The modern geometric point of view in complex function theory began with Ahlfors's classic paper [AHL] . In that work
it was demonstrated that the Schwarz lemma can be viewed as
an inequality of certain differential geometric quantities on the
disc (we will later learn that they are curvatures). This point of
view-that substantive analytic facts can be interpreted in the
language of Riemannian geometry-has developed considerably
in the last fifty years. It provides new proofs of many classical
results in complex analysis, and has led to new insights as well.