A geometrical picture of renormgroup

Tangent space

Let be a smooth manifold. Fix a point , and consider the tangent space at this point:

is the space of equivalence classes of trajectories

passing through

at the time zero, i.e.

. The equivalence relation is that

when their difference is

Let us denote this space of trajectories . By the definition of , we have a map:

Lifting tangent vectors to trajectories

Suppose that comes equipped with an action of a Lie algebra , preserving a point .

Then also acts in

Let us try to construct a map

such that

respecting the action of

Does such a map exist?