Morse Theory in Riemannian Geometry In the introduction of Cheeger and Gromoll's paper _On the structure of complete manifolds of nonnegative curvature_ the following is stated: > A point $p \in M$ is called simple if there are no geodesic loops in $M$ closed at $p$. [...] From Morse Theory it is easy to see that the existence of a simple point implies that $M$ is contractible. For me it is not clear how Morse Theory implies this result. I read Milnor's book on Morse Theory, but I don't see how the results in it on the topological structure of the path space show how to justify the said implication. So it seems to me that there are other standard applications/constructions/analogies of Morse Theory in Riemannian Geometry. I'd be glad to get a source on where to read more on morse theoretical applications in Riemannian Geometry and/or a way how to show the above implication with results found in Milnor's _Morse Theory_. Thanks in advance.

I think $M$ is assumed to be connected and complete (otherwise just puncture your manifold somewhere to remove a geodesic returning to $p$). I think you are looking for theorem 17.3 in Milnor's Morse theory. By the assumptions the homotopy type of $\Omega(M,p,p)$ is a CW complex with one cell (as there is only on geodesic from $p$ to $p$). This means in particular that $\Omega(M,p,p)$ is connected and all higher homotopy groups vanish.

But then, in view of $\pi_k(\Omega(M,p,p))\cong \pi_{k+1}(M)$ all homotopy groups of $M$ vanish as well. Thus $M$ is contractible.