Covariant derivative of a vector field along a curve

Solution 1:

In part (c), the equality should read

$$ \frac{DV}{dt}(t_0) = \nabla_{\frac{dc}{dt}|_{t_0}}(Y) = \nabla_{\dot{c}(t_0)}(Y) $$

where both sides of the equality are elements of $T_{c(t_0)}M$.

Immediately after the statement of the theorem, Do Carmo proves that given an affine connection $\nabla$ on $M$, the value of $(\nabla_X Y)(p)$ depends only on $X(p)$ and the value of $Y$ along any curve $\alpha \colon I \rightarrow M$ satisfying $\alpha(0) = p$ and $\dot{\alpha}(0) = X(p)$. In particular, the expression $\nabla_{\frac{dc}{dt}|_{t_0}}(Y)$ makes sense (as we can interpret $\nabla_{\frac{dc}{dt}|_{t_0}}(Y)$ as $(\nabla_X Y)(c(t_0))$ where $X$ is any vector field on $M$ that satisfies $X(c(t_0)) = \frac{dc}{dt}|_{t_0}$ - such vector fields exist and $(\nabla_X Y)(c(t_0))$ does not depend on a specific choice of such an $X$).