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## Summary:

- Is it possible to define a curvature tensor for dual vectors and what form would that take?

## Main Question or Discussion Point

Good day all.

Given that in Sean Carroll`s Lectures on GR he states that when calculating the covariant derivative of a 1-Form the Christoffel symbols have a negative sign as opposed to for the covariant derivative of a vector, would it be naive to think that, given the usual equation for the Riemann Curvature Tensor of a vector, we can simply multiply all Christoffel symbols by -1 to obtain the Curvature Tensor for a 1-Form? Here I am interpreting "curvature tensor" as the result of a parallel transport of a 1-form around a closed path in spacetime. I also realise that, if I am correct, the resulting geometric object may well be different to the curvature tensor for a vector. If you do the substitution you basically end up swapping the first and second terms of the Riemann Tensor for a vector. Whether I am correct or wrong another interesting point is, can we define the Bianchi Identity for whatever geometric object results?

Thanks

Paul

Given that in Sean Carroll`s Lectures on GR he states that when calculating the covariant derivative of a 1-Form the Christoffel symbols have a negative sign as opposed to for the covariant derivative of a vector, would it be naive to think that, given the usual equation for the Riemann Curvature Tensor of a vector, we can simply multiply all Christoffel symbols by -1 to obtain the Curvature Tensor for a 1-Form? Here I am interpreting "curvature tensor" as the result of a parallel transport of a 1-form around a closed path in spacetime. I also realise that, if I am correct, the resulting geometric object may well be different to the curvature tensor for a vector. If you do the substitution you basically end up swapping the first and second terms of the Riemann Tensor for a vector. Whether I am correct or wrong another interesting point is, can we define the Bianchi Identity for whatever geometric object results?

Thanks

Paul