Let $\mathbf V \colon [0,T] \times \mathbb R^d \to \mathbb R^d$ (for $T>0$) be a given, bounded smooth vector field and let $\mathbf X=\mathbf X(t,x)$ be its *flow*, i.e. the unique solution to the initial-value problem
\begin{equation}
\begin{cases}
\frac{\partial}{\partial t} \mathbf X(t,x) = \mathbf V(t,\mathbf X(t,x)) & \text{ in } (0,T) \times \mathbb R^d \\
\mathbf X(0,x) = x \quad \text{ for all } x \in \mathbb R^d.   
\end{cases}
\end{equation} 


A well-known result in standard ODE's theory says that 
$$\tag{1}
\nabla\_x \mathbf X(t,x) = \exp\bigg( \int\_0^t \nabla \mathbf V(s,\mathbf X(s,x))\,ds\bigg). 
$$



> 
> Is there an analogous formula to (1) for ODEs driven by (smooth) vector fields *on Riemannian manifolds*? In particular, does this formula involve somehow the geometry of the Riemannian manifold? 
>  A rather precise question could be: consider the $C^1$ norm of $\mathbf X$ (or even its Lipschitz constant) w.r.t. space variable $x$: does it depend on some known tensors on the manifold (e.g. curvature)?
> 
> 
> 


I have gone through books in differential geometry/differential topology (e.g. Lee, Lang) and they prove that $\mathbf X$ is smooth but do not compute explicitly the derivative.


References are very much welcome. Thanks.



Derivative of the flow for ODEs on manifolds
I think you may be interested in this remarkable, recent [paper](http://cvgmt.sns.it/media/doc/paper/3769/Ahlfors-final.pdf) by E. Brué and D. Semola (see in particular Theorem 3.11 which answers to your question in a much more general setting). 


