Symmetric Connection: Does Torsion Vanish?

[kent davidge]

Does a symmetric connection implies that torsion vanishes?

 

[Cryo]

Lovelock & Rund "Tensors, Differential forms and Variational Principles", p 75, sec 3.4, eq. 4.18 defines the torsion tensor (and proves it is a tensor) as

$S^\alpha_{\beta\gamma}=\Gamma^\alpha_{\beta\gamma}-\Gamma^\alpha_{\gamma\beta}$, where $\Gamma$ is the connection.

So yes, symmetric connection implies zero torsion

 

[kent davidge]

Lovelock & Rund "Tensors, Differential forms and Variational Principles", p 75, sec 3.4, eq. 4.18 defines the torsion tensor (and proves it is a tensor) as

$S^\alpha_{\beta\gamma}=\Gamma^\alpha_{\beta\gamma}-\Gamma^\alpha_{\gamma\beta}$, where $\Gamma$ is the connection.

So yes, symmetric connection implies zero torsion

Thanks. I was not sure about it, although I was sure about the converse, i.e., vanishing torsion implies symmetric connection.

 

[haushofer]

For a geometric explanation of vanishing torsion from a gauge perspective, see

https://www.physicsforums.com/insights/general-relativity-gauge-theory/

The torsion corresponds to the curvature of "local spacetime translations" in the gauge-theory approach to GR (and N=1 SUGRA).