$\Delta e^i =0$ where $e_i$ is geodesic.

Question

Let $e_i$ be a geodesic coordinate vector field and $e^i$ be its coframe. Then $$\Delta e^i =0$$ This is right ? If so how can we prove ?

$$\Delta e^i (e_j)=\nabla_k \nabla_k e^i(e_j) = e_k( \nabla_k e^i(e_j) )= e_k( - \Gamma_{kj}^i)$$

Answer 1

Not true. Try lines of longitude on a sphere ($\theta=\text{constant}$). Then $\omega = d\phi$ and if you compute $(d\delta+\delta d)\omega$, you don't get $0$.