General Relativity and Gravitation

, Volume 25, Issue 1, pp 55–80 | Cite as

The affine geometry of the Lanczos H-tensor formalism

  • K. S. Hammon
  • L. K. Norris
Research Articles


We identify the fiber-bundle-with-connection structure that underlies the Lanczos H-tensor formulation of Riemannian geometrical structure. We consider linear connections to be type (1,2) affine tensor fields, and we sketch the structure of the appropriate fiber bundle that is needed to describe the differential geometry of such affine tensors, namely the affine frame bundleA 1 2 M with structure groupA 1 2 (4) =GL(4) ⓈT 1 2 4 over spacetimeM. Generalized affine connections on this bundle are in 1-1 correspondence with pairs(Γ, K) onM, where thegl(4)-componentΓ denotes a linear connection and the T 1 2 4-componentK is a type (1,3) tensor field onM. We show that the Lanczos H-tensor arises from a gauge fixing condition on this geometrical structure. The resulting translation gauge, theLanczos gauge, is invariant under the transformations found earlier by Lanczos. The other Lanczos variablesQμmandq are constructed in terms of the translational component of the generalized affine connection in the Lanczos gauge. To complete the geometric reformulation we reconstruct the Lanczos Lagrangian completely in terms of affine invariant quantities. The essential field equations derived from ourA 1 2 (4)-invariant Lagrangian are the Bianchi and Bach-Lanczos identities for four-dimensional Riemannian geometry.


Field Equation Differential Geometry Geometrical Structure Structure groupA12 Fiber Bundle 
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Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • K. S. Hammon
    • 1
  • L. K. Norris
    • 2
  1. 1.Department of PhysicsNorth Carolina State UniversityRaleighUSA
  2. 2.Department of MathematicsNorth Carolina State UniversityRaleighUSA

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