I have some conceptual questions related to geodesic flows and cuvature.

1. Suppose you have one parameter group of isometries from your manifold to itself. Since isometry preserves metric then it preserves Levi-Civita connection and curvature. How would one tie this to geodesic flows\*. Is there a way to understand whether if a manifold has constant curvature by its geodesics (besides the criteria I gave below). For instance given a point $p$ on $M$, if $p$ can be connected to any other point on the manifold by a geodesic (as in sphere) then does the manifold have constant curvature? I would assume that if you have a "neighbourhood of geodesic flows" then its pullback preserves metric on that nbd. However it is not a global isometry.

\*-I know one theorem where if every geodesic circle has constant curvature then the manifold has constant curvature. 

2. My second question is where can I get some information about the set of all isometries of a manifold as a space itself? Is there a good geometry book on this topic as a reference?