how affine connection follows from Two derivative operator

Question

IN wald's GR book in chapter 3 This is stated behind the definition of affine connection :

First He showed that if we have two derivative operator $\nabla_a , \tilde\nabla_a$ (both of which consistent with the definition of derivative operator) then at point $p\in M$, for any dual vector field $\omega_b$

$$ (\nabla_a - \tilde\nabla_a)\omega_b $$

will depend only on the value of $\omega_b$ at $p$. That is if $\omega_b^\prime=\omega_b$ at p then

$(\nabla_a - \tilde\nabla_a)\omega_b=(\nabla_a - \tilde\nabla_a)\omega_b^\prime$

Then he implied from above that at $p$, $(\nabla_a - \tilde\nabla_a)$ defines a $(1,2)$ tensor like

$$(\nabla_a - \tilde\nabla_a)\omega_b=C_{ab}^c\omega_c$$

I'm unable to see how his later assertion is connected to the earlier.

Answer 1

Actually Bob is using a well known theorem:

Theorem. If $L: V \to V$ is a linear map from a finite dimensional vector space to the same space, there is a uniquely determined tensor $T \in V \otimes V^*$ such that induces $L$ as follows. If $v\in V$, $Lv$ is the contraction of the tensor product of $T$ and $v$ (the only possible contraction).

In our case, $V= T^*_pM$, with $p$ fixed, $L$ is the difference of the two covariant derivatives evaluated at $p$. Your second statement assures that $L$ is well defined as it, differently form each covariant derivative, depends on the value of its argument at $p$ and not on what happens in a neighborhood of $p$. Thus, the difference of the two derivatives at $p$ is a linear map in $T^*_pM$ with $p$ fixed. The tensor corresponding to that map in view of the quoted theorem is the mentioned tensor $C(p)$. Varying $p$ in $M$ you define a tensor field.