  In this paper, we consider time-varying real analytic vector fields as curves
on the space of real analytic vector fields. Using a suitable topology on the
space of real analytic vector fields, we study and characterize different
properties of time-varying real analytic vector fields. We study holomorphic
extensions of time-varying real analytic vector fields and show that under
suitable integrability conditions, a time-varying real analytic vector field on
a manifold can be extended to a time-varying holomorphic vector field on a
neighbourhood of that manifold. Moreover, we develop an operator setting, where
the nonlinear differential equation governing the flow of a time-varying real
analytic vector field can be considered as a linear differential equation on an
infinite dimensional locally convex vector space. Using the holomorphic
extension results, we show that the integrability of the time-varying vector
field ensures the convergence of the sequence of Picard iterations for this
linear differential equation. This gives us a series representation for the
flow of an integrable time-varying real analytic vector field. We also define
the exponential map between integrable time-varying real analytic vector fields
and their flows. Using the holomorphic extensions of time-varying real analytic
vector fields, we show that the exponential map is sequentially continuous.
