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A [[Centro-affine space|centro-affine space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b0166101.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b0166102.png" />), which may be assigned to each point of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b0166103.png" /> with an affine connection (in particular, to a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b0166104.png" />). Consider all tensors with even covariant and contravariant valencies at a point of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b0166105.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b0166106.png" />); the covariant and contravariant indices are subdivided into different pairs, for each one of which the tensor is skew-symmetric. Tensors with these two properties are called bitensors. If each skew-symmetric pair is regarded as a collective index, the number of new indices will be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b0166107.png" />. The simplest bitensor is the [[Bivector|bivector]]
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b0166108.png" /></td> </tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
If, at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b0166109.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661010.png" />,
+
A [[Centro-affine space|centro-affine space]]  $  E _ {N} $(
 +
where  $  N = n(n - 1)/2 $),
 +
which may be assigned to each point of a space  $  A _ {n} $
 +
with an affine connection (in particular, to a Riemannian space  $  V _ {n} $).
 +
Consider all tensors with even covariant and contravariant valencies at a point of the space  $  A _ {n} $(
 +
or  $  V _ {n} $);
 +
the covariant and contravariant indices are subdivided into different pairs, for each one of which the tensor is skew-symmetric. Tensors with these two properties are called bitensors. If each skew-symmetric pair is regarded as a collective index, the number of new indices will be  $  N = n(n - 1)/2 $.  
 +
The simplest bitensor is the [[Bivector|bivector]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661011.png" /></td> </tr></table>
+
$$
 +
v _ {\alpha \beta }  = - v _ {\beta \alpha }  \rightarrow  v _ {a} ,\ \
 +
\alpha , \beta =1 \dots n; \  a = 1 \dots N .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661012.png" /></td> </tr></table>
+
If, at a point  $  P $
 +
of  $  A _ {n} $,
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661013.png" />, and the set of bivectors assigned to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661014.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661015.png" />) at a given point defines a vector space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661016.png" /> such that the components satisfy the conditions
+
$$
 +
A _  \beta  ^  \alpha  = \
 +
\left (
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661017.png" /></td> </tr></table>
+
\frac{\partial  \overline{x}\; ^  \alpha  }{\partial  x  ^  \beta  }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661018.png" /></td> </tr></table>
+
\right ) _ {p} ,\ \
 +
A _ {b}  ^ {a}  = \
 +
2A _  \gamma  ^ {[ \alpha }
 +
A _  \delta  ^ { {}\beta ] }  = \
 +
A _ {[ \gamma{} }  ^ {[ \alpha{} }
 +
A _ { {}\delta ] }  ^ { {}\beta ] } ,
 +
$$
  
i.e. this set defines the centro-affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661019.png" />, called the bivector space at the given point. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661020.png" /> the bivector space may be metrized with the aid of the metric tensor
+
$$
 +
v  ^ {a}  = v ^ {\alpha \beta } ,\  v  ^ {b}  = v ^ {\gamma \delta } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661021.png" /></td> </tr></table>
+
then  $  \overline{v}\; ^ {a} = A _ {b}  ^ {a} v  ^ {b} $,
 +
and the set of bivectors assigned to  $  A _ {n} $(
 +
or  $  V _ {n} $)
 +
at a given point defines a vector space of dimension  $  N $
 +
such that the components satisfy the conditions
  
after which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661022.png" /> becomes a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661023.png" />.
+
$$
 +
\overline{v}\;  ^ {b}  = \
 +
A _ {a}  ^ {b} v  ^ {a} ,\ \
 +
v  ^ {a}  = \
 +
\overline{A}\; _ {b}  ^ {a}
 +
\overline{v}\;  ^ {b}
 +
$$
  
Bivector spaces are used in Riemannian geometry and in the general theory of relativity. The bivector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661024.png" /> is constructed at a given point of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661025.png" />, and different representations of the curvature tensor with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661028.png" /> and the second-valency bitensors with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661031.png" /> are associated, respectively. The study of the algebraic structure of the curvature tensor may then be reduced to the study of the pencil of quadratic forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661032.png" />, the second one of which is non-degenerate (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661033.png" />). The study of elementary divisors of this pair results in a classification of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661034.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661035.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661036.png" />) and if the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661037.png" /> has signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661038.png" />, then it can be shown that only three types of Einstein spaces exist.
+
$$
 +
| A _ {b}  ^ {a} |  \neq  0,\  A _ {b}  ^ {a} \overline{A}\; _ {c}  ^ {b= \delta _ {c}  ^ {a}
 +
$$
  
A bivector may be assigned to each rotation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661039.png" />; this means that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661040.png" /> there corresponds a vector, which is convenient for the study of infinitesimal transformations. Essentially, a bivector space is identical with a [[Biplanar space|biplanar space]] [[#References|[2]]].
+
i.e. this set defines the centro-affine space  $  E _ {N} $,
 +
called the bivector space at the given point. In  $  V _ {n} $
 +
the bivector space may be metrized with the aid of the metric tensor
 +
 
 +
$$
 +
g _ {ab}  = \
 +
g _ {\alpha \beta \gamma \delta }  = ^ { {roman }  def } \
 +
g _ {\alpha \gamma }  g _ {\beta \delta }  -
 +
g _ {\alpha \delta }  g _ {\beta \gamma }  ,
 +
$$
 +
 
 +
after which  $  E _ {N} $
 +
becomes a metric space  $  R _ {N} $.
 +
 
 +
Bivector spaces are used in Riemannian geometry and in the general theory of relativity. The bivector space  $  R _ {N} $
 +
is constructed at a given point of the space  $  V _ {n} $,
 +
and different representations of the curvature tensor with components  $  R _ {\alpha \beta \gamma \delta }  $,
 +
$  R _ {\gamma \delta }  ^ {\alpha \beta } $,
 +
$  R _ {\alpha \beta }  ^ {\gamma \delta } $
 +
and the second-valency bitensors with components  $  R _ {ab} $,
 +
$  R _ {b}  ^ {a} $,
 +
$  R _ {a}  ^ {b} $
 +
are associated, respectively. The study of the algebraic structure of the curvature tensor may then be reduced to the study of the pencil of quadratic forms  $  R _ {ab} - \lambda g _ {ab} $,
 +
the second one of which is non-degenerate ( $  | g _ {ab} | \neq 0 $).
 +
The study of elementary divisors of this pair results in a classification of the spaces  $  V _ {n} $.
 +
If  $  n = 4 $(
 +
$  N = 6 $)
 +
and if the form  $  g _ {\alpha \beta }  $
 +
has signature  $  (- - - +) $,
 +
then it can be shown that only three types of Einstein spaces exist.
 +
 
 +
A bivector may be assigned to each rotation in $  V _ {n} $;  
 +
this means that in $  R _ {N} $
 +
there corresponds a vector, which is convenient for the study of infinitesimal transformations. Essentially, a bivector space is identical with a [[Biplanar space|biplanar space]] [[#References|[2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.Z. Petrov, "New methods in general relativity theory" , Moscow (1966) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.P. Norden, "On complex representation of tensors of a biplanar space" , ''8'' ''Kazan. Gos. Univ. Uchen. Zap.'' , '''114''' (1954) pp. 45–53 (In Russian) {{MR|76400}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.Z. Petrov, "New methods in general relativity theory" , Moscow (1966) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.P. Norden, "On complex representation of tensors of a biplanar space" , ''8'' ''Kazan. Gos. Univ. Uchen. Zap.'' , '''114''' (1954) pp. 45–53 (In Russian) {{MR|76400}} {{ZBL|}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Consider a bivector as represented by an ordered pair of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661041.png" /> as in [[#References|[a1]]], [[#References|[a4]]] or the article [[Bivector|bivector]]. The Plücker coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661043.png" />, then constitute a bivector as in the article above. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661045.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661046.png" /> and similarly for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661047.png" />. Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661048.png" /> transform as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661049.png" />. Whence the formulas above. Here the square brackets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661050.png" /> are a notation signifying taking an alternating average. Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661051.png" />. If certain indices are to be singled out, i.e. exempt from this averaging process, this is indicated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661052.png" />. Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661053.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661054.png" />, cf. above. This is a notation introduced by R. Bach [[#References|[a5]]]. Cf. also [[Alternation|Alternation]].
+
Consider a bivector as represented by an ordered pair of vectors $  ( \mathbf u , \mathbf v ) $
 +
as in [[#References|[a1]]], [[#References|[a4]]] or the article [[Bivector|bivector]]. The Plücker coordinates of $  ( \mathbf u , \mathbf v ) $,  
 +
$  p  ^ {ij} = u  ^ {i} v  ^ {j} - u  ^ {j} v  ^ {i} $,  
 +
then constitute a bivector as in the article above. Let $  \overline{\mathbf u}\; = A \mathbf u $,  
 +
$  \overline{\mathbf v}\; = A \mathbf v $,  
 +
i.e. $  \overline{u}\;  ^ {i} = A _ {j}  ^ {i} u  ^ {j} $
 +
and similarly for $  \mathbf v $.  
 +
Then the $  p  ^ {ij} $
 +
transform as $  p  ^ {ij} = 2A _ {k}  ^ {[i} A _ {l}  ^ {j]} p  ^ {kl} $.  
 +
Whence the formulas above. Here the square brackets in $  A _ {k}  ^ {[i} A _ {l}  ^ {j]} $
 +
are a notation signifying taking an alternating average. Thus $  A _ {k}  ^ {[1} A _ {l}  ^ {2]} = (A _ {k}  ^ {1} A _ {l}  ^ {2} - A _ {k}  ^ {2} A _ {l}  ^ {1} )/2 $.  
 +
If certain indices are to be singled out, i.e. exempt from this averaging process, this is indicated by $  | {} | $.  
 +
Thus $  A _ {m} ^ {[ij  | k |  l] } = (A _ {m}  ^ {ijkl} - A _ {m}  ^ {jikl} - A _ {m}  ^ {ilkj} - A _ {m}  ^ {ljki} + A _ {m}  ^ {jlki} + A _ {m}  ^ {likj} )/6 $,  
 +
and $  g _ {ab} = 2g _ {\alpha [ \gamma{} }  g _ { {}| \beta |  \delta ] }  $,  
 +
cf. above. This is a notation introduced by R. Bach [[#References|[a5]]]. Cf. also [[Alternation|Alternation]].
  
In more modern terms what is described here is the bundle of bivectors over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016610/b01661055.png" />.
+
In more modern terms what is described here is the bundle of bivectors over $  A _ {n} $.
  
 
A centro-affine space is an affine space with a distinguished point, i.e. practically a vector space. It is not a term which is still greatly used.
 
A centro-affine space is an affine space with a distinguished point, i.e. practically a vector space. It is not a term which is still greatly used.

Latest revision as of 10:59, 29 May 2020


A centro-affine space $ E _ {N} $( where $ N = n(n - 1)/2 $), which may be assigned to each point of a space $ A _ {n} $ with an affine connection (in particular, to a Riemannian space $ V _ {n} $). Consider all tensors with even covariant and contravariant valencies at a point of the space $ A _ {n} $( or $ V _ {n} $); the covariant and contravariant indices are subdivided into different pairs, for each one of which the tensor is skew-symmetric. Tensors with these two properties are called bitensors. If each skew-symmetric pair is regarded as a collective index, the number of new indices will be $ N = n(n - 1)/2 $. The simplest bitensor is the bivector

$$ v _ {\alpha \beta } = - v _ {\beta \alpha } \rightarrow v _ {a} ,\ \ \alpha , \beta =1 \dots n; \ a = 1 \dots N . $$

If, at a point $ P $ of $ A _ {n} $,

$$ A _ \beta ^ \alpha = \ \left ( \frac{\partial \overline{x}\; ^ \alpha }{\partial x ^ \beta } \right ) _ {p} ,\ \ A _ {b} ^ {a} = \ 2A _ \gamma ^ {[ \alpha } A _ \delta ^ { {}\beta ] } = \ A _ {[ \gamma{} } ^ {[ \alpha{} } A _ { {}\delta ] } ^ { {}\beta ] } , $$

$$ v ^ {a} = v ^ {\alpha \beta } ,\ v ^ {b} = v ^ {\gamma \delta } , $$

then $ \overline{v}\; ^ {a} = A _ {b} ^ {a} v ^ {b} $, and the set of bivectors assigned to $ A _ {n} $( or $ V _ {n} $) at a given point defines a vector space of dimension $ N $ such that the components satisfy the conditions

$$ \overline{v}\; ^ {b} = \ A _ {a} ^ {b} v ^ {a} ,\ \ v ^ {a} = \ \overline{A}\; _ {b} ^ {a} \overline{v}\; ^ {b} $$

$$ | A _ {b} ^ {a} | \neq 0,\ A _ {b} ^ {a} \overline{A}\; _ {c} ^ {b} = \delta _ {c} ^ {a} $$

i.e. this set defines the centro-affine space $ E _ {N} $, called the bivector space at the given point. In $ V _ {n} $ the bivector space may be metrized with the aid of the metric tensor

$$ g _ {ab} = \ g _ {\alpha \beta \gamma \delta } = ^ { {roman } def } \ g _ {\alpha \gamma } g _ {\beta \delta } - g _ {\alpha \delta } g _ {\beta \gamma } , $$

after which $ E _ {N} $ becomes a metric space $ R _ {N} $.

Bivector spaces are used in Riemannian geometry and in the general theory of relativity. The bivector space $ R _ {N} $ is constructed at a given point of the space $ V _ {n} $, and different representations of the curvature tensor with components $ R _ {\alpha \beta \gamma \delta } $, $ R _ {\gamma \delta } ^ {\alpha \beta } $, $ R _ {\alpha \beta } ^ {\gamma \delta } $ and the second-valency bitensors with components $ R _ {ab} $, $ R _ {b} ^ {a} $, $ R _ {a} ^ {b} $ are associated, respectively. The study of the algebraic structure of the curvature tensor may then be reduced to the study of the pencil of quadratic forms $ R _ {ab} - \lambda g _ {ab} $, the second one of which is non-degenerate ( $ | g _ {ab} | \neq 0 $). The study of elementary divisors of this pair results in a classification of the spaces $ V _ {n} $. If $ n = 4 $( $ N = 6 $) and if the form $ g _ {\alpha \beta } $ has signature $ (- - - +) $, then it can be shown that only three types of Einstein spaces exist.

A bivector may be assigned to each rotation in $ V _ {n} $; this means that in $ R _ {N} $ there corresponds a vector, which is convenient for the study of infinitesimal transformations. Essentially, a bivector space is identical with a biplanar space [2].

References

[1] A.Z. Petrov, "New methods in general relativity theory" , Moscow (1966) (In Russian)
[2] A.P. Norden, "On complex representation of tensors of a biplanar space" , 8 Kazan. Gos. Univ. Uchen. Zap. , 114 (1954) pp. 45–53 (In Russian) MR76400

Comments

Consider a bivector as represented by an ordered pair of vectors $ ( \mathbf u , \mathbf v ) $ as in [a1], [a4] or the article bivector. The Plücker coordinates of $ ( \mathbf u , \mathbf v ) $, $ p ^ {ij} = u ^ {i} v ^ {j} - u ^ {j} v ^ {i} $, then constitute a bivector as in the article above. Let $ \overline{\mathbf u}\; = A \mathbf u $, $ \overline{\mathbf v}\; = A \mathbf v $, i.e. $ \overline{u}\; ^ {i} = A _ {j} ^ {i} u ^ {j} $ and similarly for $ \mathbf v $. Then the $ p ^ {ij} $ transform as $ p ^ {ij} = 2A _ {k} ^ {[i} A _ {l} ^ {j]} p ^ {kl} $. Whence the formulas above. Here the square brackets in $ A _ {k} ^ {[i} A _ {l} ^ {j]} $ are a notation signifying taking an alternating average. Thus $ A _ {k} ^ {[1} A _ {l} ^ {2]} = (A _ {k} ^ {1} A _ {l} ^ {2} - A _ {k} ^ {2} A _ {l} ^ {1} )/2 $. If certain indices are to be singled out, i.e. exempt from this averaging process, this is indicated by $ | {} | $. Thus $ A _ {m} ^ {[ij | k | l] } = (A _ {m} ^ {ijkl} - A _ {m} ^ {jikl} - A _ {m} ^ {ilkj} - A _ {m} ^ {ljki} + A _ {m} ^ {jlki} + A _ {m} ^ {likj} )/6 $, and $ g _ {ab} = 2g _ {\alpha [ \gamma{} } g _ { {}| \beta | \delta ] } $, cf. above. This is a notation introduced by R. Bach [a5]. Cf. also Alternation.

In more modern terms what is described here is the bundle of bivectors over $ A _ {n} $.

A centro-affine space is an affine space with a distinguished point, i.e. practically a vector space. It is not a term which is still greatly used.

References

[a1] E. Cartan, "Geometry of Riemannian spaces. (With notes and appendices by R. Hermann)" , Math. Sci. Press (1963) (Translated from French)
[a2] S. Gołab, "Tensor calculus" , Elsevier (1974) (Translated from Polish) Zbl 0277.53008
[a3] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[a4] J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) pp. 11ff (Translated from German) MR0066025 Zbl 0057.37803
[a5] R. Bach, "Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungsbegriffs" Math. Zeitschr. , 9 (1921) pp. 110–135 MR1544454
How to Cite This Entry:
Bivector space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bivector_space&oldid=46080
This article was adapted from an original article by A.Z. Petrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article