## Abstract

We prove that, in some situations, an induced action from a normal subgroup preserves a geometric structure. Combined with known geometric rigidity results, this result implies certain rigidity statements concerning the full diffeomorphism group of a manifold. It also provides many examples of actions on Lorentz manifolds. Combining these with a small number of well-known actions, we get the full list of connected, simply connected Lie groups admitting a locally faithful, orbit nonproper action by isometrics of a connected Lorentz manifold. We give an example of a connected nilpotent Lie group with no complicated action on a Lorentz manifold. We show that, if a connected Lie group has a normal closed subgroup isomorphic to a (two-dimensional) cylinder, then it admits a locally faithful, orbit nonproper action by isometrics of a connected Lorentz manifold.

Original language | English (US) |
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Pages (from-to) | 91-112 |

Number of pages | 22 |

Journal | Geometriae Dedicata |

Volume | 88 |

Issue number | 1-3 |

DOIs | |

State | Published - 2001 |

### Bibliographical note

Funding Information:*The author was supported in part by NSF grant DMS-9703480.

## Keywords

- Isometrics
- Lorentz manifolds
- Transformation groups