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Good morning and welcome back! Today we finally complete our discussion of Lie groups.

So far, we started with Lie group G and studied the corresponding Lie algebra

first the set of all the left invariant vector fields and so because coming from

a group to the vector fields we invoked some differentiation of sorts and today

we are going to answer the question of why these left invariant vector fields

are so interesting if you want to study Lie groups. Well you can derive them from

there but what do you learn about the group once you know the algebra. So today

we'll be concerned with the question of how to go back up there and I can

already anticipate the result at least in a neighborhood of the identity. Up

here there's an identity element. At least in a neighborhood of the identity

you can fully recover every Lie group from its Lie algebra. How big this

neighborhood is that depends and we'll look at that too and this going up there

again will be achieved by something that we will call the exponential map. So we

have a section 4.7 recovering and let's be careful some maybe all some of the

Lie group from its Lie algebra.

Alright so the first thing we're going to do well we know that down here we

actually have this isomorphism to the tangent space at the identity but now we

will also on occasion have to invoke the left invariant vector fields again to

understand this exponential map. Well but the first thing I want to provide is a

definition we could have brought much earlier and that's the following. Let M

be a smooth manifold and let Y be a vector field, a smooth vector field on M.

Then an integral curve or then a smooth curve, a smooth curve gamma and now we've

got to be a little careful with the parameter range. Let's say with the

parameter range from some real number a to some real number b into the smooth

manifold is called an integral curve.

If well if for every point along the curve that means if for any parameter

value lambda in the interval a b it is true that the tangent vector to the

curve at the point gamma of lambda remember we always indicated the

tangent vector to a curve by this symbol X curve comma at the point if this point

is actually if the tangent vector at the point is actually given by the given

vector field at the very same point through which the curve passes so that's

an integral curve. It's quite obvious that there is a unique integral curve

through each point of the manifold integral curve of a vector field Y

through each point of the manifold. Well if I say it's obvious well actually this

obvious follows from the local existence and uniqueness of solutions

to ordinary differential equations. Okay so the the idea is you have a vector

field on your manifold and you go to an arbitrary point of the manifold like

say this one where of course this it's a vector and now you can find a curve

whose tangent vectors coincide with the vectors that are already given and

locally around the point if the vector field is smooth that's important there

needs to be smooth vector field obviously then this is unique. Okay so now

that's this now there's an important definition which at first sight seems a

little well not so important but it is also plays a big role in in the

singularity theorems in general relativity but let's have this first like

this a curve gamma an integral curve an integral curve gamma of a vector field

Y is called complete if its domain so that's what I called the parameter

range before if its domain can be extended to all of R so AB well to all

of R it's complete so there is a result that says that on a compact manifold on