Tensor of Knitting machine-York

In Géométrie riemannienne, the tensor of tensor Knitting machine-York or of Knitting machine is a Tenseur mainly used in three-dimensional spaces, because in such spaces, it has the property to be null if and only if space is conformément flat.

The tensor of Knitting machine-York draws its name from the Mathématicien S Emile Cotton and James W. York. Certain results of Knitting machine were found independently by York, which justifies the use of one or the other of these names (Knitting machine and Knitting machine-York).

Formulate

The tensor of Knitting machine-York is written, in term of components,

R_ {ABC} = D_c R_ {ab} - D_b R_ {ac} + \ frac {1} {4} \ left (g_ {ac} D_b R - g_ {ab} D_c R \ right)

where G represents the metric space considered, R the scalar Courbure, R the Tenseur of Ricci and D the Dérivée covariante associated with the metric one.

Properties

By construction, the tensor of Knitting machine-York is antisymmetric compared to its last two indices and is cancelled by summation of the circular shifts of these indices:

R_ {ABC} = - R_ {acb}

R_ {ABC} + R_ {bca} + R_ {cab}

The contraction on its first two indices is also null in three-dimensional spaces:

R^a {} _ {ac} = 0

Component count independent

The tensor of Knitting machine-York has a priori 27 components independent; the antisymetry on the last two indices reduces this number to nine, and the contraction on the two first induces three additional constraints, that is to say six independent components. The constraint on the circular shifts introduced, always with three dimensions , an additional constraint, which leaves with final the five independent components, is only one of less than the Tenseur of Riemann which, with three dimensions, has six independent components.

Dual form

Because of antisymetry of the two last components of the tensor, one can associate to him his dual tensor within the meaning of the Dualité of Hodge, defined by

C^ {ab} = 2 \ epsilon^ {acd} D_b \ left (R^b_c - \ frac {1} {4} \ delta^b_c R \ right)

where ε is the Tenseur of completely antisymmetric Levi-Civita compared to its three indices.

Certain authors reserve the tensor name of of Knitting machine to the first definition of R and that of tensor of Knitting machine-York to that above.

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