Let \(M\) be a smooth manifold and let \(\nabla\) denote a connection on \(M\). Then for every smooth curve \(\gamma\colon I\to M\), the connection \(\nabla \) determines a unique operator \(D_t\) on every set of smooth tensor fields along \(\gamma\), such that the following properties hold, for smooth tensor fields \(\sigma, \tau\)

  1. Linearity in \(\mathbb{R}\): \(D_t(a\sigma+\tau)=aD_t\sigma+D_t\tau\) for \(a\in \mathbb{R}\)
  2. Product rule: \(D_t(f\sigma)=f' \sigma+fD_t\sigma\), for \(f\in C^\infty(I)\)
  3. If \(\sigma\) is extendible to a smooth tensor field \(\widetilde{\sigma}\) on the neighbourhood of the image of \(\gamma\), then \begin{equation*} D_t\sigma=\nabla_{\gamma'(t)}\widetilde{\sigma} \end{equation*}

[1, Theorem 4.24].

The proof is similar to the one for smooth vector fields.

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  1. J. Lee, Introduction to Riemannian Manifolds. Cham: Springer International Publishing, 2018. doi:10.1007/978-3-319-91755-9