Why isn't $\nabla_X(e_j)$ automatically a flat section?

Question

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (Volume 2) and An Introduction to Manifolds by Loring W. Tu (Volume 1).

I refer to Section 11.1 (page 79), Section 11.1 (page 80), Section 10.1 (page 72) and Section 10.2 (page 73).

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Section 11.1 begins with describing $\nabla$ locally.

1. First question is about Tu's description: Please explain what's going on. Why isn't $\nabla^U_X(e_j)$ (written with $U$ omitted i.e. $\nabla_X(e_j)$) automatically a flat section?

   • 1a. I think that $E|_U$ is a trivial tangent bundle, and so I think for page 80 that the first instance of the expression "$\nabla_X e_j$" should be the equation "$\nabla_X e_j$ = 0" (By definition in Example 10.3 in Section 10.1), and the second, third and fourth instances of "$\nabla_X e_j$" should be instead the expression "$\nabla_X s$" (as in $\nabla^U_X s$).

   • 1b. Technically, the first equation in page 80 is true, but I think each $\omega^i_j(X)$ is automatically the zero element of $C^{\infty}U$.

2. Second question is asking verification for my own description: Local description of connection $\nabla$ of smooth vector bundle $\pi: E \to M$

Answer 1

You are mixing up different things. One question is whether on a given vector bundle, there is some linear connection. This is discussed in Section 10, and the first step is to prove that a local trivialization of the bundle give rise to a "local connection" on the domain of the trivialization. These can the be glued to obtain a connection on the vector bundle.

Once it is clear that a bundle admits one linear connection, it follows easily that there are many of them (the form an infinite dimensional family parametrized by all sections of some bigger vector bundle). So a connection should be viewed as an "additional chosen structure" on a vector bundle.

Section 11 deals with the problem of describing some fixed connection. You can use local trivializations in this description, but the connection you are studying is not related to the flat local connection given by the chosen trivialization.