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.