International Journal of Mathematics and Mathematical Sciences
Volume 30 (2002), Issue 1, Pages 15-23
doi:10.1155/S0161171202011560
On the geometry of free loop spaces
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.
Abstract
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.