- #1

- 14,272

- 6,747

$$dl^2=d\theta^2 + \sin^2\theta\, d\varphi^2$$

where ##\theta\in [0,\pi]##. The standard interpretation of this line element is to take ##\varphi\in [0,2\pi)##, in which case the line element represents the standard metric of the sphere ##S^2##. However, from the line element itself I don't see why I could not take ##\varphi\in (-\infty,\infty)##. Is it consistent to interpret ##\varphi## as ##\varphi\in (-\infty,\infty)##, and if it is, what geometrical object the line element represent in this case? Is it a smooth manifold, or does it contain some singularities?