I'm trying to work myself through Riemannian geometry and I believe I still don't have a firm grasp on what the Levi-Civita connection actually defines. Specifically, if $\gamma(t)$ is a geodesic this means that $\nabla_{\gamma'(t)} \gamma'(t) = 0$. However if $\gamma(t)$ is just an arbitrary smooth curve, what does this expression look like? If we stick to Euclidian space, supposedly $\nabla_{\gamma'(t)} \gamma'(t) = \gamma''(t)$, or not? I've been trying to come to this conclusion using a local frame and hence writing $\gamma'(t) = \sum_{i=1}^n \gamma_i'(t) \frac{\partial}{\partial x_i}$. Per the Levi-Civita connection on $\mathbb{R}^n$ does this mean I can compute
\begin{equation} \nabla_{\gamma'(t)} \gamma'(t) = \sum_{i=1}^n \gamma_i'(t) \nabla_{\frac{\partial}{\partial x_i}} \gamma'(t) = \sum_{i=1}^n \gamma_i'(t) \sum_{j=1}^n \frac{\partial}{\partial x_i}(\gamma_j'(t)) \nabla_{\frac{\partial}{\partial x_i}} \frac{\partial}{\partial x_j} \end{equation}
At this point I'm assuming I'm doing something very wrong here but I have no idea what. For one I'm not particularly getting a straightforward expression and secondly, I thought the connections between the basis vectors were supposed to be all zero (since the Christoffel symbols in $\mathbb{R}^n$ are). How can I derive an expression for this correctly?
EDIT: In the book of do Carmo the definition of an affine connection is given as follows. A map $\nabla : \mathfrak{X}(M) \times \mathfrak{X}(M) \to \mathfrak{X}(M)$ which satisfies the following properties.
- $\nabla_{fX + gY}Z = f\nabla_XZ + g\nabla_YZ$.
- $\nabla_X(Y+Z) = \nabla_XY + \nabla_XZ$.
- $\nabla_X(fY) = f\nabla_XY + X(f)Y$.