Suppose $E$ is a $q$-dimensional real vector bundle on a smooth manifold $M$ and $\Gamma(E)$ is the set of smooth sections of $E$ on $M$. A connection on the vector bundle $E$ is a map $$ D:\Gamma(E) \to\Gamma(T^*(M)\otimes E)\tag{1} $$ which satisfies the following conditions:
- For any $s_1,s_2\in\Gamma(E)$, $D(s_1+s_2)=Ds_1+Ds_2$.
- For $s\in\Gamma(E)$ and any $\alpha\in C^\infty(M)$, $$ D(\alpha s) = d\alpha\otimes s + \alpha Ds\;. $$
Suppose $X$ is a smooth tangent vector field on $M$ and $s\in\Gamma(E)$. Let $$ D_Xs:=\langle X, Ds\rangle\;\tag{2} $$ where $\langle\;,\rangle$ represents the pairing between $T(M)$ and $T^*(M)$. Then $D_Xs$ is a section of $E$, which is called the covariant derivative of the section $s$ along $X$. This definition is given in Chern's Lectures on Differential Geometry.
By (1), $Ds$ is an element in $\Gamma(T^*(M)\otimes E)$, not $\Gamma(T^*(M))$. On the other hand, $X\in\Gamma(T(M))$. How should I understand the pairing in (2)?
In John Lee's Riemannian Manifolds, a connection in $E$ is a map $$ \nabla : T(M)\times \Gamma(E)\to \Gamma(E)\tag{3} $$ written $(X,Y)\mapsto \nabla_XY$, satisfying
- $C^\infty(M)$-linear in the first component;
- $\mathbb{R}$-linear in the second component;
- the product rule $$ \nabla_X(fY) = f\nabla_XY+(Xf)Y\;. $$
Essentially $\nabla_XY=D_XY$ in Chern's notation; we can show that (2) satisfies all the defining properties for (3).
Are there some reasons we would like to go to the more abstract definition in (1) instead of (3)?
