Anyone reading this post might be interested in my related question here. This is not a duplicate of my related question, which was about affine connections, not covariant derivatives.
Here is the definition of covariant derivative, as appears in Milnor's book Morse Theory.
DEFINITION. An affine connection at a point $p \in \text{M}$ is a function which assigns to each tangent vector $\text{X}_p \in \text{TM}_p$ and to each vector field $\text{Y}$ a new tangent vector$$\text{X}_p \vdash \text{Y} \in \text{TM}_p$$called the covariant derivative of $\text{Y}$ in the direction $\text{X}_p$.
(Note that our $\text{X} \vdash \text{Y}$ coincides with Nomizu's $\nabla_\text{X} \text{Y}$. The notation is intended to suggest that the differential operator $\text{X}$ acts on the vector field $\text{Y}$.)
This is required to be bilinear as a function of $\text{X}_p$ and $\text{Y}$. Furthermore, if$$f: \text{M} \to \mathbb{R}$$is a real valued function, and if $f\text{Y}$ denotes the vector field$$(f\text{Y})_q = f(q)\text{Y}_q$$then $\vdash$ is required to satisfy the identity$$\text{X}_p \vdash (f\text{Y}) = (\text{X}_p f)\text{Y}_p + f(p) \text{X}_p \vdash \text{Y}.$$
(As usual, $\text{X}_p$ denotes the directional derivative of $f$ in the direction of $\text{X}_p$.)
I have two questions.
- This definition of covariant derivative is quite terse here (indeed, it seems like affine connections and not covariant derivatives are being made out to be the most important thing here)—I'm just seeing text on a page and not really understanding what is going on here. Is it possible somebody could help me parse through/explain what is really being said here with regards to covariant derivative?
- Could somebody supply their intuitions behind/for covariant derivatives?
Thanks.
