-pseudo-Riemannian spaces similarly to Riemannian spaces (see [[#References|[1]]]). | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)</TD></TR></table> | ||
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====Comments==== | ====Comments==== | ||
− | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)</TD></TR></table> |
Latest revision as of 23:47, 7 May 2022
A manifold with a degenerate indefinite metric. The semi-pseudo-Riemannian manifold $ {} ^ {l _ {1} \cdots l _ {r} } V _ {n} ^ {m _ {1} \cdots m _ {r - 1 } } $
is defined as an $ n $-dimensional manifold with coordinates $ x ^ {i} $
in which there are given $ r $
line elements
$$ ds _ {a} ^ {2} = \ \sum _ { i,j= 1} ^ { {m _ a} - m _ {a- 1} } g _ {( a) ij } \ dx ^ {i + m _ {a- 1} } dx ^ {j + m _ {a- 1} } , $$
where $ 0 = m _ {0} < m _ {1} < \dots < m _ {r} = n $; $ a = 1, \dots, r $; and where the index of the quadratic form $ g _ {( a) ij } $ is $ l _ {a} $. The line element $ ds _ {a} ^ {2} $ is defined for those vectors for which all components with indices smaller than $ m _ {a - 1 } + 1 $ or larger than $ m _ {a} $ vanish. If $ l _ {1} = l _ {2} = \dots = 0 $, a semi-pseudo-Riemannian space is a semi-Riemannian space. The spaces $ V _ {n} ^ {m} $ and $ {} ^ {kl} {V _ {n} ^ {m} } $ are quasi-Riemannian spaces. The basic concepts of differential geometry (for example, curvature) are defined in semi-pseudo-Riemannian spaces similarly to Riemannian spaces (see [1]).
References
[1] | B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian) |
Comments
References
[a1] | B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian) |
Semi-pseudo-Riemannian space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-pseudo-Riemannian_space&oldid=18344