Exponential map (Lie theory)

The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups.

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.

be its Lie algebra (thought of as the tangent space to the identity element of

The typical modern definition is this: It follows easily from the chain rule that

, may be constructed as the integral curve of either the right- or left-invariant vector field associated with

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.

Thus, in the setting of matrix Lie groups, the exponential map is the restriction of the matrix exponential to the Lie algebra

is compact, it has a Riemannian metric invariant under left and right translations, then the Lie-theoretic exponential map for

coincides with the exponential map of this Riemannian metric.

, 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.

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

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

as complex manifolds, we can identify it with the tangent space

corresponds to the exponential map for the complex Lie group

The image of the exponential map always lies in the identity component of

, 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

Globally, the exponential map is not necessarily surjective.

Furthermore, the exponential map may not be a local diffeomorphism at all points.

(3) to SO(3) is not a local diffeomorphism; see also cut locus on this failure.

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.

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

(Thus, the image excludes matrices with real, negative eigenvalues, other than

Then the following diagram commutes:[8] In particular, when applied to the adjoint action of a Lie group

determines a coordinate system near the identity element e for G, as follows.

By the inverse function theorem, the exponential map

gives a structure of a real-analytic manifold to G such that the group operation