Exponential map (Lie theory)

In the theory of Lie groups, the exponential map is a map from the Lie algebra formula_1 of a Lie group formula_2 to the group which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups.
The ordinary exponential function of mathematical analysis is a special case of the exponential map when formula_2 is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects.
Definitions.
Let formula_2 be a Lie group and formula_1 be its Lie algebra (thought of as the tangent space to the identity element of formula_2). The exponential map is a map
which can be defined in several different ways. The typical modern definition is this:
It follows easily from the chain rule that formula_13. The map formula_14 may be constructed as the integral curve of either the right- or left-invariant vector field associated with formula_12. That the integral curve exists for all real parameters follows by right- or left-translating the solution near zero.
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:
where formula_17 is the identity matrix. Thus, in the setting of matrix Lie groups, the exponential map is the restriction of the matrix exponential to the Lie algebra formula_1 of formula_2.
Comparison with Riemannian exponential map.
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" .
Other definitions.
Other equivalent definitions of the Lie-group exponential are as follows:
Properties.
Elementary properties of the exponential.
For all formula_8, the map formula_41 is the unique one-parameter subgroup of formula_2 whose tangent vector at the identity is formula_12. It follows that:
More generally:
It is important to emphasize that the preceding identity does not hold in general; the assumption that formula_12 and formula_48 commute is important.
The image of the exponential map always lies in the identity component of formula_2.
The exponential near the identity.
The exponential map formula_7 is a smooth map. Its differential at zero, formula_51, is the identity map (with the usual identifications).
It follows from the inverse function theorem that the exponential map, therefore, restricts to a diffeomorphism from some neighborhood of 0 in formula_1 to a neighborhood of 1 in formula_2.
It is then not difficult to show that if "G" is connected, every element "g" of "G" is a "product" of exponentials of elements of formula_1:formula_55.
Globally, the exponential map is not necessarily surjective. Furthermore, the exponential map may not be a local diffeomorphism at all points. For example, the exponential map from formula_56(3) to SO(3) is not a local diffeomorphism; see also cut locus on this failure. See derivative of the exponential map for more information.
Surjectivity of the exponential.
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.
The image of the exponential map of the connected but non-compact group "SL"2(R) is not the whole group. Its image consists of C-diagonalizable matrices with eigenvalues either positive or with modulus 1, and of non-diagonalizable matrices with a repeated eigenvalue 1, and the matrix formula_58. (Thus, the image excludes matrices with real, negative eigenvalues, other than formula_58.)
Exponential map and homomorphisms.
Let formula_60 be a Lie group homomorphism and let formula_61 be its derivative at the identity. Then the following diagram commutes:
In particular, when applied to the adjoint action of a Lie group formula_2, since formula_63, we have the useful identity:
Logarithmic coordinates.
Given a Lie group formula_2 with Lie algebra formula_66, each choice of a basis formula_67 of formula_66 determines a coordinate system near the identity element "e" for "G", as follows. By the inverse function theorem, the exponential map formula_69 is a diffeomorphism from some neighborhood formula_70 of the origin to a neighborhood formula_71 of formula_72. Its inverse:
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.
Remark: The open cover formula_74 gives a structure of a real-analytic manifold to "G" such that the group operation formula_75 is real-analytic.