Path space and free loop space (Q550447)

The paper under review discusses the \(L^2\)-geometry of the space \(LM\) of smooth loops on a Riemannian manifold \(M\). It starts by recalling very basic facts and proceeds to a computation of the \(L^2\) covariant derivative and \(L^2\) curvature tensor of \(LM\), which is the main result claimed in the paper. Most of those geometric constructions are well-known, but perhaps scattered in the literature (see, e.g., [\textit{P. Flaschel} and \textit{W. Klingenberg}, Riemannsche Hilbertmannigfaltigkeiten. Periodische Geodätische. 282. Berlin-Heidelberg-New York: Springer-Verlag. (1972; Zbl 0238.58009), \textit{W. Klingenberg}, Lectures on closed geodesics. Berlin-Heidelberg-New York: Springer-Verlag (1978; Zbl 0397.58018), \textit{U. Schäper}; J. Geom. Phys. 11, 553--557 (1993; Zbl 0805.58017)] and references therein). No motivations or applications of these computations are given.