I was wondering about the following situation:

Suppose we have a solution to the Ricci Flow on a compact manifold $M$, $g(t)=g\_t$, on a compact time interval $[0, \delta]$, and we consider $\{(t,v)\in [0, \delta] \times TM \mid g\_t(v,v)=1\}$. The question is very simply, "Is this set compact?"

It seems like it should be, since if you fix a time $t$, then we're just looking at the sphere bundle of $M$ (which is compact), so more or less the set is like a product of the sphere bundle with a compact interval. I'm just having an issue trying to pin down that more or less part so that's it's precise.