### 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 |
---|---|

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

*Classical and Quantum Gravity*,

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

**Some Remarks on Curvature Tensor Integrability in Spacetimes.** / Hall, Graham Stanley.

Research output: Contribution to journal › Article

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

}

TY - JOUR

T1 - Some Remarks on Curvature Tensor Integrability in Spacetimes

AU - Hall, Graham Stanley

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

KW - GENERAL-RELATIVITY

KW - COLLINEATIONS

U2 - 10.1088/0264-9381/23/5/003

DO - 10.1088/0264-9381/23/5/003

M3 - Article

VL - 23

SP - 1485

EP - 1492

JO - Classical and Quantum Gravity

JF - Classical and Quantum Gravity

SN - 0264-9381

ER -