I've been studying the covariant derivative recently and ran into a couple issues with its introduction and motivation. Specifically, right now I am curious as to why we cannot simply differentiate vector fields using the smooth structure on the tangent bundle. If we can determine that a vector field is smooth, then we can differentiate it, correct?
Let $(TM, \pi, M)$ be the tangent bundle of a manifold $M$ consider a chart $(U, \phi)$. From this, we obtain an open set in the tangent bundle $\tilde U = \pi^{-1}(U)$, and a coordinate diffeomorphism from this set into $\mathbb{R}^{2n}$ that I'll call $\tilde\phi$ and write as
$$\tilde\phi(p, v_p) = (x_1(p),…, x_n(p), dx^1|_p(v_p),…,dx^n|_p(v_p))$$
Working within the Euclidean space $\tilde\phi(\tilde U)$, we can add and subtract tangent vectors at different points in the manifold. Therefore, we can differentiate with these vectors.
I am wondering where this notion falls into trouble. I've been looking for a way to motivate the need for a connection, so understanding the issue with this approach via charts would be useful. The biggest issue for me seems to be the fact that differentiation via coordinates does not provide a way to lift the result back and keep the operator linear.
That is, we can take a vector field $X:M\to TM$ and a path $\gamma:\mathbb{R}\to U$ and define the directional derivative.
$$D_\gamma X(t) =\frac{d}{dt}\left[\tilde\phi\circ X\circ \gamma \right]$$
In this way, we are differentiating a path through $\tilde\phi(\tilde U)$ and will obtain a vector as a result. However, if we try to apply $\tilde\phi^{-1}$ to lift this back into the tangent bundle and obtain another section, we find that the operation taking $X$ to its "derivative" $\tilde\phi^{-1}(D_\gamma X)$ is not even linear. This doesn't even touch upon how this sort of definition behaves under changes in coordinates.
What are your thoughts on how best to motivate the covariant derivative? The best I have seen so far for my own understanding is from Wald 1984, a book on gravitation. He postulates the property such an operator should have, and then shows that any two such operators differ only by a set of numbers $C^a_{bc}$. My only issue with the approach, is that I don't see why we can't start off by working in the coordinates that come in the definition of the manifold.
Best Answer
Note that $\mathbb{R}^n$ (and thus every finite-dimensional vector space) has a canonical connection given by considering the partial derivative operators $\partial_i$ as parallel vector fields (equivalently considering the directional derivative $U,V\mapsto\partial_UV$ as the covariant derivative operator).
Any diffeomorphism allows us to induce a connection from the domain to codomain, so any chart $\varphi:U\to\mathbb{R}^n$ on a manifold $M$ induces a local connection on $U$, which we'll call the coordinate connection, with covariant derivative $\partial$. The costruction you write above appears to be equivalent to the coordinate connection.
The trouble with coordinate connections is that they are not globally defined, and they are not intrinsic, i.e. different coordinate charts with overlapping domains will in general induce a different connection on their common domain. An example of two such charts Cartesian and polar coordinates in $\mathbb{R}^2$: One can verify that these charts induce different coordinate connections on their common domain.
Ideally, we would like to equip a Riemannian manifold with an intrinsic connection, i.e. one that is completely determined by the manifold and metric. Coordinate connections won't do, since we additionally need to specify a preferred coordinate chart. Instead the Levy-Civita connection, which is uniquely determined by the metric, is most frequently used.
Despite the drawbacks, coordinate connections are still useful in computations. It can be shown that the difference between any two connection operators is a $(2,1)$ tensor, so any (affine) connection can be written in a particular chart as $\nabla=\partial+\Gamma$, where $\Gamma$ is such a tensor.