### Abstract

If a vector field is covariantly constant then the Ricci identity shows that it annihilates the curvature tensor and all its covariant derivatives. This paper attempts a 'best possible' converse to this result in the four-dimensional Lorentz case (the spacetime of general relativity theory) by supposing a vector field annihilates the curvature tensor and a certain number of its derivatives and showing how this leads to a covariantly constant vector field. A direct proof of these results, together with a brief description of how the proof can be obtained using holonomy theory, is presented. The two- and three-dimensional cases are also remarked upon.

Original language | English |
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Pages (from-to) | 1485-1492 |

Number of pages | 7 |

Journal | Classical and Quantum Gravity |

Volume | 23 |

DOIs | |

Publication status | Published - 2006 |

### Keywords

- GENERAL-RELATIVITY
- COLLINEATIONS

## Cite this

Hall, G. S. (2006). Some Remarks on Curvature Tensor Integrability in Spacetimes.

*Classical and Quantum Gravity*,*23*, 1485-1492. https://doi.org/10.1088/0264-9381/23/5/003