Path space and free loop space

Abstract

Riemannian geometry on the space of (continuous) paths in a manifold $M$ has been studied by Cruzeiro and Malliavin. I will use concepts in path space analysis to define a Levi-Civita connection on free loop space, using the $G^0$ metric. A tangent vector $X$ at a loop $\gamma$ is a vector field along $\gamma$ such that $X(s) \in T_{\gamma(s)}M$. Following closely the calculations done by Fang, the Riemannian curvature $R^{LM}$ is given by $R^{LM}(X,Y)Z (\cdot) = R^M(X(\cdot),Y(\cdot))Z(\cdot)$.