Topological structure of the Manifold valued functions $M$ is a Riemannian manifold. What condition on $M$ for $\mathcal{C}_{[a,b]}(M)$ (the set of continuous functions of the real interval $I=[a,b]$ to $M$) to be a polish space ? For which topology ? Is it easier to consider only smooth functions ? Any reference is welcome. 
 A: This space is a complete metric space: equip $M$ with some complete riemannian metric $g$ (all riemannian metrics are complete if $M$ is compact, and I believe there always are complete metrics for non-compact manifolds) and use the associated riemannian distance $d_g$. This metric is complete (in the standard sense), and this defines a complete metric on $C(I,M)$ (of uniform convergence).
$M$ itself is separable, and you should be able to uniformly approximate any continuous function $I\to M$ by a piecewise geodesic one whose corners lie in some dense subset of $M$, and the corners only happen at dyadic times. To be precise: continuous maps $c:I\to M$ such that there is some integer $N$ such that on every interval $\left[\frac{k}{2^n},\frac{k+1}{2^n}\right]$, $c\left(\frac{k}{2^n}\right)$ and $c\left(\frac{k+1}{2^n}\right)$ are close enough for there to be a unique geodesic linking them, and $c$ is that geodesic, and the $c\left(\frac{k}{2^n}\right)$ are selected from some fixed countable dense subset of $M$.
This defines a countable dense subset of $C(I,M)$.
