International Journal of Mathematics and Mathematical Sciences
Volume 30 (2002), Issue 1, Pages 15-23

On the geometry of free loop spaces

P. Manoharan

Mathematics Department, University of Maryland, University College, 3501 University Boulevard East, Adelphi 20783, MD, USA

Received 22 January 2001; Revised 18 June 2001

Copyright © 2002 P. Manoharan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We verify the following three basic results on the free loop space LM. (1) We show that the set of all points, where the fundamental form on LM is nondegenerate, is an open subset. (2) The connections of a Fréchet bundle over LM can be extended to S1-central extensions and, in particular, there exist natural connections on the string structures. (3) The notion of Christoffel symbols and the curvature are introduced on LM and they are described in terms of Christoffel symbols of M.