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.