I am reading Chern's Lectures on Differential Geometry. Chern proves the theorem that A torsion-free affine connection is completely determined by the curvature tensor,i.e. the Theorem 2.3 on page 147 on this page.
However, I can not understand (2.20) in that link, I copied what it states below. If we consider a normal coordinate system $\alpha^{i}$ at a fixed point $O$ and also choose a natural frame at $O$, then we get a frame field $e_{i}$ and its dual $\theta^{i}$. Let $\theta^{j}_{i}$ be the restriction of the everywhere linearly independent $m^{2}$ differential 1-forms,then we have $$ \left\{ \begin{align} \theta^{i} &=\overline{\theta}^{i}+\alpha^{i}dt\\ \theta^{j}_{i} &=\overline{\theta}^{j}_{i} \end{align}\right. $$ where the $\overline{\theta}^{i}$ and $\overline{\theta}^{j}_{i}$ means they do not contain $dt$
I can not see how this is derived, can anyone help me out?