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

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 $A _ {n}$.