do Carmo's covariant derivative problem

Question

I'm reading DoCarmo's book, Riemannian Geometry and in the chapter affine connections, is talking about $\frac{DV} {dt} $ for a vector field $V$ along a curve $c$ and doesn't define it. He talks in the introduction about the case of surfaces in 3d but doesn't give a precise def for a riemannian manifold in general. Can someone tell me something about this? A precise def, a local expression.

Answer 1

On page 50 do Carmo defines the covariant derivative in Proposition 2.2, by means of three properties he lists there. He proves the proposition on page 51.