The exponential function of classical analysis given by the series,
is the solution of the differential equation
with initial value .
This classical function is defined on the real line (or the complex plane). To generalise it to other manifolds, we need two things:
- By dimensionalysis?, the argument of the function should be a tangent vector; so in the classical function , the source is really the tangent space to the target at the point .
- We need a covariant derivative to tell us what means.
So in the end we have, for any point on a differentiable manifold with an affine connection , a map , which is defined at least on a neighbourhood of in the tangent space .
Note that here comes from the initial value ; we usually take when we work in a Lie group, but otherwise we are really generalising the classical exponential function ; every solution to takes this form.
Classically, there are some other functions called ‘exponential’; given any nonzero real (or complex) number , the map (or even ) is also an exponential map. Using the natural logarithm, we can define in terms of the natural exponential map :
So while is traditionally called the ‘base’, it is really the number that matters, or even better the operation of multiplication by . This operation is an endomorphism of the real line (or complex plane), and every such endomorphism takes this form for some nonzero (and some branch of the natural logarithm, in the complex case). So we see that this generalised exponential map is simply the composite of the natural exponential map after a linear endomorphism.
Let be a differentiable manifold, let be an affine connection on , and let be a point in . Then by the general theory of differential equations, there is a unique maximally defined partial function from the tangent space to such that:
This function is the natural exponential map on at relative to . We have , where is some neighbourhood of in . If is complete? (relative to ), then will be all of .
Given any endomorphism , we can also consider the exponential map on at relative to with logarithmic base , which is simply . We say ‘logarithmic base’ since a classical exponential function with base corresponds to an exponential function whose logarithmic base is multiplication by .
Recall that a geodesic is a curve on a manifold whose velocity? is constant (as measured along that curve relative to a given affine connection). Working naïvely, we may write
pretend that this is a differential equation for a function , and take the solution
where is given by the initial value . We recognise this as being, morally, . This suggests (although we need more work for a proof) the following result:
Let be a differentiable manifold, let be an affine connection on , and let be a point in . Given a tangent vector at , there is a unique maximal geodesic on tangent to at . If is defined (which it will be whenever is complete? and may be in any case), we have . In any case, we have for sufficiently small .
In Riemannian manifolds
Let be a Riemannian manifold (or a pseudo-Riemannian manifold) and let be a point in . Then may be equipped with the Levi-Civita connection , so we define the natural Riemannian exponential map on at to be the natural exponential map on at relative to .
In Lie groups
Note: this section is under repair.
The classical exponential function or satisfies the fundamental property:
The function is a homomorphism taking addition to multiplication:
A number of proofs may be given. One rests on the combinatorial binomial identity
(which crucially depends on the fact that multiplication is commutative), whereupon
An alternative proof begins with the observation that is the solution to the system , . For each , the function is a solution to the system , , as is the function . Then by uniqueness of solutions to ordinary differential equations (over connected domains; see, e.g., here), , i.e., for all .1
Let be Lie group and let be its Lie algebra , the tangent space to the identity element . Then may be equipped with the canonical left-invariant connection or the canonical right-invariant connection . It turns out that the natural Riemannian exponential maps on at relative to and are the same; we define this to be the natural Lie exponential map on at the identity, denoted simply . Several nice properties follow:
- is defined on all of .
- is a smooth map.
- If is a smooth homomorphism from the additive group (i.e., if is an -linear map, uniquely determined by specifying ), then is a smooth homomorphism.
- For , if , then the restriction of to the subspace spanned by and is a smooth homomorphism to . In particular, is a homomorphism if is abelian (e.g., if is a commutative Lie group).
- is surjective (a regular epimorphism) if is connected and compact (and also in some other situations, such as the classical cases where is or ). See this post by Terence Tao, Proposition 1; see also the first comment which indicates an alternative proof based on the fact that maximal tori in are all conjugate to one another. Note also that the exponential map might not be surjective if the compactness assumption is dropped, as in the case of or , both of which are connected; see here for instance.
- If is compact, then it may be equipped with a Riemannian metric that is both left and right invariant (see Tao’s post linked in the previous remark); then the Lie exponential map is the same as the Riemannian exponential map at .
- If is a matrix Lie group, then is given by the classical series formula (1).
(to be expanded on)
A logarithm is a local section of an exponential map.