In a lot of textbooks on relativity the Levi-Civita connection is derived like this:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]V=V^ie_i [/tex]

[tex]dV=dV^ie_i+V^ide_i [/tex]

[tex]dV=\partial_jV^ie_idx^j+V^i \Gamma^{j}_{ir}e_j dx^r [/tex]

which after relabeling indices:

[tex]dV=(\partial_jV^i+V^k \Gamma^{i}_{kj})e_i dx^j [/tex]

so that the covariant derivative is defined as:

[tex]\nabla_j V^i=\partial_jV^i+V^k \Gamma^{i}_{kj}[/tex]

However, the connection coefficient [tex]\Gamma^{i}_{kj} [/tex] is torsion-free by definition, as [tex]de_i=\Gamma^{j}_{ir}e_j dx^r [/tex] implies that

(1) [tex]\partial_r e_i=\Gamma^{j}_{ir}e_j [/tex].

If [tex]e_i=\partial_i[/tex] then since [tex]\partial_i\partial_r=\partial_r\partial_i[/tex] then by (1):

[tex]\Gamma^{j}_{ir}e_j=\Gamma^{j}_{ri}e_j [/tex]

or that the bottom two indices are symmetric which is the torsion-free condition.

I have two questions. Is the equation [tex]\partial_r e_i=\Gamma^{j}_{ir}e_j [/tex] true in general for any connection? And also, where did the torsion-free assumption enter into the derivation above?

**Physics Forums - The Fusion of Science and Community**

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!

# Levi-civita connection assumptions

Loading...

Similar Threads - Levi civita connection | Date |
---|---|

A Is the Berry connection a Levi-Civita connection? | Jan 1, 2018 |

A Can you give an example of a non-Levi Civita connection? | Oct 30, 2017 |

Is the Levi-Civita connection unique for a given manifold? | Dec 16, 2011 |

Relationship between Chern and Levi-Civita Connections on Kahler Manifolds | Dec 4, 2009 |

Levi-Civita connection | Sep 1, 2009 |

**Physics Forums - The Fusion of Science and Community**