The exponential map on a Lie group 
is a smooth map from its Lie algebra 
to 
, defined by 
. Here 
is the unique smooth group
homomorphism with 
.
It is a one-parameter subgroup, not necessarily
a homeomorphism or an injective map (Etingof 2020, pp. 44-45).
On a Riemannian manifold 
without boundary, the exponential
map at 
is defined by 
,
where 
is the geodesic with 
and 
. Its domain consists
of those tangent vectors for which this
geodesic exists through time 1. This is an open
set containing the zero vector in 
; the exponential map is defined on the whole tangent
bundle when the Riemannian metric is a complete Riemannian metric (Burtscher 2025,
pp. 57-59). For example, on the open interval 
with its Euclidean Riemannian
metric, 
is defined precisely for 
.
The Lie exponential map and each fixed-basepoint Riemannian exponential map 
have derivative
equal to the identity map at the zero
vector. Each is therefore a local diffeomorphism
there. This does not imply global injectivity. The circle
example below has periodic exponential map.
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 Lie exponential map agrees with the Riemannian exponential
map based at the identity element. 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.
