At first it is difficult to see the geometrical analogy of the Gauge Invariance
prescription. It becomes necessary to expand our set of geometrical concepts.
First of all we need to introduce the auxiliary space of f. This happens
to be geometrically a circle. That is, f takes values from 0 to
.
From the form of the transformations for A fields we see
that since only the derivatives of f are needed, f can modified
by addition of multiples of .
Similarly, in the 
transformation,
the exponential is unchanged by such a modification.
Now we expand our notion of "space-time" as follows. At every point in space-time, we erect a circle corresponding to the auxiliary space of f. Thus in fig.s 4 and 5, just as we have shown a local inertial frame at a typical point P, so should we have a local copy of the space of f-function. This is shown in fig. 6. The statement of invariance is that the origins of the f spaces can be shifted in a space-time dependent manner without affecting the physical law.
We have to thus imagine in our space-time not only the possibility of choosing a local inertial frame, but also a particular value of the f-function, called the Gauge Function. The choice of zero of the f is analogous to the choice of space-time orientation of an inertial frame of reference. Gauge Functions f1 and f2 of two different observers can differ by an additive function of space and time.
Although there is a clear analogy between General Covariance and Gauge Invariance, there is a fundamental difference. Even in the absence of Gravity, we need to choose a frame of reference. But in the absence of Electromagnetism, the f-function is completely superfluous. Even in the absence of Gravity, space-time remains, and motion occurs. But the abstract space of the f-function is meaningless without Electromagnetism. For this reason, the two kinds of symmetry principles are classified as "external" (General Covariance), and "internal" (Gauge Invariance)
-functions