Define the first orthant of the hypercylinder: $$\mathbb{C}=\{(x_1, \ldots, x_p, x_{p+1}): \sum_{i=1}^px_i^2=1, x_i\geq0\text{ for }i=1, \ldots, p,\text{ and }x_{p+1}\in[0, L]\}.$$ Each (shortest) helix $h$ connecting two points on $\mathbb{C}$ can be uniquely characterized by the angle and height differences $\alpha$ and $l$ between the two points. For simplicity consider $p=2$ and the corresponding space of all possible helices on $\mathbb{C}$, $$\mathbb{H}=\{h=(\alpha, l):\alpha\in[0, \frac{\pi}{2}], l\in[0, L]\}$$ and for $h_1, h_2\in\mathbb{H}$ define the group operation as $$h_1\cdot h_2:=((\alpha_1+\alpha_2)\%\frac{\pi}{2}, (l_1+l_2)\%L),$$ where $\%$ denotes the remainder operator.
I am new to the domain of Lie group and was wondering if the above definition defines a Lie group structure for the space $\mathbb{H}$. Many thanks!