The following paragraph presents a very brief history of differen-
differential geometry and the notation used in these notes.
Differential geometry is probably as old as any mathematical dis-
discipline and certainly was well launched after Newton and Leibnitz
had laid the foundations of calculus. Many results concerning sur-
surfaces in 3-space were obtained by Gauss in the first half of the nine-
nineteenth centruy, and in 1854 Riemann laid the foundations for a more
abstract approach. At the end of that century, Levi-Civitae and
Ricci developed the concept of parallel translation in the classical
language of tensors. This approach received a tremendous impetus
from Einstein's work on relativity. During the early years of this
century, E. Cartan initiated research and methods that were indepen-
independent of a particular coordinate system (invariant methods). Chevalley's
book "The Theory of Lie Groups" A946) continued the clarification
of concepts and notation, and it has had a remarkable affect on the
current situation. The complete global synthesis of Cartan*s approach
was achieved when Ehresmann formulated a connexion in terms of a
fiber bundle. These notes utilize an invariant local method formulated
by Koszul.