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A space with a semi-Riemannian metric (with a degenerate metric tensor). A semi-Riemannian space is a generalization of the concept of a [[Riemannian space|Riemannian space]]. The definition of a semi-Riemannian space can be expressed in terms of the concepts used in the definition of a Riemannian space. In the definition of a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s0843101.png" /> one uses as tangent space the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s0843102.png" /> with a Euclidean metric, which is supposed to be invariant under parallel displacements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s0843103.png" /> (the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s0843104.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s0843105.png" /> is absolutely constant). If the tangent space at every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s0843106.png" /> is equipped with the structure of a [[Semi-Euclidean space|semi-Euclidean space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s0843107.png" />, then the metric of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s0843108.png" /> is degenerate, the metric tensor is also absolutely constant but is now degenerate, its matrix has rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s0843109.png" /> and has a non-singular submatrix. One defines a second degenerate metric tensor in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431010.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431011.png" />, which is called the zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431012.png" />-plane of the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431013.png" />; its matrix also possesses a non-singular submatrix, etc. The last, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431014.png" />-th metric tensor, defined in the zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431015.png" />-plane of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431016.png" />-st tensor, is a non-degenerate tensor with a non-singular matrix. Such a space is called a semi-Riemannian space, and in this case it is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431017.png" />. Analogously one defines semi-Riemannian spaces of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431018.png" />, that is, when the tangent space has the structure of a semi-pseudo-Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431019.png" />. The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431021.png" /> are called quasi-Riemannian spaces.
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As in a Riemannian space, one introduces the concept of curvature in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431022.png" />-dimensional direction. Semi-hyperbolic and semi-elliptic spaces are semi-Riemannian spaces of constant non-zero curvature, and a semi-Euclidean space is a semi-Riemannian space of constant curvature zero.
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A space with a semi-Riemannian metric (with a degenerate metric tensor). A semi-Riemannian space is a generalization of the concept of a [[Riemannian space|Riemannian space]]. The definition of a semi-Riemannian space can be expressed in terms of the concepts used in the definition of a Riemannian space. In the definition of a Riemannian space  $  V _ {n} $
 +
one uses as tangent space the space  $  \mathbf R  ^ {n} $
 +
with a Euclidean metric, which is supposed to be invariant under parallel displacements of  $  V _ {n} $(
 +
the metric tensor  $  a _ {ij} $
 +
of the space  $  V _ {n} $
 +
is absolutely constant). If the tangent space at every point of  $  V _ {n} $
 +
is equipped with the structure of a [[Semi-Euclidean space|semi-Euclidean space]]  $  R _ {n} ^ {m _ {1} \dots m _ {r - 1 }  } $,
 +
then the metric of the space  $  V _ {n} $
 +
is degenerate, the metric tensor is also absolutely constant but is now degenerate, its matrix has rank  $  m _ {1} $
 +
and has a non-singular submatrix. One defines a second degenerate metric tensor in the  $  ( n - m _ {1} ) $-
 +
plane  $  ( a _ {ij} x  ^ {j} = 0 ) $,
 +
which is called the zero  $  ( n - m _ {1} ) $-
 +
plane of the tensor  $  a _ {ij} $;
 +
its matrix also possesses a non-singular submatrix, etc. The last,  $  r $-
 +
th metric tensor, defined in the zero  $  ( n - m _ {r - 1 }  ) $-
 +
plane of the  $  ( r - 1) $-
 +
st tensor, is a non-degenerate tensor with a non-singular matrix. Such a space is called a semi-Riemannian space, and in this case it is denoted by the symbol  $  V _ {n} ^ {m _ {1} \dots m _ {r - 1 }  } $.
 +
Analogously one defines semi-Riemannian spaces of the form  $  {} ^ {l _ {1} \dots l _ {r} } V _ {n} ^ {m _ {1} \dots m _ {r - 1 }  } $,
 +
that is, when the tangent space has the structure of a semi-pseudo-Euclidean space  $  {} ^ {l _ {1} \dots l _ {r} } R _ {n} ^ {m _ {1} \dots m _ {r - 1 }  } $.
 +
The spaces  $  V _ {n}  ^ {m} $
 +
and  $  {}  ^ {kl} V _ {n}  ^ {m} $
 +
are called quasi-Riemannian spaces.
 +
 
 +
As in a Riemannian space, one introduces the concept of curvature in a $  2 $-
 +
dimensional direction. Semi-hyperbolic and semi-elliptic spaces are semi-Riemannian spaces of constant non-zero curvature, and a semi-Euclidean space is a semi-Riemannian space of constant curvature zero.
  
 
Thus, a semi-Riemannian space can be defined as a space of affine connection (without torsion) whose tangent spaces at every point are semi-Euclidean (or semi-pseudo-Euclidean), and where the metric tensor of the semi-Riemannian space is absolutely constant.
 
Thus, a semi-Riemannian space can be defined as a space of affine connection (without torsion) whose tangent spaces at every point are semi-Euclidean (or semi-pseudo-Euclidean), and where the metric tensor of the semi-Riemannian space is absolutely constant.
  
In a semi-Riemannian space, the differential geometry of lines and surfaces is constructed by analogy with the differential geometry of lines and surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431023.png" />, taking into account the special features of semi-Riemannian spaces indicated above. Surfaces of semi-hyperbolic and semi-elliptic spaces are themselves semi-Riemannian spaces. In particular, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431024.png" />-horosphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431025.png" /> in a semi-hyperbolic space is isometric to the semi-Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431026.png" />, the metric of which can be reduced to the metric of the semi-elliptic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084310/s08431027.png" />; this fact is a generalization of the isometry of a horosphere in Lobachevskii space to a Euclidean space.
+
In a semi-Riemannian space, the differential geometry of lines and surfaces is constructed by analogy with the differential geometry of lines and surfaces in $  V _ {n} $,  
 +
taking into account the special features of semi-Riemannian spaces indicated above. Surfaces of semi-hyperbolic and semi-elliptic spaces are themselves semi-Riemannian spaces. In particular, the $  m $-
 +
horosphere $  {} ^ {m+ 1 } {S _ {n} } $
 +
in a semi-hyperbolic space is isometric to the semi-Riemannian space $  V _ {n - 1 }  ^ {m, n - m - 1 } $,  
 +
the metric of which can be reduced to the metric of the semi-elliptic space $  S _ {n - m - 1 }  ^ {m} $;  
 +
this fact is a generalization of the isometry of a horosphere in Lobachevskii space to a Euclidean space.
  
 
====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>
 
 
  
 
====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><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. O'Neill,  "Semi-Riemannian geometry (with applications to relativity)" , Acad. Press  (1983)</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><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. O'Neill,  "Semi-Riemannian geometry (with applications to relativity)" , Acad. Press  (1983)</TD></TR></table>

Latest revision as of 08:13, 6 June 2020


A space with a semi-Riemannian metric (with a degenerate metric tensor). A semi-Riemannian space is a generalization of the concept of a Riemannian space. The definition of a semi-Riemannian space can be expressed in terms of the concepts used in the definition of a Riemannian space. In the definition of a Riemannian space $ V _ {n} $ one uses as tangent space the space $ \mathbf R ^ {n} $ with a Euclidean metric, which is supposed to be invariant under parallel displacements of $ V _ {n} $( the metric tensor $ a _ {ij} $ of the space $ V _ {n} $ is absolutely constant). If the tangent space at every point of $ V _ {n} $ is equipped with the structure of a semi-Euclidean space $ R _ {n} ^ {m _ {1} \dots m _ {r - 1 } } $, then the metric of the space $ V _ {n} $ is degenerate, the metric tensor is also absolutely constant but is now degenerate, its matrix has rank $ m _ {1} $ and has a non-singular submatrix. One defines a second degenerate metric tensor in the $ ( n - m _ {1} ) $- plane $ ( a _ {ij} x ^ {j} = 0 ) $, which is called the zero $ ( n - m _ {1} ) $- plane of the tensor $ a _ {ij} $; its matrix also possesses a non-singular submatrix, etc. The last, $ r $- th metric tensor, defined in the zero $ ( n - m _ {r - 1 } ) $- plane of the $ ( r - 1) $- st tensor, is a non-degenerate tensor with a non-singular matrix. Such a space is called a semi-Riemannian space, and in this case it is denoted by the symbol $ V _ {n} ^ {m _ {1} \dots m _ {r - 1 } } $. Analogously one defines semi-Riemannian spaces of the form $ {} ^ {l _ {1} \dots l _ {r} } V _ {n} ^ {m _ {1} \dots m _ {r - 1 } } $, that is, when the tangent space has the structure of a semi-pseudo-Euclidean space $ {} ^ {l _ {1} \dots l _ {r} } R _ {n} ^ {m _ {1} \dots m _ {r - 1 } } $. The spaces $ V _ {n} ^ {m} $ and $ {} ^ {kl} V _ {n} ^ {m} $ are called quasi-Riemannian spaces.

As in a Riemannian space, one introduces the concept of curvature in a $ 2 $- dimensional direction. Semi-hyperbolic and semi-elliptic spaces are semi-Riemannian spaces of constant non-zero curvature, and a semi-Euclidean space is a semi-Riemannian space of constant curvature zero.

Thus, a semi-Riemannian space can be defined as a space of affine connection (without torsion) whose tangent spaces at every point are semi-Euclidean (or semi-pseudo-Euclidean), and where the metric tensor of the semi-Riemannian space is absolutely constant.

In a semi-Riemannian space, the differential geometry of lines and surfaces is constructed by analogy with the differential geometry of lines and surfaces in $ V _ {n} $, taking into account the special features of semi-Riemannian spaces indicated above. Surfaces of semi-hyperbolic and semi-elliptic spaces are themselves semi-Riemannian spaces. In particular, the $ m $- horosphere $ {} ^ {m+ 1 } {S _ {n} } $ in a semi-hyperbolic space is isometric to the semi-Riemannian space $ V _ {n - 1 } ^ {m, n - m - 1 } $, the metric of which can be reduced to the metric of the semi-elliptic space $ S _ {n - m - 1 } ^ {m} $; this fact is a generalization of the isometry of a horosphere in Lobachevskii space to a Euclidean space.

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)
[a2] B. O'Neill, "Semi-Riemannian geometry (with applications to relativity)" , Acad. Press (1983)
How to Cite This Entry:
Semi-Riemannian space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-Riemannian_space&oldid=48653
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article