Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

To my understanding, the Levi-Civita connection is the torsion-free connection on the tangent bundle preserving a given Riemannian metric.

Furthermore, given any affine connection ∇, there is a unique torsion-free connection ∇′ with the same family of affinely parametrized geodesics. (from Wikipedia)

Does this imply that for a given Riemannian manifold there is a Levi-Civita connection that is unique for that manifold?

I am still learning this stuff so I am often having trouble seeing whether something is true or not :)

Edit: To elaborate, from the fundamental theorem of riemannian geometry, there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric.

Does this make this connection unique for that manifold? i.e. if I have a lot of manifolds can I identify them using the Levi-Civita connection?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Is the Levi-Civita connection unique for a given manifold?

Loading...

Similar Threads for Levi Civita connection |
---|

A Connection 1-forms to Christoffel symbols |

I Connections on principal bundles |

A Is the Berry connection a Levi-Civita connection? |

A Can you give an example of a non-Levi Civita connection? |

I Arbitrariness of connection and arrow on sphere |

**Physics Forums | Science Articles, Homework Help, Discussion**