# Lie derivative of dual vector

Is it possible to take the Lie derivative with respect to a dual vector (1-form) field?

Suppose I have a vector field $X$ and two dual vector fields $\omega$ and $\sigma$. Then the Lie derivative with respect to a vector field, such as $\mathcal L_X\omega$, is standard. But what about $\mathcal L_\omega X$ or $\mathcal L_\omega\sigma$? I would prefer a coordinate expression.

(My motivation is a condition for a dual frame/tetrad/basis to be a coordinate/holonomic basis, analogous to the condition $\mathcal L_{\mathbf e_\alpha}\mathbf e_\beta=0$ for a vector basis $\{\mathbf e_\alpha\}$.)

There is a similar question here.

• A short answer would be no unless you have some extra structure (a morphism $P:T^*M\rightarrow TM$). May 30, 2018 at 11:28
• Thank-you. I assume a metric is provided, and the duality is based on it. I have found one potential source: de Felice & Clarke (1990, section 2.12) have a "Frobenius theorem for 1-forms" May 31, 2018 at 13:07
• In this case, I would define $\mathcal L_\alpha = \mathcal L_{\alpha^\#}$. I recommend you Vaisman's book Lectures on the Geometry of Poisson Manifolds. It's about Poisson geometry but you can get an idea of how to proceed. May 31, 2018 at 20:08