# If X is a left invariant vector field, then L_x o x_t = x_t o L_x

#### oblixps

##### Member
Let $$L_x: G \rightarrow G$$ be the left multiplication map that sends any point y to xy. Let $$x_t$$ be the flow of the left invariant vector field X. I want to show that $$L_x \circ x_t = x_t \circ L_x$$.

I let q be any point in G, and let $$x_{t}(q) = y$$. Also, $$L_{x}(y) = xy$$. So I have $$L_x \circ x_{t} (q) = xy$$ and i want to show that $$x_t \circ L_{x} (q) = x_{t}(xq) = xy$$.

i'm having trouble showing the last statement that $$x_{t}(xq) = xy$$. Since X is a left invariant vector field, i know that $$dL_{x}X_{q} = X_{xq}$$ and $$\frac{\partial x_t(q)}{\partial t} = X_{q}$$ since x_t is the flow of X. i can't figure out how to use these to get to my desired result however.

would someone mind offering hints on how to proceed?

#### oblixps

##### Member
I came across the hint that equivalently, i should show that $$x_t = L_x \circ x_t \circ L_{x}^{-1}$$ by showing that $$L_x \circ x_t \circ L_{x}^{-1}$$ is the flow of $$dL_x \circ X \circ L_{x}^{-1}$$. So i want to show that $$\frac{\partial}{\partial t} (L_x \circ x_t \circ L_{x}^{-1}) = dL_x \circ X \circ L_{x}^{-1}$$.

I am having a little trouble with the chain rule and was wondering if someone could help out. so far, $$\frac{\partial}{\partial t} (L_x \circ x_t \circ L_{x}^{-1}) = d(L_x \circ x_t \circ L_{x}^{-1})(\frac{\partial}{\partial t}) = (dL_x \circ dx_t \circ dL_{x}^{-1})(\frac{\partial}{\partial t})$$ but i am not sure how to simplify this.

i have tried $$(dL_{x}^{-1})(\frac{\partial}{\partial t}) = (L_{x}^{-1} \circ \gamma)'$$ where $$\gamma$$ is the curve that goes through the vector $$\frac{\partial}{\partial t}$$. i also tried using the fact that $$(dL_{x}^{-1})(\frac{\partial}{\partial t})f = \frac{\partial}{\partial t}(f \circ L_{x}^{-1})$$ so $$(dL_x \circ dx_t \circ dL_{x}^{-1})(\frac{\partial}{\partial t}) = dL_x \circ X \circ (f \circ L_{x}^{-1})$$ but now i have an f in there i can't get rid of.

i am still a beginner on differential geometry so these manipulations are relatively new to me. any help on this would be greatly appreciated.