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

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

called covariant derivative along \(\gamma\), satisfying, for \(V,W\in \mathfrak{X}(\gamma)\),

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*}

[1, Theorem 4.24].

Links Link to heading

• connection
• smooth curve on a manifold

Results Link to heading

• representation in local coordinates
• extension on tensor fields along a curve
• derivatives of a smooth function along a curve
• higher order covariant derivatives along a geodesic

Related concepts Link to heading

• acceleration of a curve
• parallel vector field
• geodesic

References Link to heading

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