# Exponential map (Lie theory)

which can be defined in several different ways. The typical modern definition is this:

We have a more concrete definition in the case of a matrix Lie group. The exponential map coincides with the matrix exponential and is given by the ordinary series expansion:

If *G* is compact, it has a Riemannian metric invariant under left *and* right translations, and the Lie-theoretic exponential map for *G* coincides with the exponential map of this Riemannian metric.

For a general *G*, there will not exist a Riemannian metric invariant under both left and right translations. Although there is always a Riemannian metric invariant under, say, left translations, the exponential map in the sense of Riemannian geometry for a left-invariant metric will *not* in general agree with the exponential map in the Lie group sense. That is to say, if *G* is a Lie group equipped with a left- but not right-invariant metric, the geodesics through the identity will not be one-parameter subgroups of *G*^{[citation needed]}.

Other equivalent definitions of the Lie-group exponential are as follows:

In these important special cases, the exponential map is known to always be surjective:

For groups not satisfying any of the above conditions, the exponential map may or may not be surjective.

is then a coordinate system on *U*. It is called by various names such as logarithmic coordinates, exponential coordinates or normal coordinates. See the closed-subgroup theorem for an example of how they are used in applications.