Bivector space
A centro-affine space (where
), which may be assigned to each point of a space
with an affine connection (in particular, to a Riemannian space
). Consider all tensors with even covariant and contravariant valencies at a point of the space
(or
); 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
. The simplest bitensor is the bivector
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If, at a point of
,
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then , and the set of bivectors assigned to
(or
) at a given point defines a vector space of dimension
such that the components satisfy the conditions
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i.e. this set defines the centro-affine space , called the bivector space at the given point. In
the bivector space may be metrized with the aid of the metric tensor
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after which becomes a metric space
.
Bivector spaces are used in Riemannian geometry and in the general theory of relativity. The bivector space is constructed at a given point of the space
, and different representations of the curvature tensor with components
,
,
and the second-valency bitensors with components
,
,
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
, the second one of which is non-degenerate (
). The study of elementary divisors of this pair results in a classification of the spaces
. If
(
) and if the form
has signature
, then it can be shown that only three types of Einstein spaces exist.
A bivector may be assigned to each rotation in ; this means that in
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 as in [a1], [a4] or the article bivector. The Plücker coordinates of
,
, then constitute a bivector as in the article above. Let
,
, i.e.
and similarly for
. Then the
transform as
. Whence the formulas above. Here the square brackets in
are a notation signifying taking an alternating average. Thus
. If certain indices are to be singled out, i.e. exempt from this averaging process, this is indicated by
. Thus
, and
, 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 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 |
Bivector space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bivector_space&oldid=14487