Lie group actions and submanifolds

by mnb96
Tags: actions, submanifolds
 P: 604 Hello, Let's suppose that I have a Lie group G parametrized by one real scalar t and acting on ℝ2. Is it generally correct to say that the orbits of the points of ℝ2 under the group action are one-dimensional submanifolds of ℝ2, because G is parametrized by one single scalar? If so, how can I prove this statement? Thanks.
 HW Helper Sci Advisor P: 2,020 What if the action has a fixed point at p? Will the orbit through p be one-dimensional? If your G is acting smoothly on R^2, you can at least say that the orbit through each point is an immersed submanifold of R^2; it will generally be either 1- or 0-dimensional.
 P: 604 thanks a lot! you are right. I am just thinking of the action of the rotation group SO(2) on ℝ2; clearly the point at (0,0) will remain unchanged, hence its orbit is 0-dimensional. Do you have any hint to suggest in order to prove these facts? I mean, proving that the orbits are immersed submanifolds of R^2.
HW Helper