In Euclidean space, if I have a smooth dynamical system $\dot{x}=F(x)$, it's linearization about a solution $x(t)$ is $\dot{v}(t) = DF(x(t))v(t)$, where $DF(x)$ is the Jacobian matrix of $F$ at $x$.
I am curious on how linearization works on smooth manifolds. Suppose $(M,\nabla)$ is a smooth manifold with affine connection $\nabla$ and $\dot{p}=F(p)$ describes a dynamical system on $M$, where $F:M \to TM$ is a smooth vector field. What exactly is the "right way" to define the linearization of the system about a solution $p(t)$? By right, I guess the most intrinsic-to-the-manifold way.
In the Euclidean set up, recall $DF(x)v$ can be interpreted as the directional derivative of $F$ along $v$ at $x$. This leads me to say that the linearization of $\dot{p}=F(p)$ about the solution $p(t)$ is $\dot{v}(t) = \nabla_{v(t)} F|_{p(t)}$. However, this seems a bit odd. Under those dynamics (if they are even well-defined) $v(t) \in T_{p(t)}M$. So, $v(t)$ is a tangent vector constantly varying through the tangent spaces. Is this the right way to do it? Is this even well-defined?
I guess if $(E_i)$ is a local frame and $(\Gamma_{ij}^k)$ are the Christoffel symbols, we can write locally $\dot{v}^k = v(F^k)+v^iF^j(p)\Gamma_{ij}^k(p)$.