SIGMA 9 (2013), 012, 5 pages      arXiv:1206.5229      https://doi.org/10.3842/SIGMA.2013.012
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Specialized Orthonormal Frames and Embedding

Frank B. Estabrook
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109 USA

Received October 09, 2012, in final form February 12, 2013; Published online February 15, 2013

Abstract
We discuss some specializations of the frames of flat orthonormal frame bundles over geometries of indefinite signature, and the resulting symmetries of families of embedded Riemannian or pseudo-Riemannian geometries. The specializations are closed sets of linear constraints on the connection 1-forms of the framing. The embeddings can be isometric, as in minimal surfaces or Regge-Teitelboim gravity, or torsion-free, as in Einstein vacuum gravity. Involutive exterior differential systems are given, and their Cartan character tables calculated to express the well-posedness of the underlying partial differential embedding and specialization equations.

Key words: embedding; orthonormal frames; Cartan theory.

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References

1. Estabrook F.B., Mathematical structure of tetrad equations for vacuum relativity, Phys. Rev. D 71 (2005), 044004, 5 pages, gr-qc/0411029.
2. Estabrook F.B., The Hilbert Lagrangian and isometric embedding: tetrad formulation of Regge-Teitelboim gravity, J. Math. Phys. 51 (2010), 042502, 10 pages, arXiv:0908.0365.
3. Estabrook F.B., Wahlquist H.D., Moving frame formulations of 4-geometries having isometries, Classical Quantum Gravity 13 (1996), 1333-1338.
4. Ivey T.A., Landsberg J.M., Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, Vol. 61, Amer. Math. Soc., Providence, RI, 2003.
5. Paston S.A., Sheykin A.A., Embeddings for the Schwarzschild metric: classification and new results, Classical Quantum Gravity 29 (2012), 095022, 17 pages, arXiv:1202.1204.
6. Stephani H., Kramer D., MacCallum M., Hoenselaers C., Herlt E., Exact solutions of Einstein's field equations, 2nd ed., Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2003.