Bivector

A class $ [ \mathbf u , \mathbf v ] $ of ordered pairs $ \mathbf u , \mathbf v $ of vectors of an affine space $ A $, starting at a common origin (considered in a basis of the underlying space). A bivector is considered to be equal to zero if its constituent vectors $ \mathbf u $ and $ \mathbf v $ are collinear. A non-zero bivector generates a unique two-dimensional space in $ A $, its carrier. Two bivectors are said to be parallel if their carrier planes are parallel. If $ A $ has finite dimension $ n $, and $ (u ^ {1} \dots u ^ {n} ) $ are the contravariant coordinates of $ \mathbf u $, while $ (v ^ {1} \dots v ^ {n} ) $ are the contravariant coordinates of $ \mathbf v $, calculated with respect to some basis $ e = ( \mathbf e _ {1} \dots \mathbf e _ {n} ) $ of the underlying space of $ A $, then the quantities

$$ a _ {ij} = \ \left | \begin{array}{l} u ^ {i} \\ u ^ {j} \end{array} \ \begin{array}{l} v ^ {i} \\ v ^ {j} \end{array} \ \right | = 2! u ^ {[i} v ^ {j]} ,\ \ 1 \leq i, j \leq n, $$

are called the Plücker coordinates of the pair $ \mathbf u , \mathbf v $. Two pairs of vectors are in the same class if their Plücker coordinates with respect to some basis coincide (they will then be equal in any basis). The coordinates of the class are then called the coordinates of the bivector $ [ \mathbf u , \mathbf v ] $ with respect to the basis $ e $. These coordinates are skew-symmetric with respect to their indices; they contain $ ( {} _ {2} ^ {n} ) $ independent coordinates. Under a transition to another basis of $ A $, the coordinates of a bivector behave as coordinates of a twice-contravariant tensor. A bivector is also called a free bivector. In the presence of a scalar product in $ A $, a number of metrical concepts of vector algebra can be extended to bivectors. The measure of a bivector is the area of the parallelogram formed by the vectors $ \mathbf u , \mathbf v , - \mathbf u , - \mathbf v $, the origin of each one being located in the end of the preceding one. This only depends on the class, not on the representatives $ \mathbf u , \mathbf v $. The scalar product of two bivectors is the number equal to the product of the measures of the factors by the cosine of the angle between their two carrier planes. This product is a bilinear form of the coordinates of the factors, the coefficients of which are defined by the metric tensor of the space $ A $ alone.

If the dimension of $ A $ is 3, the bivector $ [ \mathbf u , \mathbf v ] $ may be identified with a vector of $ A $ which, in the presence of a scalar product, is called the vector product of the vectors $ \mathbf u , \mathbf v $.

In tensor calculus a bivector is an arbitrary contravariant skew-symmetric tensor of valency 2 (i.e. a tensor of type $ (2, 0) $). Each such tensor may be represented as a sum of tensors, to which correspond non-zero bivectors in the above sense with different carrier planes. They define the sheets of the bivector. The rank of the skew-symmetric matrix of dimension $ n \times n $ consisting of the coordinates of a bivector is an even number $ 2 \rho $, where $ \rho $ is the number of sheets of the bivector. In a real affine space $ A $ this matrix is similar to the matrix

$$ \left \| \begin{array}{ccccc} J _ {1} &{} &{} &{} & 0 \\ {} &\cdot &{} &{} &{} \\ {} &{} &J _ \rho &{} &{} \\ {} &{} &{} &\cdot &{} \\ 0 &{} &{} &{} & 0 \\ \end{array} \ \right \| $$

with the blocks

$$ J _ {j} = \ \left \| \begin{array}{rc} 0 & 1 \\ -1 & 0 \\ \end{array} \ \right \| ,\ \ 1 \leq j \leq \rho . $$

See also Exterior product; Poly-vector; Plücker coordinates.

References

[1]	J.A. Schouten, "Tensor analysis for physicists" , Cambridge Univ. Press (1951)

Comments

Assign to a non-zero bivector $ ( \mathbf u , \mathbf v ) $ the plane it generates, i.e. the corresponding point of the Grassmannian of 2 planes in (the underlying vector space of) $ A $. Then the Plücker coordinates of this element in the Grassmann manifold can be identified with the Plücker coordinates of the bivector.

References

[a1]	E. Cartan, "Geometry of Riemannian spaces. (With notes and appendices by R. Hermann)" , Math. Sci. Press (1983) (Translated from French)
[a2]	S. Gołab, "Tensor calculus" , Elsevier (Translated from Polish)
[a3]	P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)