Let \(M\) be a [smooth manifold](smooth manifold.md), \(\nabla\) a connection in \(TM\) and \(\gamma\colon I\to M\) a maximal geodesic with initial point \(p\in M\) and \(\gamma'(0)\in T_pM\). Then the \(n\)-th derivative of a smooth real-valued function \(f\) along \(\gamma\) at \(0\in I\) is given by
\begin{equation*} (f\circ \gamma)^{(n)}=\nabla^n_{\gamma'}f. \end{equation*}Note, we use the convention \(\nabla^n_{\gamma'}f = \nabla^n_{\gamma',\ldots ,\gamma'} f\).
Proof
We prove the statement by an induction argument. For \(n=1\) see (0x66e28ddc) . Assume the statement is true for some \(n\in \mathbb{N}\). Then
\begin{align*} (f\circ \gamma)^{(n+1)}&=((f\circ \gamma)^{(n)})'\\ &=(\nabla^n_{\gamma'}f)'\\ &=\nabla_{\gamma'}(\nabla^n_{\gamma'} f)\\ &=\nabla^{n+1}_{\gamma'} f, \end{align*}where we sequentially used the induction assumption, the formula in (0x66e28ddc) and (0x66e29b2a) .