A flow on a manifold $M$ is a smooth map $\phi:M\times\mathbb{R}\to M$ such that $\phi(x, 0) = x$, for all $x\in M$, and $\phi(\phi(x, s), t) = \phi(x, s + t)$, for all $s,t\in\mathbb{R}$. We denote $\phi(x, t)$ by $\phi^t(x)$. In particular, the geodesic flow on the unit tangent bundle $T^1M$ on a Riemannian manifold $(M, g)$ is the flow on $T^1M$, $G:T^1 M\times\mathbb{R}\to T^1M$, given by $G^t((p ,v)) = \psi_{(p ,v)}'(t)$, for $(p, v)\in T_pM\subset TM$, where $\psi_{(p, v)}$ denotes the unique geodesic on $M$ satisfying $\psi_{(p, v)} = p$, and $\psi_{(p, v)}'(0) = v$. I have the following questions regarding these definitions:
- Does $\phi$ necessarily need to be defined for all $t\in\mathbb{R}$? That is, how does the definition of a flow on $M$ account for the possibility of there existing some $x\in M$, and $T\in\mathbb{R}$, such that, if $t>T$, then $\phi(x, t)\notin M$, or would this fact simply mean that $\phi$ is not a flow?
- Similarly to the above question, how does the definition of geodesic flow account for the possibility of there existing some $(p ,v)\in T^1M$, and $T\in\mathbb{R}$, such that, if $t>T$, then $\psi_{(p, v)}(t)\notin M$?
The second of the above questions seems to relate in a way to the completeness of the manifold $M$. In particular, it seems as if we can only define the geodesic flow on a manifold $M$ if the manifold is geodesically complete. Is this conjecture correct? If it is correct, then, since it is known that hyperbolic space, $\mathbf{H}^2$, is geodesically complete, it follows that the geodesic flow on $T^1\mathbf{H}^2$ exists. My final question is: how can we prove that the geodesic flow, either on $T^1\mathbf{H}^2$, or on $T^1M$, in general, is actually a flow? Of course, that $G^0 = \mathrm{id}:T^1M\to T^1M$ is clear, but how can we show that $G^s\circ G^t = G^{s + t}$, for all $s,t\in\mathbb{R}$?
Thank you in advance!