Bivector space

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