Question about covariant derivative on manifolds. I am currently learning about the covariant derivative on smooth manifolds with the following definition:

Definition. A covariant derivative of vector fields is an operation that associates to two vector fields $X$ and $Y$ a new vector field $\nabla_XY$ and satisfies the following properties.

*

*$\mathcal{C}^{\infty}(M)$-linearity with respect to $X$:
$$
\nabla_{X_1+X_2}Y = \nabla_{X_1}Y + \nabla_{X_2}Y,\quad \nabla_{fX}Y = f\nabla_XY;
$$

*additivity and the Leibniz rule with respect to $Y$:
$$
\nabla_X(Y_1+y_2) = \nabla_XY_1+\nabla_XY_2,\quad \nabla_X(fY) = D_Xf\cdot Y + f\nabla_XY.
$$

Then there is this theorem, for which I do not understand the proof. 
How am I supposed to understand the transfer the vectorfield via $\varphi$ part? Where does the covariant derivative on $U_i$ come from? Can you please explain me, what the first part with the chart means in exact terms? Thank you
 A: Presumably, the author means that $\nabla^i$ is the pullback through $\varphi_i$ of the covariant derivative $\bar \nabla$ on $\Bbb R^n$ defined by taking the directional derivatives in the following way.
If $\bar X= \sum_j \bar X^j\partial_j$ and $\bar Y = \sum_k \bar Y^k \partial_k$ are vectors fields on $\Bbb R^n$, then $\bar \nabla_{\bar X} \bar Y = \sum_{j,k} \bar X^j \frac{\partial \bar Y^k}{\partial x^j} \partial_k$.
Let $X$ and $Y$ be vector fields on $M$.
The pullback covariant derivative $\nabla^i = {\varphi_i}^*\bar\nabla$ is given by the formula $\nabla^i_XY= {\varphi_i}^*\left(\bar \nabla_{{\varphi_i}_*X}{\varphi_i}_*Y\right)$, and therefore, the covariant derivative $\nabla$ on $M$ is given by
$$
\nabla_XY = \left(\sum_i\lambda_i\nabla^i\right)_XY= \sum_i\lambda_i {\varphi_i}^*\left(\bar\nabla_{{\varphi_i}_*X}{\varphi_i}_*Y\right).
$$
You can check that it indeed defines a covariant derivative since the pullback $\varphi_i^*$ and the pushforward ${\varphi_i}_*$ are $\mathcal{C}^{\infty}(M;\Bbb R)$-linear.
In addition, I would like to say that this construction works on all manifolds, not only compact ones (as suggests the text OP has linked).
Indeed, we only need to have a partition of unity associated to a locally finite open cover.
Since all manifolds possess such a partition of unity, the same proof works.
Finally, let us mention that there is a quicker but more conceptual way to prove the existence of a covariant derivative on any manifold.
By the Whitney's embedding Theorem, any manifold embeds into some euclidean space $\Bbb R^N$.
The orthogonal projection of the directional derivative of vector fields on the tangent space yields such a covariant derivative.
It is the famous Levi-Civita connection, which is in fact associated to the Riemannian metric induced by the embedding.

In case your question is more about the definition of the pushforward or the pullback through a diffeomorphism $\varphi\colon U\to V$, they are defined by
$$
(\varphi_*X)(q) = d\varphi(\varphi^{-1}(q)) X(\varphi^{-1}(q))
$$
for $q\in V$ and $X$ a vector field on $U$, and
$$
(\varphi^* Y)(p) = [d\varphi(p)]^{-1} Y(\varphi(p))
$$
for $p\in U$ and $Y$ a vector field on $V$.
