If we have a complete vector field $X : M \to TM$ on a smooth manifold $M$, it determines a flow $\varphi^t:M\to M$ for all $t$ in $\mathbb{R}$.
I thought about a way to "discretize" this and I came up with the following setup. Given a geodesically complete Riemannian manifold $(M, g)$ and a smooth vector field $X:M \to TM$ on the manifold, you can use the exponential map $\exp: TM \to M$ to define the following discrete-time dynamical system $$p_{t+1} = \exp\circ X(p_{t}) =: F(p_{t}),$$ where $p_{0} \in M$.
What is this called and how should I start searching for it in the literature?