Let $ (M,g) $ be a compact Riemannian manifold, and let the path space $ \Omega $ of $ M $ be the set of equivalence class of smooth maps $ \gamma : [0,1] \rightarrow M $ (equivalent under re-parametrization). What is that natural notion of smoothness on this space and of functions $ \Omega \rightarrow \mathbb{R} $? Where can I read about this?
I have some vague ideas about this. What I would like to capture is: given a sufficiently nice coordinate system $ U \in \mathbb{R}^n $, the space of paths that are straight line segments can be thought of as $ U \times U $, and the notion of smooth function on the space of line segments is just notion of smooth functions on $ U \times U $.