# Covariant Derivative Commutation

• PLuz
In summary, the conversation discusses the general formula for commuting covariant derivatives, specifically for a (r,s)-tensor field. The formula provided in the link is compared to the formula for a vector field and it is noted that there is a difference in the minus signs, which could be accounted for in the convention used to define the Riemann tensor. It is believed that the formula is correct for a coordinate basis, but there may be additional terms for a non-coordinate basis.
PLuz
Hello,

Can anyone tell me the general formula for commuting covariant derivatives, I mean, given a (r,s)-tensor field what is the formula to commute covariant derivatives?

I found a formula http://pt.scribd.com/doc/25834757/21/Commuting-covariant-derivatives page 25, Eq.6.18 but it doesn't seem right, since for a vector field one would write:

$(\nabla_{\alpha} \nabla_{\beta}- \nabla_{\beta}\nabla_{\alpha})U^{\gamma}=R^{\gamma}\hspace{.5 mm}_{\delta \alpha \beta}U^{\delta}$

And according to the formula in the link it would be, for a vector field$(\nabla_{\alpha} \nabla_{\beta}- \nabla_{\beta}\nabla_{\alpha})U^{\gamma}=-R^{\gamma}\hspace{.5 mm}_{\delta \alpha \beta}U^{\delta}$

Thank you

I see only a difference in minus signs between the two expressions which can be accounted for in the convention used to define the Riemann tensor. I believe this formula is correct (at least for a coordinate basis, I cannot be sure if there are more terms for a non-coordinate basis).

For non-coordinate (aka anholonomic) basis, compute the Riemann tensor (it doesn't matter which sign convention you use) with the connection coefficients as provided in MTW page 210, formula 8.24b.

## What is a covariant derivative?

A covariant derivative is a mathematical operation that allows for the differentiation of a vector field along a curve in a curved space. It takes into account the curvature of the space and the orientation of the curve, unlike a regular derivative which only considers the orientation of the curve.

## How does the covariant derivative differ from the regular derivative?

The covariant derivative takes into account the curvature of the space and the orientation of the curve, while the regular derivative only considers the orientation of the curve. This makes the covariant derivative more suitable for use in curved spaces.

## What is the commutation of covariant derivatives?

The commutation of covariant derivatives is the measure of how two covariant derivatives of a vector field commute with each other. In simpler terms, it is the difference between taking the covariant derivative along one path and then along another path, versus taking the covariant derivative along the second path and then along the first path.

## Why is the commutation of covariant derivatives important?

The commutation of covariant derivatives is important because it tells us how the curvature of a space affects the differentiation of vector fields. It also helps us understand the properties of geodesic curves, which are the shortest paths between two points on a curved surface.

## How is the commutation of covariant derivatives calculated?

The commutation of covariant derivatives is calculated using the Christoffel symbols, which are numbers that represent the curvature of a space. These symbols are used to define the covariant derivative and its commutation with other covariant derivatives.

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