Problem
Problem 4.38 Consider the product $T^{1} \times \mathbb{R}^{+}$of the one-dimensional torus by the multiplicative group of strictly positive numbers (that group is called the group of similarities of the plane). Let $(\theta, x)$ denote local coordinates. Show that the vector field
$$
\frac{\partial}{\partial \theta}+x \frac{\partial}{\partial x}
$$
is left-invariant.