Let \(M\) be a smooth manifold and let \(\nabla\) be the connection in \(TM\). Then for each smooth curve \(\gamma\colon I\to M\), the connection determines a unique operator

\begin{equation*} D_t\colon \mathfrak{X}(\gamma)\to \mathfrak{X}(\gamma), \end{equation*}

called covariant derivative along curves, satisfying

  1. Linearity in \(\mathbb{R}\): \(D_t(aV+W)=aD_tV+D_tW\) for \(a\in \mathbb{R}\)
  2. Product rule: \(D_t(fV)=f' V+fD_tV\), for \(f\in C^\infty(I)\)
  3. If \(V\in \mathfrak{X}(\gamma)\) is extendible to a smooth vector field \(\widetilde{V}\) on the neighbourhood of the image of \(\gamma\), then \begin{equation*} D_tV=\nabla_{\gamma'(t)}\widetilde{V}. \end{equation*}
Remarks
  • There is an analogous operator of smooth tensor fields along \(\gamma\).

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