Cheeger-Gromoll Splitting in Peterson's text

Question

So I don't see how the distance function allows us to conclude that $M=U_0\times \mathbb{R}$. Could someone explain it in more detail.

Answer 1

Look at the level sets of $r$, i.e. at the sets $\{x: r(x)=c\} =: U_c$ The condition on the Hessian allows to conclude that the curves normal to these level sets are complete geodesics and that the level sets have no focal points. The diffeomorphism $U_0\times \mathbb{R} \rightarrow M$ can be written down explicitly by looking at the exponential map restricted to the normal bundle of any of the hypersurfaces $U_c$, e.g. of $U_0$.