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