heres an example: the set G of all lines through the origin of (x,y,z) space. since each such line is determined by any other point, consider the three planes x=1, y=1, z=1.
each line through the origin contains a point with at least one non zero coordinate, hence with some coordinate equal to 1, so each such line meets at least one of those planes in a unique point.
thus the set of all lines in G is covered by three sets each isomorphic to a plane. hence G is a 2 dimensional manifold with three coordinate charts. moreover, there is a 2:1 surjection from the unit sphere onto G, since each point of the sphere determines one line through (0,0,0), and each such line meets the sphere twice.