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Covariant derivatives on null submanifolds. (English) Zbl 1235.83016

Summary: The degenerate nature of the metric on null hypersurfaces makes it difficult to define a covariant derivative on null submanifolds. Recent approaches using decomposition to define a covariant derivative on null hypersurfaces are investigated, with examples demonstrating the limitations of the methods. Motivated by Geroch’s work on asymptotically flat spacetimes, conformal transformations are used to construct a covariant derivative on null hypersurfaces, and a condition on the Ricci tensor is given to determine when this construction can be used. Several examples are given, including the construction of a covariant derivative operator for the class of spherically symmetric hypersurfaces.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
53Z05 Applications of differential geometry to physics

References:

[1] Geroch R.: Asymptotic structure of space-time. In: Esposito, F.P., Witten, L. (eds) Asymptotic Structure of Space-Time, pp. 1–105. Plenum Press, New York and London (1976)
[2] Spivak M.: A Comprehensive Introduction to Differential Geometry. Vol. 3, 2nd edn. Publish or Perish, Houston (1979) · Zbl 0439.53003
[3] Duggal K.L., Bejancu A.: Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications. Kluwer, Dordrecht (1996) · Zbl 0848.53001
[4] Hickethier, D.: Covariant derivatives on null submanifolds. PhD Thesis, Oregon State University (2010) (Available online at: http://hdl.handle.net/1957/19547 .) · Zbl 1235.83016
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