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.