General Relativity and Gravitation

, Volume 5, Issue 5, pp 555–572 | Cite as

Space tensors in general relativity I: Spatial tensor algebra and analysis

  • Enrico Massa
Research Articles

Abstract

A pair (M, Γ) is defined as a Riemannian manifold M of normal hyperbolic type carrying a distinguished time-like congruence Γ. The spatial tensor algebraD associated with the pair (M, Γ) is discussed. A general definition of the concept of spatial tensor analysis over (M, Γ) is then proposed. Basically, this includes a spatial covariant differentiation\(\tilde \nabla \) and a time-derivative\(\tilde \nabla _T \), both acting onD and commuting with the process of raising and lowering the tensor indices. The torsion tensor fields of the pair\(\left( {\tilde \nabla ,\tilde \nabla _T } \right)\) are discussed, as well as the corresponding structural equations. The existence of a distinguished spatial tensor analysis over (M, Γ) is finally established, and the resulting mathematical structure is examined in detail.

Keywords

Riemannian Manifold Fundamental Form Differential Form Tensor Field Natural Basis 

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Copyright information

© Plenum Publishing Company Limited 1974

Authors and Affiliations

  • Enrico Massa
    • 1
  1. 1.Istituto Matematico dell'Università di GenovaGenovaItaly

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