  We show that universal transport coefficients of the fractional quantum Hall
effect (FQHE) can be understood as a response to variations of spatial
geometry. Some transport properties are essentially governed by the
gravitational anomaly. We develop a general method to compute correlation
functions of FQH states in a curved space, where local transformation
properties of these states are examined through local geometric variations. We
introduce the notion of a generating functional and relate it to geometric
invariant functionals recently studied in geometry. We develop two
complementary methods to study the geometry of the FQHE. One method is based on
iterating a Ward identity, while the other is based on a field theoretical
formulation of the FQHE through a path integral formalism.
