exponential map

The exponential map

For a Lie group with Lie algebra , the exponential map is

where is the 1-parameter subgroup with .

In other words, to find the exponential map, we take an element of the Lie algebra, and we convert it to the left-invariant vector field . Then we can take the flow generated by and look at which is the integral curve starting at with and evaluate it at .

Why 1?

Why do we evaluate on the somewhat arbitrary seeming number ?

It comes from the property of 1-parameter subgroup as integral curves for left-invariant vector fields. We know that

Where is the integral curve starting at for a left invariant vector field identified by an element .

So if we use this formula and evaluate where then we get

Thus, the exponential function traces out integral curves of left-invariant vector fields that go through the identity element of the Lie group. We can write flows in terms of the exponential map. The flow which is the flow of , the left-invariant vector field corresponding to is

Properties

• If then
• The exponential map respects Lie group homomorphisms. The following diagram commutes for Lie group homomorphism

• For a matrix Lie algebra, the exponential map is the matrix exponential (proof here).