A vector $V$ is parallel $\iff$ $\nabla V=0$
Where $\nabla V$ is the total covariant derivative of $V$ and $V$ is said to be parallel if it is parallel along every curve.
How do we take the total covariant derivative of a vector field? If we write $V = V_i \frac{\partial}{\partial_i}$ then I know how to take take the regular covariant deriviative, but it's weird for me to take the total covariant derivative of just one vector field, I feel like when I take the total derivative of tensor fields it is like a generalized product rule. Can somebody show me the details explicitly please? Thanks!
You can consider the covariant derivative as an operator $\nabla : \Gamma( TM) \to \Gamma(T^*M) \otimes \Gamma(TM)$. That is, it takes a vector field and gives you a vector field tensor a one-form. The one-form describes the direction you are taking the covariant derivative in. Assuming $\nabla$ is the Levi-Civita connection, we can write this in local coordinates as $$\nabla V = d(V^j) \otimes\frac{\partial}{\partial x^j} +V^j \Gamma^k_{ij} \;dx^i \otimes \frac{\partial}{\partial x^k}$$ where $d$ is the exterior derivative and $\Gamma^k_{ij}$ are the Christoffel symbols. So for example, $$\nabla_{\frac{\partial}{\partial x^s}}V = \frac{\partial V^j}{\partial x^s} \frac{\partial }{\partial x^j} + V^j\Gamma^k_{sj} \frac{\partial}{\partial x^k}$$