Here is a homework question: Let $M$ be a $C^\infty$ differentiable manifold with the affine connection $\nabla$, $\{e_i\}$ be a local frame, $\{\omega^i\}$ be its dual frame, $\{\omega_{j}^i\}$ be the connection $1$-forms.
(1) Show that the geodesic equation is
$$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\omega_k}{\mathrm{d}t}\right)+\frac{\omega_i}{\mathrm{d}t}\frac{\omega_{i}^k}{\mathrm{d} t}=0$$
I am confused with the notations $\frac{\omega}{\mathrm{d}t}$ where $\omega$ is an 1-form. Any explanation of references are welcomed. THANKS!