Given tangent space of submanifold of Lie group, is it possible to recover the submanifold?

Question

I have computed the tangent space of certain submanifolds (the unstable manifolds) of a Lie group at a particular point.

I know that the exponential map lets us move between the Lie algebra and the Lie group, and that the Lie algebra is the tangent space of the Lie group at the identity. I am also aware that the tangent space of the Lie group at any point other than the identity is just the point multiplied by the Lie algebra.

So, is there a way to obtain the submanifold from just the tangent space? For example, I have that the tangent space of one of these submanifolds is one dimensional and I have a basis for it. Is there a way to recover the corresponding 1-dimensional submanifold of the Lie group from this information?

Answer 1

In general, you cannot recover a submanifold of a Lie group from a single tangent space. However, if your submanifold is a closed subgroup, then the tangent space at any point will allow you to recover the Lie algebra of the subgroup. This will tell you the connected component of the subgroup that contains the group identity. If the subgroup is disconnected, then it remains to describe the non-identity components.

Did you have just a single tangent space, or did you have all of them?