Covariant derivative

Remarks

Clause 1 is well defined because $\nabla_X s$ is $C^\infty(M)$-linear in $X$, so its value at $p$ depends only on $X_p$. The existence and uniqueness in clause 2 follow by writing $V(t) = \sum_i V^i(t) e_i(\gamma(t))$ in a local frame: then $D_t V = \sum_i \big(\dot V^i + \sum_j \omega^i_j(\gamma'(t)) V^j\big) e_i$, where $\nabla e_j = \sum_i \omega^i_j \otimes e_i$. Example: for the flat connection on $\R^n$ and $E = T\R^n$, $D_t V = \dot V$ is the ordinary derivative of the components; so $D_t \gamma' = \ddot \gamma$ is the acceleration. Non-example: on the round sphere $S^2 \subset \R^3$ with its Levi-Civita connection, $D_t \gamma'$ is the tangential part of $\ddot \gamma$, not $\ddot\gamma$ itself. Along a great circle $\ddot \gamma$ is normal to the sphere, so $D_t \gamma' = 0$ although $\ddot \gamma \ne 0$. A connection on $TM$ extends uniquely to covariant derivatives of all tensor fields that obey the product rule for $\otimes$, commute with contractions and act on functions by $\nabla_X f = X f$. In index notation $\nabla_i Y^k = \partial_i Y^k + \Gamma^k_{ij} Y^j$, often written $Y^k{}_{;i}$.