Show that Riemannian manifold $M$ is contractible if the loop space $\Omega_{p,p}$ is contractible.
The problem is from the following material. It contends that the result is by standard Morse theory. Please help me to understand details in this argument. Thanks!
On Complete Open Manifolds of Positive Curvature, Detlef Gromoll and Wolfgang Meyer, Annals of Mathematics, Second Series, Vol. 90, No. 1 (Jul., 1969), pp. 83 (the last paragraph)
Probably you can write down an explicit contraction, but recall that $\pi_{k-1}(\Omega_{p,p}(M))\cong \pi_{k}(M)$. This comes from the path space fibration $\Omega_{p,p}(M)\rightarrow P(M)\rightarrow M$. This implies that $\pi_k(M)=0$ for all $k>0$. If $M$ is connected, this says that $M$ is weakly contractible. Manifolds are $CW$-complexes, hence $M$ is contractible by Whitehead's Theorem.