Path space of a manifold

Remarks

The path space is a Hilbert manifold. Choose a complete Riemannian metric on $M$. For a smooth path $\gamma$ from $p$ to $q$, let $T_\gamma$ be the Hilbert space of $H^1$ vector fields $\xi$ along $\gamma$ with $\xi(0) = \xi(1) = 0$. Small $\xi \in T_\gamma$ give paths $t \mapsto \exp_{\gamma(t)}(\xi(t))$ from $p$ to $q$, and these maps are charts with smooth chart changes. Every $H^1$ path is close to a smooth one, so these charts cover $\Omega_{p,q} M$, and each model space $T_\gamma$ is isomorphic to $H^1_0([0, 1]; \R^n)$.

On this Hilbert manifold the energy $E(\gamma) = \frac{1}{2} \int_0^1 \abs{\gamma'(t)}^2 \, dt$ is a smooth function whose critical points are the geodesics from $p$ to $q$. The inclusion of piecewise smooth or smooth paths into $\Omega_{p,q} M$ is a homotopy equivalence. For connected $M$ the space is homotopy equivalent to the based loop space of $M$.