This course will be divided into two halves. The first half will discuss issues related to the Cheeger-Gromov theory and its generalizations in Riemannian geometry. Given this, the second half is concerned with applications of this theory and point of view to current issues in General Relativity.
This course is quite advanced, and will lead into current research topics in these areas. A number of open problems will be presented.
The basic reference for the course is the following set of lecture notes: (PS file) or (DVI file) This includes numerous other references.
As reference textbooks, I'd suggest:
For Riemannian geometry: P. Petersen, Riemannian Geometry, Grad. Texts. in Math., Springer Verlag
For General Relativity: R. M. Wald, General Relativity, Univ. Chicago Press
This is probably the single best book for people coming with a background in geometry. Other excellent books - classics in the area - are:
Hawking and Ellis, The Large Scale Structure of Space-Time, Cambridge Univ. Press,
Misner-Thorne-Wheeler, Gravitation, W.H.Freeman, San Francisco.
W. Rinder, Essential Relativity, Springer Verlag
A wonderful source of material on relativity, from the popular to the current state of the art on many aspects of research you can find at: