On a Lie group, exp is a map from the Lie algebra to its Lie
group. If you think of the Lie algebra as the
tangent space to the identity of the Lie
group, exp(
)
is defined to be 
,
where 
is the unique Lie group homeomorphism
from the real numbers to the Lie
group such that its velocity at time 0 is 
.
On a Riemannian manifold, exp is a map from the tangent bundle of the manifold
to the manifold, and exp(
) is defined to be 
, where 
is the unique geodesic traveling
through the base-point of 
such that its velocity at time 0 is 
.
The three notions of exp (exp from complex analysis, exp from Lie groups, and exp from Riemannian geometry)
are all linked together, the strongest link being between the Lie
groups and Riemannian geometry definition. If 
is a compact Lie group, it admits
a left and right invariant Riemannian metric.
With respect to that metric, the two exp maps agree on their common domain. In other
words, one-parameter subgroups are geodesics. In the case of the manifold 
, the circle,
if we think of the tangent space to 1 as being the imaginary
axis (y-axis) in the complex
plane, then
 |  |  |
(1)
|
 |  |  |
(2)
|
and so the three concepts of the exponential all agree in this case.
